Universal and Operational Benchmarking of Quantum Memories

TL;DR

This paper introduces a universal, operational benchmark for quantum memories that quantifies their robustness, resource requirements, and potential to outperform classical strategies across various settings, demonstrated on IBM hardware.

Contribution

The authors develop a new benchmark that is universally applicable, efficiently computable, and experimentally accessible for evaluating quantum memories across different platforms.

Findings

Benchmark effectively quantifies quantum memory capabilities.

Demonstrated on IBM Q hardware with error-suppression techniques.

Provides insights into non-Markovian noise effects.

Abstract

Quantum memory -- the capacity to store and faithfully recover unknown quantum states -- is essential for quantum-enhanced technology. There is thus a pressing need for operationally meaningful means to benchmark candidate memories across diverse physical platforms. Here we introduce a universal benchmark distinguished by its relevance across multiple key operational settings, exactly quantifying (1) the memory's robustness to noise, (2) the number of noiseless qubits needed for its synthesis, (3) its potential to speed up statistical sampling tasks, and (4) performance advantage in non-local games beyond classical limits. The measure is analytically computable for low-dimensional systems and can be efficiently bounded in experiment without tomography. We thus illustrate quantum memory as a meaningful resource, with our benchmark reflecting both its cost of creation and what it can accomplish. We demonstrate the benchmark on the five-qubit IBM Q hardware, and apply it to witness efficacy of error-suppression techniques and quantify non-Markovian noise. We thus present an experimentally accessible, practically meaningful, and universally relevant quantifier of a memory's capability to preserve quantum advantage.

Tables

Table 1. Table 1: Dephasing channel

Setting	p=1/4			p=1/2			p=3/4			p=1
	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory
${\sigma }_1$, $O_1$	0.2937	0.2495	0.2500	0.5137	0.5088	0.5000	0.7682	0.7471	0.7500	0.9595	1.0000	1.0000
${\sigma }_2$, $O_2$	0.7261	0.7436	0.7500	0.5009	0.4993	0.5000	0.2668	0.2533	0.2500	0.0532	0.0000	0.0000
${\sigma }_3$, $O_3$	0.9647	1.0000	1.0000	0.9735	1.0000	1.0000	0.9720	1.0000	1.0000	0.9736	1.0000	1.0000
$|0⟩$, $O_4$	0.7904	0.7778	0.7887	0.7723	0.7905	0.7887	0.7468	0.7834	0.7877	0.7222	0.7800	0.7887
$|1⟩$, $O_4$	0.2008	0.2111	0.2113	0.2599	0.2148	0.2113	0.3126	0.2139	0.2113	0.3380	0.2089	0.2113
Score	0.5400	0.5138	0.5000	0.9584	1.0049	1.0000	1.4219	1.4961	1.5000	1.8278	1.9924	2.0000

Table 2. Table 2: Erasure channel

Setting	p=1/4			p=1/2			p=3/4			p=1
	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory
${\sigma }_1$, $O_1$	0.2671	0.2499	0.2500	0.4946	0.5046	0.5000	0.7301	0.7510	0.7500	0.9452	1.0000	1.0000
${\sigma }_2$, $O_2$	0.0073	0.0000	0.0000	0.0190	0.0000	0.0000	0.0226	0.0000	0.0000	0.0374	0.0000	0.0000
${\sigma }_3$, $O_3$	0.2732	0.2523	0.2500	0.4978	0.5066	0.5000	0.7366	0.7480	0.7500	0.9403	1.0000	1.0000
$|0⟩$, $O_4$	0.2153	0.2011	0.1972	0.3979	0.3934	0.3943	0.5721	0.5900	0.5915	0.7394	0.7915	0.7887
$|1⟩$, $O_4$	0.0641	0.0521	0.0528	0.1216	0.1055	0.1057	0.1780	0.1527	0.1585	0.2205	0.20789	0.2113
$|0⟩$, $|2⟩$	0.6655	0.7448	0.7500	0.4419	0.5043	0.5000	0.2181	0.2491	0.2500	0.0118	0.00000	0.0000
$|1⟩$, $|2⟩$	0.6681	0.7490	0.7500	0.4398	0.4987	0.5000	0.2134	0.2494	0.2500	0.0116	0.0000	0.0000
Score	1.2463	1.2462	1.2500	1.4486	1.5136	1.5000	1.6894	1.7546	1.7500	1.8845	2.0005	2.0000

Table 3. Table 3: Stochastic damping channel

Setting	p=1/4			p=1/2			p=3/4			p=1
	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory	IBMQ	QASM	Theory
${\sigma }_1$, $O_1$	0.6351	0.6265	0.6250	0.7439	0.7491	0.7500	0.8623	0.8746	0.8750	0.9694	1.0000	1.0000
${\sigma }_2$, $O_2$	0.3811	0.3857	0.3750	0.2844	0.2471	0.2500	0.1615	0.1229	0.1250	0.0474	0.0000	0.0000
${\sigma }_3$, $O_3$	0.9866	1.0000	1.0000	0.9793	1.0000	1.0000	0.9796	1.0000	1.0000	0.9742	1.0000	1.0000
${\sigma }_4$, $O_4$	0.3135	0.2438	0.2500	0.5387	0.5009	0.5000	0.7563	0.7533	0.7500	0.9668	1.0000	1.0000
$|0⟩$, $O_5$	0.4740	0.4988	0.5000	0.4639	0.5043	0.5000	0.4375	0.4952	0.5000	0.4264	0.4917	0.5000
$|1⟩$, $O_5$	0.5287	0.5026	0.5000	0.5548	0.5093	0.5000	0.5781	0.4963	0.5000	0.6006	0.4998	0.5000
Score	1.0699	1.0695	1.0757	1.2570	1.3025	1.309	1.5667	1.6443	1.6353	1.8734	2.0060	2.0000

Equations

$$\mathcal{R}(\mathcal{N})$$

$$\mathcal{P}(\mathcal{N},\mathcal{G})=\sum _{i,j}{\alpha }_{ij}\mathrm{Tr}[\mathcal{N}({\sigma }_i)O_j].$$

$$\underset{\mathcal{G}}{\max }\frac{P(\mathcal{N},\mathcal{G})}{{\max }_{\mathcal{M}\in \mathrm{EB}}\mathcal{P}(\mathcal{M},\mathcal{G})}=\mathcal{R}(\mathcal{N})+1.$$

$$\mathcal{R}(\mathcal{N})\ge d_A\max \mathrm{eig}({\Phi }_{\mathcal{N}})-1,$$

$$\mathcal{I}(T)={\int }_0^{T}dt\max \left\{0,\frac{d\mathcal{R}({\mathcal{N}}_t)}{dt}\right\}.$$

$$\mathcal{R}(\mathcal{N})=\min \left\{s\ge 0:\exists \mathcal{M}\in \mathrm{EB},\text{s.t.}\frac{\mathcal{N}+s\mathcal{M}}{1+s}\in \mathrm{EB}\right\},$$

$$\frac{1}{s+1}\mathcal{N}+\frac{s}{s+1}\mathcal{M}={\mathcal{M}}^{\prime }\in \mathrm{EB}.$$

$$\frac{1}{s+1}\mathcal{N}+\frac{s}{s+1}\mathcal{M}$$

$$R_{\text{syn}}^{\epsilon }(\mathcal{N})=\min \left\{d:\exists \Lambda \in \mathcal{O},\Vert \Lambda ({\mathcal{I}}_d)-\mathcal{N}{\Vert }_{\diamond }\le \epsilon \right\},$$

$${\mathcal{R}}^{\epsilon }(\mathcal{N})=\underset{\Vert {\mathcal{N}}^{\prime }-\mathcal{N}{\Vert }_{\diamond }\le \epsilon }{\min }\mathcal{R}({\mathcal{N}}^{\prime }).$$

$$R_{\text{syn}}^{\epsilon }(\mathcal{N})=1+\lceil {\mathcal{R}}^{\epsilon }(\mathcal{N})\rceil .$$

$$1+{\mathcal{R}}^{\epsilon }(\mathcal{N})$$

$$\le 1+\mathcal{R}({\mathcal{N}}^{\prime })$$

$$=1+\mathcal{R}(\Lambda ({\mathcal{I}}_d))$$

$$\le 1+\mathcal{R}({\mathcal{I}}_d)$$

$$=d.$$

$$\Lambda ({\Phi }_{\mathcal{C}})=\mathrm{Tr}\left({\varphi }^{+}{\Phi }_{\mathcal{C}}\right){\Phi }_{{\mathcal{N}}^{\prime }}+\mathrm{Tr}\left((I-{\varphi }^{+}){\Phi }_{\mathcal{C}}\right){\Phi }_{\mathcal{M}},$$

$$\frac{1}{d_c}{\mathcal{N}}^{\prime }+\frac{d_c-1}{d_c}\mathcal{M}={\mathcal{M}}^{\prime }.$$

$$\Lambda ({\Phi }_{\mathcal{C}})=q{\Phi }_{{\mathcal{M}}^{\prime }}+(1-q){\Phi }_{\mathcal{M}},$$

$${\mathcal{C}}_{\min }(\mathcal{N})=\underset{\mathcal{N}={\sum }_i{c}_i{\mathcal{M}}_i}{\min }\mathcal{C}(\{c_i,{\mathcal{M}}_i\}).$$

$${\mathcal{C}}_{\min }(\mathcal{N})=(1+2\mathcal{R}(\mathcal{N}){)}^2.$$

$$\mathcal{N}=\sum _{i:c_i\ge 0}|c_i^{+}|{\mathcal{M}}_i-\sum _{i:c_i<0}|c_i^{-}|{\mathcal{M}}_i,{\mathcal{M}}_i\in \mathrm{EB},$$

$$\mathcal{N}=(s+1){\mathcal{M}}^{\prime }-s\mathcal{M},$$

$$\mathcal{P}(\mathcal{M},{\mathcal{G}}^{\prime })\ge 0\forall \mathcal{M}\in \mathrm{EB}.$$

$$\underset{{\mathcal{G}}^{\prime }}{\max }\frac{\mathcal{P}(\mathcal{N},{\mathcal{G}}^{\prime })}{{\max }_{\mathcal{M}\in \mathrm{EB}}\mathcal{P}(\mathcal{M},{\mathcal{G}}^{\prime })}=1+\mathcal{R}(\mathcal{N}).$$

$$\mathcal{R}(\mathcal{N})+1=$$

$$x_1-x_2={\Phi }_{\mathcal{N}},$$

$$x_1,x_2\in \mathrm{cone}(\mathrm{Choi}(\mathrm{EB})),$$

$$\mathrm{OPT}=$$

$$W=W^{†},$$

$$\mathrm{Tr}[{\Phi }_{{\mathcal{M}}^{\prime }}W]\in [0,1],\forall {\mathcal{M}}^{\prime }\in \mathrm{EB}.$$

$$\mathcal{P}(\mathcal{M},{\mathcal{G}}^{\prime })\in [0,1]$$

$$\mathcal{P}(\mathcal{N},{\mathcal{G}}^{\prime })$$

$$=d\sum _{i,j}{\alpha }_{ij}\mathrm{Tr}[{\Phi }_{\mathcal{N}}({\sigma }_i^T\otimes O_j)],$$

$$:=\mathrm{Tr}[{\Phi }_{\mathcal{N}}W],$$

Full text

Memories are essential for information processing, from communication to sensing and computation. In the context of quantum technologies, such memories must also faithfully preserve the uniquely quantum properties that enable quantum advantages, including quantum correlations and coherent superpositions Duan et al. (2001). This has motivated extensive work in experimental realisations across numerous physical platforms Zhong et al. (2015); Wang et al. (2017), and presents a pressing need to find operationally meaningful means to compare quantum memories across diverse physical and functional settings. In contrast, present approaches towards detecting and benchmarking the quantum properties of memories are often ad-hoc, involving experimentally taxing process tomography, or only furnishing binary measures of performance based on tests of entanglement and coherence preservation Chuang and Nielsen (1997); Poyatos et al. (1997); D’Ariano and Lo Presti (2001); Namiki (2008); Häseler and Lütkenhaus (2010); Macchiavello and Rossi (2013); Rosset et al. (2018); Simnacher et al. (2019).

Our work addresses these issues by envisioning memory as a physical resource. We provide a means to quantify this resource by asking: how much noise can a quantum memory sustain before it is unable to preserve uniquely quantum aspects of information? Defining this as the robustness of a quantum memory (RQM), we demonstrate that the quantifier has diverse operational relevance in benchmarking the quantum advantages enabled by a memory — from speed-up in statistical sampling to nonlocal quantum games (see Fig. 1). We prove that RQM behaves like a physical resource measure, representing the number of copies of a pure idealised qubit memory that are required to synthesise the target memory. We show the measure to be exactly computable for many relevant cases, and introduce efficient general bounds through experimental and numerical methods. The quantifier is, in particular, experimentally accessible without full tomography, enabling immediate applications in benchmarking different memory platforms and error sources, as well as providing a witness for non-Markovianity. We experimentally test our benchmark on the five-qubit IBM Q hardware for different types of error, demonstrating its versatility. In addition, the generality of our methods within the broad physical framework of quantum resource theories Horodecki and Oppenheim (2013); Coecke et al. (2016); Chitambar and Gour (2019) ensures that many of our operational interpretations of the RQM can also extend to the study of more general quantum processes Kuo et al. (2018); Gour (2019); Takagi and Regula (2019); Liu and Yuan (2020); Liu and Winter (2019); Takagi et al. (2020), including general resource theories of quantum channels, gate-based quantum circuits, and dynamics of many-body physics. Our work thus presents an operationally meaningful, accessible, and practical performance-based measure for benchmarking quantum processors that is immediately relevant in today’s laboratories.

Framework. Any quantum memory can be viewed as a channel in time — mapping an input state we wish to encode into a state we will eventually retrieve in the future. An ideal memory preserves all information, such that proper post-processing operations on the output state can always undo the effects of the channel. Such channels preserve all state overlaps, in the sense that any pair of distinguishable input states remain distinguishable at output. In contrast, this is not possible with classical memories that store only classical data. To distinguish orthogonal states in some basis $\ket{k}$, we are forced to measure in this basis and record only the classical measurement outcome $k$. Such a measure-and-prepare process will never distinguish $\ket{0}+\ket{1}$ from $\ket{0}-\ket{1}$. In fact, this procedure exactly encompasses the class of all entanglement-breaking (EB) channels Horodecki et al. (2003); Holevo (2008): if we store one part of an entangled bipartite state within classical memory, the output is always separable. As such, classical memories are mathematically synonymous with EB channels.

To systematically characterise how well a general memory preserves quantum information, we consider how robust it is against noise. We define robustness of quantum memories (RQM) as the minimal amount of a classical memory that needs to be mixed with the target memory $\mathcal{N}$ such that the resultant probabilistic mixture is also classical:

[TABLE]

where the minimisation is over the set of all entanglement-breaking channels $\mathrm{EB}$. We explicitly prove that the robustness measure is a bona fide resource measure of quantum memories, satisfying all necessary operational properties. Crucially, we show that the robustness satisfies monotonicity — a memory’s RQM can never increase under any resource non-generating (RNG) transformation, that is, any physical transformation of quantum channels that maps EB channels only to EB channels. We thus refer to such transformations as free within the resource theory of quantum memories.

Commonly encountered free transformations include pre- or post-processing with an arbitrary channel or, more generally, the class of so-called classically correlated transformations Rosset et al. (2018).

Operational interpretations. We illustrate the operational relevance of RQM in three distinct settings. The first is memory synthesis. From the perspective of physical resources, one important task is to synthesise a target resource by expending a number of ideal resources, which can be thought of as the “currency” in this process. Intuitively, a more resourceful object would be harder to synthesise and hence require more ideal resources, allowing us to understand the required number of ideal memories as the resource cost of a given memory. In entanglement theory, an analogous concept involves determining the minimum number of Bell pairs that are required to engineer a particular entangled state using free operations (entanglement cost) Buscemi and Datta (2011); Brandao and Datta (2011). For quantum memories, we consider an ideal qubit memory $\mathcal{I}_{2}$ as the identity channel that perfectly preserves any qubit state. The task of single-shot memory synthesis is then to convert $n$ copies of ideal qubit memories $\mathcal{I}_{2}^{\otimes n}$ to the target memory $\mathcal{N}$ via a free transformation. We show that the robustness measure lower bounds the number $n$ of the requisite ideal memories, i.e., $n\geq\lceil\log_{2}(\lceil\mathcal{R}(\mathcal{N})\rceil+1)\rceil$. Therefore, a larger robustness indicates that the memory requires more ideal resources to synthesise. Furthermore, we show that there always exists an optimal RNG transformation that saturates this lower bound, and thus the robustness tightly captures the optimal resource cost for this task. We summarise our first result as follows.

Theorem 1. The minimal number of ideal qubit memories required to synthesise a memory $\mathcal{N}$ is $n=\lceil\log_{2}(\lceil\mathcal{R}(\mathcal{N})\rceil+1)\rceil$.

In Methods, we further consider imperfect memory synthesis by allowing an error $\varepsilon$ and show that the optimal resource cost is characterised by a smoothed robustness measure with smoothing parameter $\varepsilon$. Theorem 1 thus corresponds to the special case of $\varepsilon=0$.

In the second task, we consider classical simulation of quantum memories. The motivation here is analogous to computational speed-up — the observational statistics of any quantum algorithm can be simulated on a classical computer, albeit at an exponential overhead. Similarly, one strategy for simulating quantum memories is to perform full tomography of the input state and store the resulting classical density matrix. Then, at the output of the memory, the input state $\rho$ is reconstructed and any observational statistics on $\rho$ can be directly obtained. This method clearly requires an exponential amount of input samples for an $n$-qubit memory — and thus results in an exponential overhead in resources and speed.

Formally, the functional behaviour of any memory is fully described by how its observational statistics vary as a function of input, i.e., the set of expectation values $\mathrm{Tr}(O\mathcal{N}(\rho))$, for each possible observable $O$ and input state $\rho$. In order to simulate a quantum channel using only classical memories, we aim to estimate this quantity to some fixed additive error with at most some fixed admissible failure probability by taking samples — inputting $\rho$ in a classical memory, measuring $O$, and repeating to get expectation value estimates. Intuitively, the more non-classical a memory is, the more classical samples will be required to simulate its statistics effectively. We thus define the simulation overhead $C$ as the increase in the number of samples required when using only classical memories, versus having access to $\mathcal{N}$ itself. We then prove that the optimal overhead is given exactly by the RQM of the quantum memory.

Theorem 2. The minimal overhead — in terms of extra runs or input samples needed — to simulate the observation statistics of a quantum memory $\mathcal{N}$ is given by $\mathcal{C}_{\min}=(1+2\mathcal{R}(\mathcal{N}))^{2}$.

For entanglement-breaking channels $\mathcal{M}$, the robustness $\mathcal{R}(\mathcal{M})$ vanishes and hence $\mathcal{C}_{\min}(\mathcal{M})=1$, aligning with the intuition that classical memories require no extra simulation cost. For $n$ ideal qubit memories, $\mathcal{R}({\mathcal{I}}_{2}^{\otimes n})=2^{n}-1$ and hence the classical simulation overhead scales exponentially with $n$.

In the third setting, we consider the capability of quantum memories to provide advantages in a class of two-player nonlocal quantum games. Related games of this type have previously been employed in understanding features of Bell nonlocality Buscemi (2012) and detecting quantum memories Rosset et al. (2018). Consider then a set of states $\{\sigma_{i}\}$, from which one party (Alice) selects one state uniformly at random and encodes it in a memory $\mathcal{N}$. Her counterpart Bob is given this memory and tasked with guessing which of the states $\{\sigma_{i}\}$ was encoded by performing a measurement $\{O_{j}\}$. The probability that Bob guesses $\sigma_{j}$ when the input state is $\sigma_{i}$ is given by $\mathrm{Tr}[\mathcal{N}(\sigma_{i})O_{j}]$. Thus, by associating with each such guess a coefficient $\alpha_{ij}\in\mathbb{R}$ we can define the payoff of the game — this can be used to give different weights to corresponding states, or to penalise certain guesses. The performance of the two players in the game defined by $\mathcal{G}=\{\{\alpha_{ij}\},\{\sigma_{i}\},\{O_{j}\}\}$ is then evaluated using the average payoff function,

[TABLE]

Such games can be considered as a generalisation of the task of quantum state discrimination, as can be seen by taking $\alpha_{ij}=\delta_{ij}p_{i}$ for some probability distribution $\{p_{i}\}$. We see that the players’ maximum achievable performance is limited by Bob’s capacity to discern Alice’s inputs, and thus each such game serves as a gauge for the memory quality of $\mathcal{N}$. In order to establish a quantitative benchmark for the resourcefulness of a given memory, we can then compute the best advantage it can provide in the same game $\mathcal{G}$ over all classical memories. To make such a problem well-defined, we will constrain ourselves to games for which the payoff $\mathcal{P}(\mathcal{M},\mathcal{G})$ is non-negative. In the Methods we then show that the maximal capabilities of a quantum memory in this setting are exactly measured by the robustness.

Theorem 3. The advantage that a quantum memory $\mathcal{N}$ can provide over classical memories in all nonlocal quantum games is given by

[TABLE]

We will shortly see that such games, in addition to showcasing another operational aspect of the robustness, allow us to efficiently bound $\mathcal{R}(\mathcal{N})$ in many relevant cases.

Computability and measurability. We can efficiently detect and bound the robustness of a memory through the performance of the memory in game scenarios. Specifically, consider games $\mathcal{G}$ such that all classical memories achieve a pay-off in the range $[0,1]$. By Thm. 3 we know that any such game $\mathcal{G}$ provides a lower bound $\mathcal{P}(\mathcal{N},\mathcal{G})-1$ on $\mathcal{R}(\mathcal{N})$, akin to an entanglement witness quantitatively bounding measures of entanglement Eisert et al. (2007). This provides a physically accessible way of bounding the robustness measure by performing measurements on a chosen ensemble of states, and in particular there always exists a choice of a quantum game $\mathcal{G}$ such that $\mathcal{P}(\mathcal{N},\mathcal{G})-1$ is exactly equal to $\mathcal{R}(\mathcal{N})$. This approach makes the measure accessible also in experimental settings, avoiding costly full process tomography. We use this method to explicitly compute the robustness of some typical quantum memories in Fig. 4(a), with detailed construction of the quantum games deferred to the Supplemental Materials.

In addition to the above linear witness method, we also give non-linear witnesses of a memory $\mathcal{N}$ based on the moments of its Choi state. Consider channels $\mathcal{N}$ with input dimension $d$ and $k=0,1,\dots,\infty$, in Supplementary Materials, we prove $\mathcal{R}(\mathcal{N})\geq d^{\frac{k-1}{k}}\left(\mathrm{Tr}\left[\left(\Phi_{\mathcal{N}}\right)^{k}\right]\right)^{\frac{1}{k}}-1$, where $\Phi_{\mathcal{N}}$ is the Choi state of $\mathcal{N}$. Higher values of $k$ provide tighter lower bounds, which can be measured in experiment by implementing a generalised swap test on $k$ copies of the channel. In the limit $k\rightarrow\infty$ we obtain the strongest bound, which depends only on the maximal eigenvalue of the Choi state. Remarkably, the bound is actually tight for all qubit-to-qubit and qutrit-to-qubit channels.

Theorem 4. The RQM of any quantum memory $\mathcal{N}$ with input dimension $d_{A}$ and output dimension $d_{B}$ can be lower bounded by

[TABLE]

and equality holds when $d_{A}\leq 3$ and $d_{B}=2$.

We stress that this provides an exact and easily computable expression for the robustness for low-dimensional channels. This contrasts with related measures of entanglement of quantum states such as the robustness of entanglement Vidal and Tarrach (1999), for which no general expression exists even in $2\times 2$-dimensional systems.

Given a full description of the memory, we can also provide efficiently computable numerical bounds on the robustness via a semi-definite program, which we show to be tight in many relevant cases. We leave the detailed discussion to Supplementary Materials.

Applications. The robustness of quantum memories, being information theoretical in nature, applies across all physical and operational settings. This enables its immediate applicability to many present studies of quantum memory. For example, non-Markovianity and mitigation of errors resulting from non-Markovianity are widely studied problems in the context of quantum memories. RQM can be used both to identify the former, and measure the efficacy of the latter.

In particular, considering a memory $N_{t}$ that stores states from time [math] to $t\geq 0$, we can quantify its non-Markovianity as

[TABLE]

For any Markovian process $\mathcal{N}_{t}$, the robustness measure $\mathcal{R}(\mathcal{N}_{t})$ is a decreasing function of time owing to monotonicity of $\mathcal{R}$ (see Methods). Thus $\mathcal{I}(T)=0$ for any Markovian process $\mathcal{N}_{t}$, and nonzero values of $\mathcal{I}(T)$ directly quantify the memory’s non-Markovianity in a similar way to Ref. Rivas et al. (2010). Meanwhile, the goal of any error-mitigation procedure is to preserve encoded qubits. Thus, the characterisation of an increase in the RQM of relevant encoded sub-spaces provides a universal measure of the efficacy for any such behaviour.

In Fig. 4(b), we illustrate these ideas using a single-qubit memory subject to unwanted coupling from a qubit bath. The robustness of quantum memories degrades over time (yellow-starred line) - but has a revival around $t=1$, indicating non-Markovianity. Indeed, plotting $\mathcal{I}(t)$, we see clear signatures of non-Markovian effects arise at this moment (cyan-crossed line). Meanwhile, the green-dotted line quantifies how dynamical decoupling improves this memory through increased RQM. This improvement has a direct operational interpretation. For example, the approximate 4-fold increase in robustness around $t=0.8$ indicates that a quantum protocol that runs on a dynamically decoupled quantum memory could be much harder to simulate than its counterpart.

Experiment. We experimentally verify our benchmarking method on the ‘ibmq-ourense’ processor on the IBM Q cloud. We first consider a proof-of-principle verification of the scheme by estimating the RQM of three types of single-qubit noise channels — the dephasing channel, stochastic damping channel, and erasure channels. We synthesise the noise channels by entangling the target state with ancillary qubits. For example, the dephasing channel $\Delta_{p}(\rho_{A})=p\rho_{A}+(1-p)Z\rho_{A}Z$ can be realised by the circuit in the dashed box of Fig. 3(a), where we input an ancillary state $\ket{0}_{E}$, rotate it with $R^{Y}_{\theta}=\exp(-i\theta Y/2)$, and apply a controlled-Z. Here $\theta=2\arccos(\sqrt{p})$ and $Y$ is Pauli-Y matrix. We exploit the quantum game approach to estimate the RQM of the three types of noise channels. We choose a normalised quantum game $\mathcal{G}$ with the maximal payoff for EB channels of $\max_{\mathcal{M}\in\mathrm{EB}}\mathcal{P}(\mathcal{M},\mathcal{G})=1$, so that the robustness of memory $\mathcal{N}$ can be lower bounded by $\mathcal{R}(\mathcal{N})\geq\mathcal{P}(\mathcal{N},\mathcal{G})-1$. For each input-output setting $(\sigma_{i},O_{j})$, we measure the probability $p(j|i)=\mathrm{Tr}[O_{j}\mathcal{N}(\sigma_{i})]$ with 8192 experimental runs. The payoff is obtained as a linear combination of the probabilities $\mathcal{P}(\mathcal{N},\mathcal{G})=\sum_{i,j}\alpha_{i,j}p(j|i)$ with real coefficients $\alpha_{i,j}$. As shown in Fig. 3(b), the experimental data (circles, upper and lower triangles) aligns well with the theoretical result (solid lines), with a deviation of less than $0.13$. The deviation mostly results from the inherent noise in the hardware, especially the notable two-qubit gate error and the readout error.

Next, we show that the RQM can be applied to benchmark quantum gates and quantum circuits. Conventional quantum process benchmarking approaches, such as randomised benchmarking Emerson et al. (2005); Knill et al. (2008), generally focus on characterising the similarity between the noisy circuit and the target circuit. In contrast, our method is concerned with the capability of the noisy quantum processor in preserving quantum information, which can be thus regarded as an alternative operational approach for benchmarking processes. In the experiment, we focus on the two-qubit controlled-X (CX) gate, a standard gate used for entangling qubits. We sequentially apply $n$ (up to six) CX gates with interchanged control and target qubits for two adjacent gates. For example, one, three, and six CX gates correspond to the CX gate, the swap gate, and the identity gate, respectively.

Assuming that the dominant error is due to depolarising or dephasing effects, we estimate the RQM of each circuit via the correspondingly designed quantum game. As shown in Fig. 3(c), we can see that although the robustness with one CX is $2.667\pm 0.106$, it only slowly decreases to $2.497\pm 0.115$ for six CX gates. Our results thus indicate that while the CX gate is imperfect (with an average 0.0340 decrease of robustness for each CX gate), the dominant noise of the two qubit circuit may instead stem from imperfect state preparation and measurement (roughly leading to a 0.3 decrease in robustness). We also note that the large robustness loss of a single CX gate might also be due to the existence of other errors, which would imply that the choice of the quantum game could be further optimised. However, whenever the quantum game gives a large lower bound for the robustness, this is sufficient to ensure that the quantum process performs well in preserving quantum information. To demonstrate this, we consider the circuit ${CX}^{0}_{2}\cdot{CX}^{0}_{1}$ for preparing the three-qubit GHZ state. We lower bound the robustness as $5.837\pm 0.548$, verifying that the three-qubit noisy circuit can preserve more quantum information than all two-qubit circuits, whose robustness is upper bounded by 3. We leave the detailed experimental results and analysis to the Supplementary Materials.

Discussion. In this work, we introduced an operationally meaningful, practically measurable, and platform-independent benchmarking method for quantum memories. We defined the robustness of quantum memories and showed it to be an operational measure of the quality of a memory in three different practical settings. The greater the robustness of a memory, the more ideal qubit memories are needed to synthesise the memory; the more classical resources are required to simulate its observational statistics; and the better the memory is at two-player nonlocal quantum games based on state discrimination. The measure can be evaluated exactly in low-dimensional systems, and efficiently approximated both numerically by semi-definite programming and experimentally through measuring suitable observables. This thus constitutes a promising means to quantify the quantum mechanical aspects of information storage, and provides practical tools for benchmarking quantum memories across different experimental platforms and operational settings. The theory is applicable across different physical platforms exhibiting any known type of error source, as we experimentally confirm on the five-qubit IBM Q hardware. With the development of near-term noisy intermediate-scale quantum technologies Preskill (2018); Arute et al. (2019), we anticipate that our quantifier can become an industry standard for benchmarking quantum devices.

From a theoretical perspective, our work also constitutes a significant development in the resource theory of quantum memories. The only previously known general measure of this resource involved a performance optimisation over a large class of possible quantum games Rosset et al. (2018), thus making it difficult to evaluate, experimentally inaccessible, and obscuring a direct quantitative connection to tasks of practical relevance — the robustness explicitly addresses all of these issues. Furthermore, the generality of the resource-theoretic framework ensures that the tools developed here for quantum memories can be naturally extended to other settings, including purity, coherence, entanglement of channels Bennett et al. (2003); Kaur and Wilde (2017); Liu and Yuan (2020); Theurer et al. (2019); Gour and Scandolo (2019); Bäuml et al. (2019); Yuan (2019); Gour and Wilde (2018), and the magic of operations Seddon and Campbell (2019); Wang et al. (2019), etc. Another direction is to consider infinite-dimensional quantum systems, such as the optical modes of light. Finally, memories are essentially a question of reversibility, and thus have a natural connection to heat dissipation in thermodynamics Goold et al. (2016); Bérut et al. (2012). Indeed, recent results show connections between free energy and information encoding Narasimhachar et al. (2019), and thus present a natural direction towards understanding what thermodynamic consequences quantum memory quantifiers may have.

Methods

Here we present properties of the robustness measure, formal statements of Theorems 1-4 and sketch their proofs. Full version of the proofs and details on the numerical simulations can be found in Supplementary Materials.

Properties of RQM. Recall the definition of RQM

[TABLE]

where our chosen set of free channels are the entanglement-breaking (EB) channels. Define free transformation $\mathcal{O}$ as the set of physical transformations on quantum channels (super-channels) that map EB channels to EB channels, i.e. $\mathcal{O}=\{\Lambda:\Lambda(\mathcal{M})\in\mathrm{EB},\forall\mathcal{M}\in\mathrm{EB}\}$. This class includes, for instance, the family of classically correlated transformations, which were considered in Rosset et al. (2018) as a physically-motivated class of free transformations under which quantum memories can be manipulated. In particular, transformations $\Lambda(\mathcal{N})=\mathcal{M}_{1}\circ\mathcal{N}\circ\mathcal{M}_{2}$ with arbitrary pre- and post-processing channels $\mathcal{M}_{1},\mathcal{M}_{2}$ are free. We show that RQM satisfies the following properties.

Non-negativity. $\mathcal{R}(\mathcal{N})\geq 0$ with equality if and only if $\mathcal{N}\in\mathrm{EB}$.

Monotonicity. $\mathcal{R}$ does not increase under any free transformation, $\mathcal{R}(\Lambda(\mathcal{N}))\leq\mathcal{R}(\mathcal{N})$ for arbitrary $\mathcal{N}$ and $\Lambda\in\mathcal{O}$.

Convexity. $\mathcal{R}$ does not increase by mixing channels, $\mathcal{R}\left(\sum_{i}p_{i}\mathcal{N}_{i}\right)\leq\sum_{i}p_{i}\mathcal{R}(\mathcal{N}_{i})$.

Additional properties such as bounds under tensor product of channels are presented in Supplementary Materials.

Proof of properties. Non-negativity follows directly from the definition. For monotonicity, suppose $s=\mathcal{R}(\mathcal{N})$ with the minimisation achieved by $\mathcal{M}$ such that

[TABLE]

Apply an arbitrary free transformation $\Lambda$ on both sides and using linearity, we obtain $\frac{1}{s+1}\Lambda(\mathcal{N})+\frac{s}{s+1}\Lambda(\mathcal{M})=\Lambda\left(\mathcal{M}^{\prime}\right)\in\mathrm{EB}$. Therefore by definition $\mathcal{R}(\Lambda(\mathcal{N}))\leq s=\mathcal{R}(\mathcal{N})$. For convexity, suppose $s_{i}=\mathcal{R}(\mathcal{N}_{i})$ with the minimisation achieved by $\mathcal{M}_{i}$ for each $i$ and let $\mathcal{M}_{i}^{\prime}=(\mathcal{N}_{i}+s_{i}\mathcal{M}_{i})/(1+s_{i})$. Let $s=\sum_{i}p_{i}s_{i}$, $\mathcal{N}=\sum_{i}p_{i}\mathcal{N}_{i}$ and $\mathcal{M}=\frac{1}{s}\sum_{i}p_{i}s_{i}\mathcal{M}_{i}$, then by convexity of the set of EB channels

[TABLE]

therefore by definition we have $\mathcal{R}\left(\sum_{i}p_{i}\mathcal{N}_{i}\right)=\mathcal{R}(\mathcal{N})\leq s=\sum_{i}p_{i}s_{i}=\sum_{i}p_{i}\mathcal{R}(\mathcal{N}_{i})$.

Single-shot memory synthesis. Here we study a more general scenario, imperfect memory synthesis, which allows a small error between the synthesised memory and the target memory. The resource cost for this task is defined as the minimal dimension required for the ideal qudit memory $\mathcal{I}_{d}$,

[TABLE]

where $\|\cdot\|_{\diamond}$ denotes the diamond norm, which describes the distance of two channels. We also include a smooth parameter $\varepsilon$ of the cost which tolerates an arbitrary amount of error in the synthesis protocol. The case with $\varepsilon=0$ corresponds to the case with exact synthesis. When considering the ideal qubit memory $\mathcal{I}_{2}$ as the unit optimal resource, the minimal number of ideal qubit memories $\mathcal{I}_{2}^{\otimes n}$ required for memory synthesis is given by $n=\lceil\log_{2}(R_{\textrm{syn}}^{\varepsilon}(\mathcal{N}))\rceil$

Correspondingly we define a smoothed version of the robustness measure by minimising over a small neighbourhood of quantum channels,

[TABLE]

We prove that the smoothed robustness measure exactly quantifies the resource cost for imperfect single-shot memory synthesis.

Formal statement of Theorem 1. For any quantum channel $\mathcal{N}$ and any $0\leq\varepsilon<1$, the resource cost for single-shot memory synthesis satisfies

[TABLE]

Note that by setting $\varepsilon=0$ we recover the result for perfect memory synthesis stated in the main text.

Proof. We start by proving $R_{\textrm{syn}}^{\varepsilon}(\mathcal{N})\geq 1+\lceil\mathcal{R}^{\varepsilon}(\mathcal{N})\rceil$. The first step is to show that the robustness of the identity channel is $\mathcal{R}(\mathcal{I}_{d})=d-1$. The proof of this fact is omitted here. Next we show that the desired inequality can be proven using the monotonicity property. For an arbitrary memory synthesis protocol $\Lambda(\mathcal{I}_{d})=\mathcal{N}^{\prime}$ where $\|\mathcal{N}^{\prime}-\mathcal{N}\|_{\diamond}\leq\varepsilon$, we have

[TABLE]

Here the second line follows by definition and the fourth line follows from monotonicity. As the above inequality holds for all memory synthesis protocols, it also holds for the optimal protocol. Also notice that dimensions are integers. Thus we derive that $R_{\textrm{syn}}^{\varepsilon}(\mathcal{N})\geq 1+\lceil\mathcal{R}^{\varepsilon}(\mathcal{N})\rceil$.

To prove the other side $R_{\textrm{syn}}^{\varepsilon}(\mathcal{N})\leq 1+\lceil\mathcal{R}^{\varepsilon}(\mathcal{N})\rceil$, suppose the channel achieves the mimum of Eq. (10) is $\mathcal{N}^{\prime}$, and let $d_{c}=1+\lceil\mathcal{R}(\mathcal{N}^{\prime})\rceil$. To prove the desired inequality, it suffices to show that $\exists\Lambda\in\mathcal{O}$ such that $\Lambda(\mathcal{I}_{d_{c}})=\mathcal{N}^{\prime}$. Indeed such a $\Lambda$ is a protocol that achieves the required accuracy using resource $1+\lceil\mathcal{R}^{\varepsilon}(\mathcal{N})\rceil$, thus the optimal protocol should only use less resource.

Next we explicitly construct such a free transformation $\Lambda$, which transforms a quantum channel to another channel. As there is a one-to-one correspondence between Choi states and quantum channels, we give this construction based on transformation of the Choi state:

[TABLE]

where $\Phi$ denotes the Choi state of the subscript channel and $\phi^{+}$ is the maximally entangled state. In the full proof we show that $\Lambda$ is a valid physical transformation, i.e. a quantum super-channel.

As it is easy to verify that $\Lambda(\mathcal{I}_{d_{c}})=\mathcal{N}^{\prime}$, it only remains to show that $\Lambda$ is a free transformation, which maps EB channels to EB channels. To do this, first notice that as $d_{c}\geq 1+\mathcal{R}(\mathcal{N}^{\prime})$, there exists $\mathcal{M},\mathcal{M}^{\prime}\in\mathrm{EB}$ such that

[TABLE]

Then we can rewrite Eq. (79) as

[TABLE]

with $q=d_{c}\mathrm{Tr}\left[\phi^{+}\Phi_{\mathcal{C}}\right]$. When $\mathcal{C}$ is an EB channel, $\Phi_{\mathcal{C}}$ is a separable state, and we have $0\leq q\leq 1$. Thus $\Lambda(\Phi_{\mathcal{C}})$ is a separable Choi state that corresponds to an EB channel, which means that $\Lambda$ is a free transformation and concludes the proof.

Simulating observational statistics. Observe that the general simulation strategy is to find a set of free memories $\{\mathcal{M}_{i}\}\subseteq\mathrm{EB}$ such that the target memory can be linearly expanded as $\mathcal{N}=\sum_{i}c_{i}\mathcal{M}_{i},c_{i}\in\mathbb{R}$. By using $\mathcal{M}_{i}$ and measuring $\mathrm{Tr}[O\mathcal{M}_{i}(\rho)]$, we can obtain the target statistics as $\mathrm{Tr}[O\mathcal{N}(\rho)]=\sum_{i}c_{i}\mathrm{Tr}[O\mathcal{M}_{i}(\rho)]$ . Thus, compared with having access to $\mathcal{N}$ and directly measuring $O$, the classical simulation introduces an extra sampling overhead with a multiplicative factor $\|c\|_{1}^{2}=\left(\sum_{i}|c_{i}|\right)^{2}$. In particular, suppose we aim to estimate $\mathrm{Tr}[O\mathcal{N}(\rho)]$ to an additive error $\varepsilon$ with failure probability $\delta$. Due to Hoeffding’s inequality Pashayan et al. (2015), when having access to $\mathcal{N}$ we need $T_{0}\propto 1/{\varepsilon^{2}}\log(\delta^{-1})$ samples to achieve this estimate to desired precision, and when only having access to free resources in a specific decomposition $\mathcal{N}=\sum_{i}c_{i}\mathcal{M}_{i}$, we need $T\propto{\|c\|_{1}^{2}}/{\varepsilon^{2}}\log(\delta^{-1})$ samples. The simulation overhead is thus given by $\|c\|_{1}^{2}\propto T/T_{0}$.

By minimising the simulation overhead over all possible expansions, we obtain the optimal simulation cost

[TABLE]

Our second result shows that this optimal cost is quantified by the robustness measure.

Formal statement of Theorem 2. For any quantum channel $\mathcal{N}$, the optimal cost for the observational simulation of $\mathcal{N}$ using EB channels is given by

[TABLE]

Proof. For any linear expansion $\mathcal{N}=\sum_{i}c_{i}\mathcal{M}_{i}$, denote the positive and negative coefficients of $c_{i}$ by $c_{i}^{+}$ and $c_{i}^{-}$, respectively. Then we have

[TABLE]

with $\|c\|=\sum_{i:c_{i}\geq 0}|c_{i}^{+}|+\sum_{i:c_{i}<0}|c_{i}^{-}|$. As the channel is trace preserving, taking trace on both sides we get $\sum_{i:c_{i}\geq 0}|c_{i}^{+}|-\sum_{i:c_{i}<0}|c_{i}^{-}|=1$. Denote $s=\sum_{i:c_{i}<0}|c_{i}^{-}|$, hence with $\|c\|_{1}=2s+1$, $\mathcal{M}=\sum_{i:c_{i}<0}|c_{i}^{-}|\mathcal{M}_{i}/s$, and $\mathcal{M}^{\prime}=\sum_{i:c_{i}\geq 0}|c_{i}^{+}|\mathcal{M}_{i}/(1+s)$, we have

[TABLE]

where by convexity of $\mathrm{EB}$ we have $\mathcal{M},\mathcal{M}^{\prime}\in\mathrm{EB}$. Therefore finding the optimal expansion is equivalent to finding the smallest $s$ such that Eq. (19) holds, which by definition equals to the robustness, i.e. $s_{\min}=\mathcal{R}(\mathcal{N})$. Then we conclude that $\mathcal{C}_{\min}(\mathcal{N})=(1+2\mathcal{R}(\mathcal{N}))^{2}$.

Nonlocal games. Consider a quantum game $\mathcal{G}$ defined by the tuple $\mathcal{G}=(\{\alpha_{ij}\},\{\sigma_{i}\},\{O_{j}\})$, where $\sigma_{i}$ are input states, $\{O_{j}\}$ is a positive observable valued measures at the output, and $\alpha_{i}\in\mathbb{R}$ are the coefficients which define the particular game. The maximal performance in the game $\mathcal{G}$ enabled by a channel $\mathcal{N}$ is quantified by the payoff function $\mathcal{P}(\mathcal{N},\mathcal{G})=\sum_{ij}\alpha_{ij}\mathrm{Tr}[O_{j}\mathcal{N}(\sigma_{i})]$. Theorem 3 establishes the connection between the advantage of a quantum channel in the game scenario over all EB channels and the robustness measure. To ensure that the optimisation problem is well-defined and bounded, we will optimise over games which give a non-negative payoff for classical memories, which include standard state discrimination tasks.

Formal statement of Theorem 3. Let $\mathcal{G}^{\prime}$ denote games such that all EB channels achieve a non-negative payoff, that is,

[TABLE]

Then the maximal advantage of a quantum channel $\mathcal{N}$ over all EB channels, maximised over all such games, is given by the robustness:

[TABLE]

Proof. The proof is based on duality in conic optimisation (see Ref. Takagi and Regula (2019) and references therein). First we write the robustness as an optimisation problem

[TABLE]

where $\Phi_{\mathcal{N}}$ is the Choi state of $\mathcal{N}$, $\mathrm{Choi}(\mathrm{EB})$ denotes the Choi states of EB channels, i.e. bipartite separable Choi states, and $\mathrm{cone}(\cdot)$ represents the unnormalised version. This can be written in standard form of conic programming, based on which we can write the dual form of this optimisation problem. The dual form can be simplified as

[TABLE]

We can verify that these primal and dual forms satisfy the condition for strong duality, therefore $\mathrm{OPT}=1+\mathcal{R}(\mathcal{N})$, and it remains to show that $\mathrm{OPT}$ equals the maximal advantage in games.

As the constraints in the dual form (23) are linear, without loss of generality we can rescale the optimisation so that we only need to consider games $\mathcal{G}^{\prime}$ that satisfy

[TABLE]

for any $\mathcal{M}\in\mathrm{EB}$. We can then write

[TABLE]

where $d$ is the input dimension of $\mathcal{N}$ and $W=d\sum_{i,j}\alpha_{ij}\sigma_{i}^{T}\otimes O_{j}$. Using this representation, the maximal advantage can be written as an optimisation problem equivalent to (23). In particular, since any Hermitian matrix can be expressed in the form of $W$ for some real coefficients $\{\alpha_{ij}\}$, any witness $W$ in (23) can be used to construct a corresponding game $\mathcal{G}^{\prime}$, and conversely any game $\mathcal{G}^{\prime}$ satisfying the optimisation constraints gives rise to a valid witness $W$ in (23). We thus have

[TABLE]

concluding the proof.

Computability and bounds. It is known that the description of the set of separable states is NP-hard in the dimension of the system Gurvits (2003), and indeed this property extends to the set of entanglement-breaking channels Gharibian (2010), making it intractable to describe in general. Nonetheless, we can solve the problem of quantifying the RQM in relevant cases, as well as establish universally applicable bounds. As described in the main text, suitably constructing nonlocal games $\mathcal{G}$ can provide such lower bounds, which can indeed be tight. More generally, one can employ the positive partial transpose (PPT) criterion Horodecki et al. (1996) to provide an efficiently computable semidefinite programming relaxation of the problem, often providing non-trivial and useful bounds on the value of the RQM. We leave a detailed discussion of these methods to the Supplemental Material. In the case of low-dimensional channels, which is of particular relevance in many near-term technological applications, we can go further than numerical bounds and establish an analytical description of the RQM.

Formal statement of Theorem 4. For any channel $\mathcal{N}$ with input dimension $d_{A}$ and output dimension $d_{B}$, its RQM satisfies

[TABLE]

and equality holds when $d_{A}\leq 3$ and $d_{B}=2$.

Proof. The idea behind the proof is to employ the reduction criterion for separability Horodecki and Horodecki (1999); Cerf et al. (1999), which can be used to show that any entanglement-breaking channel $\mathcal{M}:A\to B$ satisfies $\Phi_{\mathcal{M}}\leq\frac{1}{d_{A}}I_{AB}$. Therefore, the set of channels satisfying this criterion provides a relaxation of the set of EB channels, and we can define a bound on the RQM by computing the minimal robustness with respect to this set. A suitable decomposition of a channel $\mathcal{N}$ can then be used to show that, in fact, this bound is given exactly by the larger of $d_{A}\max\operatorname{eig}(\Phi_{\mathcal{N}})-1$ and [math]. In the case of $d_{A}\leq 3$ and $d_{B}=2$, the reduction criterion is also a sufficient condition for separability, which ensures that the robustness $\mathcal{R}(\mathcal{N})$ matches the lower bound.

Experiment details. The processor has five qubits with $T_{1}$ and $T_{2}$ ranging from $25\sim 110\mu s$, single-qubit gate error $3.3\sim 6.5\times 10^{-4}$, two-qubit gate error $1.0\sim 1.5\times 10^{-2}$, and read-out error $1.9\sim 4.5\times 10^{-2}$. Our experiments are run on the first three qubits, which have the highest gate fidelities, and the circuits are implemented with Qiskit Aleksandrowicz et al. (2019).

Acknowledgements. We are grateful to Ryuji Takagi for making us aware of an error in a preliminary version of this manuscript. We acknowledge Simon Benjamin and Earl Campbell for insightful discussions. This work is supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Technology (EP/M013243/1), the National Natural Science Foundation of China Grants No. 11875173 and No. 11674193, and the National Key R&D Program of China Grants No. 2017YFA0303900 and No. 2017YFA0304004, the National Research Foundation of Singapore Fellowship No. NRF-NRFF2016-02 and the National Research Foundation and L’Agence Nationale de la Recherche joint Project No. NRF2017-NRFANR004 VanQuTe, the MOE Tier 1 Grant MOE$2019$-T1-$002$-$015$, the Foundational Questions Institute (FQXi) large grant FQXi-RFP-IPW-1903. Finally we acknowledge use of the IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM, the IBM Q team, the National Foundation of Singapore or the Ministry of Education of Singapore.

Author contributions. X.Y., Y.L., and B.R. devised the main conceptual and proof ideas. X.Y. and Q.Z. performed the numerical simulation and the experiment on the IBM Q cloud. All authors contributed to the development of the theory and the writing of the manuscript.

Data availability. The authors declare that all data supporting this study are contained within the article and its supplementary files.

Competing interests. The authors declare no competing financial interests.

I Resource framework

We first review the framework of resource theory of memories introduced in Ref. Rosset et al. (2018).

I.1 Resource theories of memories

Focusing on two chronologically ordered systems $A$ and $B$, a quantum memory is described by a channel $\mathcal{N}^{A\rightarrow B}$ that maps system $A$ to $B$, i.e., a completely positive trace-preserving (CPTP) linear map from $\mathcal{D}(\mathcal{H}_{A})$ to $\mathcal{D}(\mathcal{H}_{B})$. Here, $\mathcal{H}$ represents the Hilbert space and $\mathcal{D}(\mathcal{H})$ represents the set of states. The resource theory of quantum memories $\mathbf{C}=(\mathcal{F},\mathcal{O},\mathcal{R})$ is a tuple with the free memory set $\mathcal{F}$, free transformations $\mathcal{O}$ and resource measure $\mathcal{R}$. The resource theory provides a framework to systematically study the properties of quantum memories. In this work, we mainly focus on the capability of preserving quantum information of memories. In this section, we introduce definitions of the free memory set $\mathcal{F}$ and free transformations $\mathcal{O}$, and leave the discussion of resource measures $\mathcal{R}$ in the next section.

Free memories $\mathcal{F}$: entanglement breaking (EB) or equivalently measure-and-prepare channels,

[TABLE]

Here $\{M_{i}^{A}\}$ is a POVM satisfying $M^{A}_{i}\geq 0$ and $\sum_{i}M^{A}_{i}=I^{A}$. The reason that we choose EB channels to be free is because only classical information is stored and forwarded from system $A$ to system $B$. The channels that have the maximal resource are isometric channels as they are reversible.

We consider the normalised Choi state of a channel $\mathcal{N}$,

[TABLE]

with maximally entangled state ${\Phi^{+}_{AA^{\prime}}}=1/{d}\sum_{ij}\ket{ii}\bra{jj}$. Here $d$ is the dimension of the input system $A^{\prime}$. For a linear CPTP map $\mathcal{N}$, the corresponding Choi state is a normalised quantum state satisfying $\mathrm{Tr}_{B}[\Phi^{AB}_{\mathcal{N}}]=I_{A}/d$. Conversely, for any normalised bipartite quantum state $\Phi^{AB}$ which satisfies $\mathrm{Tr}_{B}[\Phi^{AB}]=I_{A}/d$, there is a unique quantum channel $\mathcal{N}$ whose Choi state equals to $\Phi^{AB}$ and can be espressed as

[TABLE]

We have the following properties for the free memory set $\mathcal{F}$:

1. (a)

The set of entanglement-breaking channels is convex. If a channel $\mathcal{M}$ is of the form

[TABLE]

with $\psi_{i}$ pure and $\{P_{i}\}$ mutually orthogonal rank one projections, then it is an extreme element of the set of entanglement-breaking channels Horodecki et al. (2003). 2. (b)

A quantum channel is EB if and only if its Choi state is a separable state. Equivalently, a free channel admits a Kraus decomposition as $\mathcal{M}(\rho)=\sum_{i}K_{i}\rho K_{i}^{\dagger}$ where each $K_{i}$ is rank one Horodecki et al. (2003). 2. 2.

Free super-operations $\mathcal{O}$: any super-operation that only transmits classical information is free,

[TABLE]

Here $\mathcal{U}^{A^{\prime}\rightarrow AE}$ and $\mathcal{V}^{BE\rightarrow B^{\prime}}$ are arbitrary quantum channels and $\Delta^{E}$ is a dephasing channel that enforces system $E$ to be classical.

For mathematical simplicity, we can also consider free operations as resource non-generating super-operations $\Lambda$, which map EB channels to EB channels,

[TABLE]

Meanwhile, it is not hard to see that free super-operations are also convex.

Quantum channels can be represented with Choi states and transformations of quantum channels, i.e., super-operations, can be similarly regarded as special linear transformations of Choi states. Given the Choi state $\Phi^{AB}_{\mathcal{N}}$ of channel $\mathcal{N}^{A\rightarrow B}$, a super-operation $\Lambda(\mathcal{N})$ can be equivalently described by a linear map acting on the Choi state. That is, suppose the Choi state of $\Lambda(\mathcal{N})$ is $\Phi^{AB}_{\Lambda(\mathcal{N})}$, we have

[TABLE]

where $\phi_{\Lambda}$ can be understood as a linear map from state $\Phi^{AB}_{\mathcal{N}}$ to state $\Phi^{A^{\prime}B^{\prime}}_{\Lambda(\mathcal{N})}$.

II Robustness of memory

In this section, we introduce robustness measures of memories and study their properties. We also define the generalised robustness and study its properties.

II.1 Definition

The robustness of memories is defined as

[TABLE]

with a minimisation over all possible EB channels. We also define the generalised robustness as

[TABLE]

with a minimisation over all channels. Note that the robustness measures can be equivalently defined based on the Choi state of channels,

[TABLE]

Note also that related measures have appeared in general resource theories of states Vidal and Tarrach (1999); Brandão and Gour (2015); Regula (2018); Anshu et al. (2018) as well as channels Díaz et al. (2018); Takagi and Regula (2019); Liu and Winter (2019), where their properties were studied.

We also define the logarithmic robustness as

[TABLE]

and the smoothed logarithmic robustness as

[TABLE]

Similarly, the max-entropy of a memory can be defined as

[TABLE]

and the smoothed version as

[TABLE]

Note that, as the smoothing is defined based on the diamond norm of channels, the smoothed measures cannot be obtained by smoothing the Choi state, i.e.,

[TABLE]

II.2 Properties

Here, we focus on properties of the robustness measures. We prove it for $\mathcal{R}(\mathcal{N})$ and the related measures. Unless otherwise mentioned, the same proof for $\mathcal{R}$ also holds for $\mathcal{R}_{G}$.

Non-negativity. For any channel $\mathcal{N}$, we have

[TABLE]

More specifically, we also have the following:

• •

$\mathcal{R}(\mathcal{N})=0\Leftrightarrow\mathcal{N}\in\mathcal{F}$.

• •

$\mathrm{LR}(\mathcal{N})=0\Leftrightarrow\mathcal{N}\in\mathcal{F}$.

• •

$\mathrm{LR}^{\varepsilon}(\mathcal{N})=0$ for all $\mathcal{N}\in\mathcal{F}$.

This follows directly from the definition. 2. 2.

Monotonicity. For a resource non-generating super-operation $\Lambda$, we have

[TABLE]

Proof.

We first prove $\mathcal{R}(\Lambda(\mathcal{N}))\leq\mathcal{R}(\mathcal{N})$. Suppose the minimisation of $s=R(\mathcal{N})$ is achieved with channel $\mathcal{M}$, we have

[TABLE]

Applying the resource non-generating super-operation $\Phi$ to both sides of the above equation, we have

[TABLE]

Since $\Lambda(\mathcal{M})\in\mathcal{F}$, we conclude that $\mathcal{R}(\Lambda(\mathcal{N}))\leq s=\mathcal{R}(\mathcal{N})$. As $\mathrm{LR}(\mathcal{N})=\log_{2}(1+\mathcal{R}(\mathcal{N}))$, we also have $\mathrm{LR}(\Lambda(\mathcal{N}))\leq\mathrm{LR}(\mathcal{N})$.

To prove $\mathrm{LR}^{\varepsilon}(\Lambda(\mathcal{N}))\leq\mathrm{LR}^{\varepsilon}(\mathcal{N})$, suppose $\mathcal{N}^{\prime}$ achieves the minimisation of the smooth of $\mathrm{LR}^{\varepsilon}(\mathcal{N})$, so that $\mathrm{LR}^{\varepsilon}(\mathcal{N})=\mathrm{LR}(\mathcal{N}^{\prime})$. As $\|\Lambda(\mathcal{N}^{\prime})-\Lambda(\mathcal{N})\|_{\diamond}\leq\|\mathcal{N}^{\prime}-\mathcal{N}\|_{\diamond}\leq\varepsilon$, we have

[TABLE]

With monotonicity, we also have that the measures are invariant under reversible transformations. ∎

As a special case of the monotonicity property, we have that the robustness measure of sequentially connected memories is upper bounded by the minimal robustness of each memory. That is,

[TABLE]

This also holds for the logarithmic robustness and the smoothed logarithmic robustness. 3. 3.

Convexity. For a set of memories $\{\mathcal{N}_{i}\}$ with probability distribution $\{p_{i}\}$ satisfying $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$, the averaged resource measure cannot be increased via mixing memories, i.e.,

[TABLE]

Proof.

For all $i$, suppose the minimisation of $s_{i}=R(\mathcal{N}_{i})$ is achieved with $\mathcal{M}_{i}$, that is,

[TABLE]

Let $s=\sum_{i}p_{i}s_{i}$, $\mathcal{N}=\sum_{i}p_{i}\mathcal{N}_{i}$ and $\mathcal{M}=\frac{1}{s}\sum_{i}p_{i}s_{i}\mathcal{M}_{i}$, then

[TABLE]

Since $\mathcal{M}\in\mathcal{F}$, we have

[TABLE]

∎ 4. 4.

Relation under tensor product. For two channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, we have

[TABLE]

For the logarithmic robustness and max-entropy, we also have

[TABLE]

or the tighter relation

[TABLE]

Proof.

For $i=1,2$, suppose the minimisation of $s_{i}=\mathcal{R}(\mathcal{N}_{i})$ is achieved with $\mathcal{M}_{i}$, that is,

[TABLE]

Then

[TABLE]

Let $s=(s_{1}+1)(s_{2}+1)+s_{1}s_{2}-1$, then $\mathcal{N}_{1}\otimes\mathcal{N}_{2}$ can be expressed as

[TABLE]

where

[TABLE]

and

[TABLE]

In conclusion,

[TABLE]

For the generalised robustness, we have

[TABLE]

Therefore

[TABLE]

To prove the lower bound, we define the partial trace of a quantum channel as

[TABLE]

By definition (28), the partial trace of a bipartite EB channel is also an EB channel.

Suppose the minimisation of $s=\mathcal{R}(\mathcal{N}_{1}^{A}\otimes\mathcal{N}_{2}^{B})$ is achieved with $\mathcal{M}^{AB}$, that is,

[TABLE]

where superscript denotes input systems. Now, take partial trace on system B, we get

[TABLE]

thus $\mathcal{R}(\mathcal{N}_{1})\leq s$. Symmetrically, we also have $\mathcal{R}(\mathcal{N}_{2})\leq s$, so the lower bound $\max\{\mathcal{R}(\mathcal{N}_{1}),\mathcal{R}(\mathcal{N}_{2})\}\leq\mathcal{R}(\mathcal{N}_{1}\otimes\mathcal{N}_{2})$ is obtained.

∎ 5. 5.

Stability. For any channel $\mathcal{N}$ and any EB channel $\mathcal{M}$,

[TABLE]

Proof.

As appending a free channel is a resource non-generating operation, we have $\mathcal{R}(\mathcal{N}\otimes\mathcal{M})\leq\mathcal{R}(\mathcal{N})$. Then, from property 4 we also have $\mathcal{R}(\mathcal{N}\otimes\mathcal{M})\geq\mathcal{R}(\mathcal{N})$, thus the equality is obtained. ∎

III Single-shot memory dilution

III.1 Preliminary

Before proving the main results, we first obtain some preliminary Lemmas which are useful for the following discussions.

Lemma 1**.**

The logarithmic robustness of an ideal $d$-dimensional quantum memory $\mathcal{I}_{d}$ is

[TABLE]

Proof.

It is equivalent to show that $\mathcal{R}(\mathcal{I}_{d})=d-1$. We first prove the upper bound $\mathcal{R}(\mathcal{I}_{d})\leq d-1$. Notice that the Choi state of $\mathcal{I}_{d}$ is the maximally entangled state $\ket{\Phi^{+}_{AB}}=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}\ket{i}_{A}\otimes\ket{i}_{B}$. Using the same technique as in Vidal and Tarrach (1999), we show that $\Phi^{+}_{AB}$ can be expressed as separable states

[TABLE]

where $\rho^{-}=\frac{1}{d(d-1)}\sum_{i\neq j}\ket{ij}_{AB}\bra{ij}_{AB}$ and $\rho^{+}=\frac{\Phi^{+}_{AB}+(d-1)\rho^{-}}{d}$. It is shown in Vidal and Tarrach (1999) that these are both separable states. It remains to show that these correspond to Choi states of EB channels. This is true since

[TABLE]

By definition of robustness of memory, we have $\mathcal{R}(\mathcal{I}_{d})\leq d-1$. Also, the lower bound $\mathcal{R}(\mathcal{I}_{d})\geq d-1$ is trivial, because if $\mathcal{R}(\mathcal{I}_{d})<d-1$, then using the same construction, we can show that the robustness of entanglement of $\Phi^{+}_{AB}$ is less than $d-1$, violating the known result in Vidal and Tarrach (1999). ∎

Lemma 2**.**

For any pure state $\ket{\psi}$ and quantum channels $\mathcal{N}_{1},\mathcal{N}_{2}$ in the same space, the linear map

[TABLE]

is a quantum super-channel represented by operation on Choi states. Here, $\mathcal{C}$ is an input quantum channel, $\Phi_{\mathcal{C}}$ is its Choi state, and $\Lambda(\Phi_{\mathcal{C}})$ is the Choi state of the output channel.

Proof.

For an input quantum state $\omega$, the output channel acts on $\omega$ as

[TABLE]

In order to show that $\Lambda$ is a quantum super-channel, we only need to show that $\Lambda(\mathcal{C})$ can be decomposed into three steps: pre-processing, action with ancillary system, and post-processing, which are constructed as follows:

• •

pre-processing: the input state is appended with a maximally entangled state

[TABLE]

• •

action with ancillary system: the channel $\mathcal{C}$ acts on system B while identity acts on systems A and C

[TABLE]

• •

post-processing: suppose the channels $\mathcal{N}_{1},\mathcal{N}_{2}$ have Kraus operators $\{K_{j}\}_{j},\{T_{k}\}_{k}$, respectively. We construct a post-processing channel with Kraus operators

[TABLE]

To see that this is a valid quantum channel, we have

[TABLE]

Also, we can see that the output of this channel is

[TABLE]

∎

III.2 Single-shot memory dilution

We consider the problem of single-shot dilution or channel simulation under resource non-generating super-operations, which is defined as

[TABLE]

Note that compared to the definition in Methods of the main text, we used the logarithm of the dimension to represent the number of qubit memories required for the task.

Theorem 1**.**

For any quantum channel $\mathcal{N}$, its single-shot memory dilution rate is

[TABLE]

Proof.

First we prove the left hand side $\mathrm{LR}^{\varepsilon}(\mathcal{N})\leq R_{c}^{1,\varepsilon}(\mathcal{N})$. Suppose there exists a memory dilution protocol $\Lambda$ such that

[TABLE]

With the monotonicity of the logarithmic robustness, we have

[TABLE]

Here, the last line follows from Lemma 1. Since the above equation holds for any dilution protocol, we conclude that $\mathrm{LR}^{\varepsilon}(\mathcal{N})\leq R_{c}^{1,\varepsilon}(\mathcal{N})$.

Next, we prove the right hand side $R_{c}^{1,\varepsilon}(\mathcal{N})\leq\mathrm{LR}^{\varepsilon}(\mathcal{N})+1$.

Suppose the optimisation in $\mathrm{LR}^{\varepsilon}(\mathcal{N})$ is achieved with $\mathcal{N}^{\prime}$, such that $\mathrm{LR}^{\varepsilon}(\mathcal{N})=\mathrm{LR}(\mathcal{N}^{\prime})=\log_{2}(1+\mathcal{R}(\mathcal{N}^{\prime}))$. Denote $d_{c}=\lceil R(\mathcal{N}^{\prime})\rceil+1$. Then, there exists $\mathcal{M},\mathcal{M}^{\prime}\in\mathcal{F}$ such that

[TABLE]

We construct a linear map as

[TABLE]

where systems $C$ and $D$ have dimension $d_{c}$. By Lemma 2, we know that $\Lambda$ is a super-channel.

Next, we verify that $\Lambda$ is a free resource non-generating super-operation. We first rewrite Eq. (79) as

[TABLE]

with $q=d_{c}\mathrm{Tr}\left(\Phi^{+}_{CD}\Phi_{\mathcal{C}}^{CD}\right)$. When $\mathcal{C}$ is an EB channel, $\Phi_{\mathcal{C}}^{CD}$ is a separable state, and we have $0\leq q\leq 1$. Thus $\Lambda(\Phi_{\mathcal{C}}^{CD})$ is a separable Choi state that corresponds to an EB channel.

Lastly, we verify that when inputting the identity channel, the output channel $\Lambda(\mathcal{I}^{C\rightarrow D})=\mathcal{N}^{\prime A\rightarrow B}$ is $\varepsilon$-close to the target channel $\mathcal{N}$. Therefore we have

[TABLE]

∎

Remark 1**.**

Define the smooth robustness of quantum memory as

[TABLE]

Since dimensions are integers, the single-shot memory dilution rate can be exactly characterised as

[TABLE]

The above can be compared with the one-shot characterisation of dilution in resource theories of states Brandao and Datta (2011); Zhao et al. (2018); Liu and Winter (2019), which yields related results but is not applicable to the study of manipulation of quantum channels.

IV Simulating quantum memories with classical resources

The task is to simulate a target quantum memory $\mathcal{N}$ with free EB memories and free super-operations. For an unknown input state $\rho$, the output state is $\mathcal{N}(\rho)$. Suppose after another operation $\mathcal{U}$ is applied to the output state, we read out the system by the measuring the average value $\braket{O}_{\mathcal{U}\circ\mathcal{N}(\rho)}=\mathrm{Tr}[\mathcal{U}\circ\mathcal{N}(\rho)O]$. The simulation scheme works as follows.

As we require the simulation scheme to be independent of $\rho$, $\mathcal{U}$, and $O$, it is equivalent to have $\mathcal{N}$ as a linear expansion of EB channels $\mathcal{M}_{i}$, i.e.,

[TABLE]

with coefficients $c_{i}\in\mathbb{R}$ possibly negative. This linear decomposition always exists as the set of EB channels forms a complete basis for the space of quantum channels. 2. 2.

To obtain the averaged value $\braket{O}_{\mathcal{U}\circ\mathcal{N}(\rho)}$, we first re-express it as $\braket{O}_{\mathcal{U}\circ\mathcal{N}(\rho)}=\|c\|_{1}\braket{\overline{O}}_{\mathcal{U}\circ\mathcal{N}(\rho)}$ with a normalised observable

[TABLE]

an overhead $\|c\|_{1}=\sum_{i}|c_{i}|$, and a normalised probability distribution $\{p_{i}=|c_{i}|/\|c\|_{1}\}$.

We can obtain $\braket{\overline{O}}_{\mathcal{U}\circ\mathcal{N}(\rho)}$ by averaging $\textrm{sign}(c_{i})\braket{O}_{\mathcal{U}\circ\mathcal{M}_{i}(\rho)}$ with probability $p_{i}$, which is described as follows:

1. (a)

We randomly generate $i$ according to the probability distribution $\{p_{i}\}$. 2. (b)

As each $\mathcal{M}_{i}$ is EB, we have $\mathcal{M}_{i}(\rho)=\sum_{j}\mathrm{Tr}[\rho M_{i}^{j}]\sigma^{j}_{i}$ with POVM $\{M_{i}^{j}\geq 0\}$ satisfying $\sum_{j}M_{i}^{j}=I$. 3. (c)

To get $\braket{O}_{\mathcal{U}\circ\mathcal{M}_{i}(\rho)}$, we first measure $\rho$ with the POVM $\{M_{i}^{j}\geq 0\}$, obtaining outcome $j$ with probability $\mathrm{Tr}[\rho M_{i}^{j}]$. Then we prepare $\sigma^{j}_{i}$, apply $\mathcal{U}$, and measure the observable $O$ to have $\mathrm{Tr}[\mathcal{U}(\sigma^{j}_{i})O]$. The value $\braket{O}_{\mathcal{U}\circ\mathcal{M}_{i}(\rho)}$ can be evaluated by averaging the measurement results $\mathrm{Tr}[\mathcal{U}(\sigma^{j}_{i})O]$ over all outcomes $j$. 4. (d)

Finally, by multiplying $\textrm{sign}(c_{i})$ to $\braket{O}_{\mathcal{U}\circ\mathcal{M}_{i}(\rho)}$ and averaging over all $\mathcal{M}_{i}$ with probability $p_{i}$, we recover the normalised average expectation value $\braket{\overline{O}}_{\mathcal{U}\circ\mathcal{N}(\rho)}$. 3. 3.

The target averaged value $\braket{O}_{\mathcal{U}\circ\mathcal{N}(\rho)}$ is obtained by multiplying the constant overhead $\|c\|_{1}$ to $\braket{\overline{O}}_{\mathcal{U}\circ\mathcal{N}(\rho)}$.

Suppose we aim to estimate $\braket{O}_{\mathcal{N}(\rho)}$ to an additive error $\varepsilon$ with probability $\delta$, we need the number of samples to be

[TABLE]

according to the Hoeffding inequality. Meanwhile, given the channel $\mathcal{N}$ itself, the number of samples needed is $T_{0}\propto 1/{\varepsilon^{2}}\log(\delta^{-1})$. Thus the simulation cost can be quantified by the overhead

[TABLE]

We can further minimise the simulation cost over all possible decomposition strategies of Eq. (84). Denote the positive and negative coefficients of $c_{i}$ by $c_{i}^{+}$ and $c_{i}^{-}$, respectively. Then we have

[TABLE]

with $\|c\|=\sum_{i:c_{i}\geq 0}|c_{i}^{+}|+\sum_{i:c_{i}<0}|c_{i}^{-}|$. As the channel is trace preserving, we also have $\sum_{i:c_{i}\geq 0}|c_{i}^{+}|-\sum_{i:c_{i}<0}|c_{i}^{-}|=1$. Denote $s=\sum_{i:c_{i}<0}|c_{i}^{-}|$ hence with $\|c\|=2s+1$, $\mathcal{M}=\sum_{i:c_{i}<0}|c_{i}^{-}|\mathcal{M}_{i}$, and $\mathcal{M}^{\prime}=\sum_{i:c_{i}>0}|c_{i}^{+}|\mathcal{M}_{i}$, we have

[TABLE]

Optimising over all possible decomposition is equivalent to optimising over all $\mathcal{M},\mathcal{M}^{\prime}\in\mathcal{F}$, which coincides with the definition of the robustness of memories. Therefore we have

[TABLE]

Therefore, the robustness of quantum memory quantifies the cost of simulating the memory with free EB memories.

The result of this section can be regarded as an extension of the framework of negativity-based simulators of Pashayan et al. (2015); Howard and Campbell (2017), which were recently adapted to the study of general resource theories of states Seddon et al. (2020), and applied in other settings to investigate the properties of specific channel resources Seddon and Campbell (2019); Wang et al. (2019).

V Quantum game for testing the power of a memory

In this section, we discuss how to use quantum games to test the power of a memory and how the optimal strategy is related to the robustness of memories. We consider quantum games similar to Ref. Rosset et al. (2018) by firstly inputting a general set of input states $\{\sigma_{i}\}$ to the channel. In Ref. Rosset et al. (2018), the output state is measured together with an ancillary set of states, so that the test can be independent of whether the measurement is faithfully implemented. In this work, instead of requiring such a measurement-device-independent feature, we directly measure the output states with a general set of observables $\{O_{j}\}$ by assuming that the measurement is trusted. We can define a general pay-off function as

[TABLE]

with real coefficients $\alpha_{i,j}$, where we use $\mathcal{G}=(\{\sigma_{i}\},\{O_{j}\},\{\alpha_{i,j}\})$ to denote a particular game. When the coefficients $\alpha_{i,j}$ are selected randomly, the pay-off function can be arbitrary, and in particular does not have to be non-negative. Instead, we first constrain ourselves to the case where the coefficients are selected such that the pay-off function is non-negative for any channel, $\mathcal{P}(\mathcal{N},\mathcal{G})\geq 0$. Then the maximal pay-off of the game is

[TABLE]

where the maximisation is over all games $\mathcal{G}\in\mathcal{S}_{G}$ with $\mathcal{S}_{G}=\{\mathcal{G}:\mathcal{P}(\mathcal{N},\mathcal{G})\geq 0,\,\mathcal{P}(\mathcal{M},\mathcal{G})\leq 1,\,\forall\mathcal{N}\in\mathrm{CPTP},\mathcal{M}\in\mathcal{F}\}$.

To prove it, we first briefly review the duality in conic optimisation. We follow the description of Ref. Takagi and Regula (2019) and refer to the references therein for more details. Given real complete normed vector spaces $\mathcal{W}$ and $\mathcal{W}^{\prime}$, a conic optimisation problem is defined as

[TABLE]

where $A\in\mathcal{W}^{*},y\in\mathcal{W}^{\prime}$, $\Lambda:\mathcal{W}\rightarrow\mathcal{W}^{\prime}$ is a linear function, and $\mathcal{K}\subseteq\mathcal{W}$ is a closed and convex cone. The dual form of the optimisation is given by

[TABLE]

where

[TABLE]

The primal and dual problems are equivalent if Slater’s condition is satisfied: that is there exists a feasible solution $x$ such that $x$ is in the (relative) interior of $\mathcal{K}$.

Now we prove Eq. (92).

Proof.

We first write $\mathcal{R}_{G}(\mathcal{N})$ in terms of its Choi state as

[TABLE]

This can be equivalently recast as

[TABLE]

where we use $\mathcal{V}$ ($\mathcal{F}$) to represent bipartite (separable) Choi states, while $\mathrm{cone}(\mathcal{F})$ and $\mathrm{cone}(\mathcal{V})$ represent their unnormalised versions. Now define $\mathcal{W}=\mathrm{cone}(\mathcal{V})\oplus\mathrm{cone}(\mathcal{V})$, $\mathcal{W}^{\prime}=\mathcal{V}$, $\mathcal{K}=\mathrm{cone}(\mathcal{F})\oplus\mathrm{cone}(\mathcal{V})$, $\Lambda\left(x_{1}\oplus x_{2}\right)=x_{1}-x_{2}$, $A=I\oplus 0$, $y=\Phi^{+}_{\mathcal{N}}$, then $\mathcal{R}_{G}(\mathcal{N})$ can be represented as the form of Eq. (93). Note that $\mathcal{W}^{*}$ is the set of Hermitian operators. The dual form of the optimisation gives

[TABLE]

where

[TABLE]

Note that the above condition is further equivalent to

[TABLE]

This is because when we have $\mathrm{Tr}[x_{1}]\geq\mathrm{Tr}[Wx_{1}]-\mathrm{Tr}[Wx_{2}],\,\forall x_{1}\in\mathrm{cone}(\mathcal{F}),\,\forall x_{2}\in\mathrm{cone}(\mathcal{V})$, we can set $x_{2}=0$ to have $\mathrm{Tr}[Wx_{1}]\leq\mathrm{Tr}[x_{1}],\,\forall x_{1}\in\mathrm{cone}(\mathcal{F})$. Similarly, we set $x_{1}=0$ and get $\mathrm{Tr}[Wx_{2}]\geq 0,\,\forall x_{2}\in\mathrm{cone}(\mathcal{V})$. On the other hand, it is straightfward to verify that Eq. (100) implies Eq. (99). Therefore, the dual form can be written as

[TABLE]

which can be expressed with quantum channels

[TABLE]

Slater’s condition holds as we can choose, say, $W=I/2$. Thus we have strong duality, $\mathcal{R}_{G}^{d}(\mathcal{N})=\mathcal{R}_{G}(\mathcal{N})$.

Now we show that the dual form is equivalent to the maximal pay-off of the quantum game. We first write the pay-off function in terms of the Choi state of the channel as

[TABLE]

where $\sigma_{i}^{T}$ is the transpose of $\sigma_{i}$ and $W$ is a Hermitian operator

[TABLE]

Denote $\mathcal{P}_{\max}(\mathcal{N})=\max_{\mathcal{G}\in\mathcal{S}_{G}}\mathcal{P}(\mathcal{N},\mathcal{G})$ with $\mathcal{S}_{G}=\{\mathcal{G}:\mathcal{P}(\mathcal{N},\mathcal{G})\geq 0,\,\mathcal{P}(\mathcal{M},\mathcal{G})\leq 1,\,\forall\mathcal{N}\in\mathrm{CPTP},\mathcal{M}\in\mathcal{F}\}$, the optimisation over all games in $\mathcal{S}_{G}$ is equivalent to optimise over all Hermitian operators $W$ that satisfies $\mathrm{Tr}[\Phi^{+}_{\mathcal{M}}W]\geq 0$ for all channels $\mathcal{M}$ and $\mathrm{Tr}[\Phi^{+}_{\mathcal{M}^{\prime}}W]\leq 1$ for entanglement breaking channels. Therefore, we have

[TABLE]

∎

The robustness $\mathcal{R}(\mathcal{N})$ can be understood very similarly in this context. The maximal pay-off of the game is

[TABLE]

where the maximisation is over all games $\mathcal{G}\in\mathcal{S}$ with $\mathcal{S}=\{\mathcal{G}:\mathcal{P}(\mathcal{M},\mathcal{G})\in[0,1],\,\forall\mathcal{M}\in\mathcal{F}\}$.

Proof.

We follow the proof for the generalised robustness. We first write the robustness measure in the standard form of the primal optimisation problem as

[TABLE]

where we define components in the standard form Eq. (93) as $\mathcal{W}=\mathrm{cone}(\mathcal{V})\oplus\mathrm{cone}(\mathcal{V})$, $\mathcal{W}^{\prime}=\mathcal{V}$, $\mathcal{K}=\mathrm{cone}(\mathcal{F})\oplus\mathrm{cone}(\mathcal{F})$, $\Lambda\left(x_{1}\oplus x_{2}\right)=x_{1}-x_{2}$, $A=I\oplus 0$, $y=\Phi^{+}_{\mathcal{N}}$.

The dual form of the optimisation gives

[TABLE]

where

[TABLE]

Note that the above condition is further equivalent to

[TABLE]

We can express it with quantum channels as

[TABLE]

which is exactly the maximal pay-off function $\mathcal{P}_{\max}(\mathcal{N})=\max_{\mathcal{G}\in\mathcal{S}}\mathcal{P}(\mathcal{N},\mathcal{G})$. Since Slater’s condition also holds in this case as we can choose $W=I/2$, we conclude that strong duality holds, thus $\mathcal{P}_{\max}(\mathcal{N})=\mathcal{R}(\mathcal{N})+1.$

∎

By considering games $\mathcal{G}\in\mathcal{S}^{\prime}$ with $\mathcal{S}^{\prime}=\{\mathcal{G}:\mathcal{P}(\mathcal{M},\mathcal{G})\geq 0,\,\forall\mathcal{M}\in\mathcal{F}\}$, we thus get the result presented in the main text,

[TABLE]

The characterisation of $\mathcal{R}_{G}$ in terms of performance in nonlocal games can be compared with Takagi et al. (2019); Takagi and Regula (2019), where this quantity was related to the advantage in discrimination tasks in general resource theories of states and channels. The setting considered in that work is different, however, most importantly since the measurements there are performed on the joint Choi state of channels rather than their output states, making them significantly more difficult to perform in practice.

VI Numerical and analytical calculation

VI.1 Relaxing to PPT-inducing channels

The robustness and generalised robustness measures of memories are both convex optimisation problems due to the convexity property. They can be numerically calculated by focusing on the Choi states as

[TABLE]

and

[TABLE]

Instead of directly solving this optimisation problem, we reduce the restrictions that $\Phi^{+}_{\mathcal{M}},\Phi^{+}_{\mathcal{M}^{\prime}}$ are separable Choi states to ones that they are positive partial transpose (PPT) Choi states. We denote the robustness measures against PPT-inducing channels as $\mathcal{R}^{*}(\mathcal{N})$ and $\mathcal{R}^{*}_{G}(\mathcal{N})$, defined as

[TABLE]

and

[TABLE]

As the set of PPT Choi states is larger than the set of separable Choi states, the optimisation always give lower bounds

[TABLE]

The bound is tight for channels with qubit inputs and outputs or ones with qubit inputs and qutrit outputs or vice versa. In general cases, the optimisation with respect to PPT Choi states can be efficiently solved as semidefinite programs. In the case of qubit outputs, we will in fact establish an analytical form of the measures.

[TABLE]

and

[TABLE]

In Fig. 4(a) we show several the robustness of examples channels.

Meanwhile the expression for the robustness of quantum memories $\mathcal{N}$ is similar to the robustness of entanglement of Choi matrix $\Phi^{+}_{\mathcal{N}}$, which is defined as

[TABLE]

and

[TABLE]

Note that $\Phi^{+}_{\mathcal{M}},\Phi^{+}_{\mathcal{M}^{\prime}}$ belong to the subset of separable states whose partial trace is identity due to the property of Choi states. Thus the robustness of entanglement of the Choi state lower bounds the robustness of quantum memory, $RE(\Phi^{+}_{\mathcal{N}})\leq\mathcal{R}(\mathcal{N})$ and $RE_{G}(\Phi^{+}_{\mathcal{N}})\leq\mathcal{R}_{G}(\mathcal{N})$. Similarly, we can also relax the separable state set with PPT to efficiently calculate $RE^{*}(\Phi^{+}_{\mathcal{N}})$ and $RE^{*}_{G}(\Phi^{+}_{\mathcal{N}})$, so that we have $RE^{*}(\Phi^{+}_{\mathcal{N}})\leq RE(\Phi^{+}_{\mathcal{N}})$ and $RE^{*}_{G}(\Phi^{+}_{\mathcal{N}})\leq RE_{G}(\Phi^{+}_{\mathcal{N}})$. By randomly choosing quantum channels, one can find that the quantities $RE^{*}(\Phi^{+}_{\mathcal{N}})$ and $\mathcal{R}^{*}(\mathcal{N})$ are different, as shown in Fig. 4(b) and (d).

Note that for the numerical examples in the main text, we focus on quantum channels with input dimension $d_{\text{in}}$ and output dimension $d_{\text{out}}$ such that $d_{\text{in}}\times d_{\text{out}}\leq 6$, in which case the SDP relaxation provided here is actually tight, due to the well-known correspondence between SEP and PPT in low dimensions Horodecki et al. (1996).

VI.2 Examples of quantum game

The pay-off of a quantum game $\mathcal{G}=(\{\sigma_{i}\},\{O_{j}\},\{\alpha_{i,j}\})$ can be expressed as

[TABLE]

where $\Phi_{\mathcal{N}}^{+}=\mathcal{I}\otimes\mathcal{N}(\Phi^{+})$ is the Choi state of $\mathcal{N}$, $\Phi^{+}=1/d\sum_{ij}\ket{ii}\bra{jj}$ is the maximally entangled state, $d$ is the dimension of the input system, and the game operator is

[TABLE]

For the robustness of quantum memories, we consider games that satisfy

[TABLE]

VI.2.1 Depolarising channel $\Delta_{p}(\rho)=p\rho+(1-p)I/2$

For the depolarising channel $\Delta_{p}(\rho)=p\rho+(1-p)I/2$, the corresponding Choi state is the Werner state,

[TABLE]

The robustness of $\Delta_{p}(\rho)$ can be obtained by applying the entanglement witness considered in our previous work Xu et al. (2014). Here, we construct the game $\mathcal{G}=(\{\sigma_{i}\},\{O_{j}\},\{\alpha_{i,j}\})$ such that

[TABLE]

Because $2\mathrm{Tr}[\Phi^{+}\sigma]\leq 1$ for all bipartite qubit separable states $\sigma$, we have $P(\mathcal{M},\mathcal{G})\in[0,1]$, $\forall\mathcal{M}\in\mathcal{F}$. We can also show that $W$ is the optimal witness of $\mathcal{R}_{G}(\Delta_{p})$ by proving

[TABLE]

This is equivalent to finding a separable Choi state $\Phi_{\mathcal{M}}^{+}$ that satisfies

[TABLE]

This is satisfied by choosing $\Phi_{\mathcal{M}}^{+}=\frac{1}{3}\Phi^{+}+\frac{1}{6}I$.

Now we show how to decompose the witness into input states and measurements. Suppose we choose $\{\sigma_{i}\},\{O_{j}\}$ as

[TABLE]

with $O_{i}=\sigma_{i}^{T}$, the corresponding coefficients are

[TABLE]

For the qubit depolarising channel $\Delta_{p}(\rho)=p\rho+(1-p)I/2$, the pay-off of this game is

[TABLE]

This provides a lower bound

[TABLE]

The equal sign is always achieved, as verified from the numerical calculation in Fig. 4(a).

VI.2.2 Erasure channels $\mathcal{E}_{p}(\rho)=p\rho+(1-p)|{2}\rangle\langle{2}|$

Considering the erasure channels $\mathcal{E}_{p}(\rho)=p\rho+(1-p)|{2}\rangle\langle{2}|$ with $\ket{2}$ orthogonal to $\{\ket{0},\ket{1}\}$, the Choi state is

[TABLE]

Now we choose the game with a witness

[TABLE]

and we can prove that $P(\mathcal{M},\mathcal{G})\in[0,1]$, $\forall\mathcal{M}\in\mathcal{F}$ by showing $\mathrm{Tr}[W\sigma]\in[0,1]$ for all separable states $\sigma$. We sketch the proof here. Consider a projective measurement $\{P_{01}=\ket{0}\bra{0}+\ket{1}\bra{1},P_{2}=\ket{2}\bra{2}\}$ on the second system, then

[TABLE]

Here $p_{01}=\mathrm{Tr}[P_{01}\sigma_{01}P_{01}]$, $p_{2}=\mathrm{Tr}[P_{2}\sigma_{2}P_{2}]$, $\sigma_{01}=P_{01}\sigma P_{01}/p_{01}$, and $\sigma_{2}=P_{2}\sigma P_{2}/p_{2}$. The last line follows from $\mathrm{Tr}[2\Phi^{+}\sigma_{01}]\leq 1$ and $p_{01}+p_{2}=1$.

As $W\geq 0$, $\mathrm{Tr}[W\sigma]\geq 0$. We can check that

[TABLE]

which is consistent with the numerical calculation,

[TABLE]

We can also prove that the estimate from the game witness is tight. That is, we need to find a separable Choi state $\Phi_{\mathcal{M}}^{+}$ that satisfies

[TABLE]

Here, we choose $\Phi_{\mathcal{M}}^{+}=q(\frac{1}{3}\Phi^{+}+\frac{1}{6}I_{2}\otimes I_{2})+(1-q)\frac{I}{2}\otimes\ket{2}\bra{2}$ with $I_{2}=\ket{0}\bra{0}+\ket{1}\bra{1}$. To satisfy the above inequality, we choose $q=\frac{2p}{p+1}$ when $p\geq 1/3$ and $q=\frac{1-p}{p+1}$ when $p\leq 1/3$.

The decomposition of the witness is similar to the one for the depolarising channel. The only difference is that we need to consider an additional measurement to take into account of the term $I\otimes\ket{2}\bra{2}$. That is, with the same inputs given in Eq. (129), we consider the game with input states

[TABLE]

measurements

[TABLE]

and coefficients

[TABLE]

VI.2.3 stochastic damping channels $\mathcal{D}_{p}(\rho)=p\rho+(1-p)|{0}\rangle\langle{0}|$

Note that the robustness of quantum memories of the two above examples are equal to the robustness of entanglement of the corresponding Choi states. For the stochastic damping channel, we found them different as shown in numerical examples of Fig. 4(c) of the main text.

The Choi state of the stochastic damping channels $\mathcal{D}_{p}(\rho)=p\rho+(1-p)|0\rangle\langle 0|$ is

[TABLE]

Suppose the spectral decomposition of this matrix is

[TABLE]

where $\lambda_{0}\geq\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ and $\psi_{i}$ is the density matrix of eigenstate. We can compute that

[TABLE]

as well as $\ket{\psi_{0}}=\alpha\ket{00}+\beta\ket{11}$ with real coefficients $\alpha$ and $\beta$ depending on $p$,

[TABLE]

Now construct the game with a witness $W$ defined by

[TABLE]

Note that the witness does not satisfy $\mathrm{Tr}[W\sigma]\in[0,1]$ for all separable states $\sigma$ as $\max_{\sigma\in\mathrm{SEP}}\mathrm{Tr}[W\sigma]=2\max\{|\alpha|^{2},|\beta|^{2}\}$ can be larger than 1. However, Choi states of EB channels are only a subset of separable states, and we can indeed show that $\mathrm{Tr}[W\sigma]\in[0,1]$ for all separable Choi states $\sigma$. That is, for all separable states $\sigma_{AB}$ satisfying $\mathrm{Tr}_{B}[\sigma_{AB}]=I_{A}/2$, we want to prove

[TABLE]

We prove it by considering a stronger scenario for general EB channels with input dimension $d$ and we also consider an optimisation over all possible pure states $\psi_{0}$,

[TABLE]

This is equivalent to the following Lemma.

Lemma 3**.**

For any EB channel with input dimension $d$, the maximal eigenvalue of its Choi state $\sigma_{AB}$ is upper bounded by $1/d$,

[TABLE]

Proof.

The maximal eigenvalue of $\sigma_{AB}$ can be obtained by considering its Schatten $\infty$-norm. For an EB channel, its Choi state can always be expressed as

[TABLE]

with $\sum_{i}p_{i}\psi_{i}=I/d$, $\sum_{i}p_{i}=1$, and $p_{i}\geq 0$. Therefore we have

[TABLE]

Here the inequality follows from $\mathrm{Tr}\left[\phi_{i_{1}}\phi_{i_{2}}\dots\phi_{i_{n}}\right]\leq 1$.

∎

The decomposition of $\psi_{0}$ can be expressed as

[TABLE]

Thus we can choose

[TABLE]

and $\sigma_{i}=O_{i}^{T}$with coefficients

[TABLE]

And finally we obtain the lower bound

[TABLE]

VI.3 Tight lower bound via moments of Choi states

Consider the Choi state $\Phi_{\mathcal{N}}^{+}$ of a channel $\mathcal{N}$, we can either lower bound the robustness via a witness measurement or via purity measurement. We prove Lemma 1 in the main text.

Lemma 4**.**

For any EB channel $\mathcal{M}$ with input dimension $d$ and $k=0,1,\dots,\infty$, we have

[TABLE]

For any channel $\mathcal{N}$ with input dimension $d$, its RQM can be lower bounded by

[TABLE]

Proof.

The first half of this Lemma, Eq. (156), can be proven by following the proof of Lemma 3.

For the second half, we show that $\mathcal{R}_{G}(\mathcal{N})\geq d^{\frac{k-1}{k}}\left(\mathrm{Tr}\left[\left(\Phi_{\mathcal{N}}^{+}\right)^{k}\right]\right)^{\frac{1}{k}}-1$. Suppose the optimal decomposition is

[TABLE]

with $\mathcal{M}^{\prime}\in\mathrm{EB}$ and $\mathcal{M}\in\mathrm{CPTP}$. Then

[TABLE]

and we obtain a lower bound for the robustness

[TABLE]

∎

The comparison of the lower bounds with different moments of Choi states is shown in Fig. 4(d).

We will show that the bound obtained by taking $k\to\infty$ is in fact tight for low-dimensional channels, and furthermore exactly characterises the PPT robustness $\mathcal{R}^{*}$ for all channels with qubit output.

Theorem 2**.**

Consider a channel ${\mathcal{N}}:A\to B$ such that $d_{A}\leq 3$ and $d_{B}=2$. Then

[TABLE]

More generally, for any channel such that $d_{B}=2$, the PPT robustness satisfies

[TABLE]

Proof.

We will employ the reduction criterion for separability Horodecki and Horodecki (1999); Cerf et al. (1999), which states that a bipartite state $\rho_{AB}$ with $d_{A}\leq 3$ and $d_{B}=2$ is separable if and only if

[TABLE]

Notice that when $\rho_{AB}=\Phi^{+}_{\mathcal{N}}$ is the Choi state of a channel ${\mathcal{N}}:A\to B$, this reduces to

[TABLE]

that is, $\max\operatorname{eig}\Phi^{+}_{\mathcal{N}}\leq\frac{1}{d_{A}}$. The SDPs for the robustness measures then become

[TABLE]

The inequality ${\mathcal{R}}({\mathcal{N}})\geq{\mathcal{R}}_{G}({\mathcal{N}})\geq d_{A}\max\operatorname{eig}\Phi^{+}_{\mathcal{N}}$ is obvious as $\Phi^{+}_{\mathcal{N}}\leq\lambda\Phi^{+}_{\mathcal{M}}\leq\frac{\lambda}{d_{A}}I$ for any feasible $\lambda$ and $\Phi^{+}_{\mathcal{M}}$. On the other hand, let $\mu$ denote the largest eigenvalue of $\Phi^{+}_{\mathcal{N}}$. If $\mu\leq\frac{1}{d_{A}}$, then $\Phi^{+}_{\mathcal{N}}$ itself is a feasible solution and the equality ${\mathcal{R}}({\mathcal{N}})={\mathcal{R}}_{G}({\mathcal{N}})=0$ is trivial, so assume that $\mu\in(\frac{1}{d_{A}},1]$. Denoting by $\Delta$ the completely depolarising channel with $\Phi^{+}_{\Delta}=I/d_{A}d_{B}$, we define the channels ${\mathcal{M}}_{\pm}$ as the convex combinations

[TABLE]

Noting that $\mu d_{A}\geq 1$ and thus $\mu d_{A}d_{B}-1\geq 2\mu d_{A}-1\geq\mu d_{A}$, it follows that $\Phi^{+}_{{\mathcal{M}}_{\pm}}\leq\frac{1}{d_{A}}I$, which means that the above constitutes a valid feasible solution for the robustness ${\mathcal{R}}({\mathcal{N}})$ with $\lambda=\mu d_{A}$. We conclude that ${\mathcal{R}}_{G}({\mathcal{N}})\leq{\mathcal{R}}({\mathcal{N}})\leq\mu d_{A}=d_{A}\max\operatorname{eig}\Phi^{+}_{\mathcal{N}}$, and so the quantities must all be equal.

The second part of the Proposition follows since the action of the positive map $\rho\to(\mathrm{Tr}\rho)I-\rho$, on which the reduction criterion is based, is unitarily equivalent to the transpose map when acting on a 2-dimensional system Horodecki and Horodecki (1999); Cerf et al. (1999). This means that the reduction criterion and the PPT criterion are equivalent when $d_{B}=2$. ∎

Altogether, the results establish Theorem 4 in the main text.

As an immediate consequence of the results, we notice that the multiplicativity of the lower bound coupled with the sub-multiplicativity of $R_{G}$ (see Sec. II.2) yields exact multiplicativity for $R_{G}$ or, equivalently, additivity for $D_{\max}$ in the case of low-dimension channels.

Corollary 1**.**

For any channel $\mathcal{N}:A\to B$ with $d_{B}=2$, we have

[TABLE]

In particular, when $d_{A}\leq 3$, it holds that

[TABLE]

or equivalently $D_{\max}(\mathcal{N}^{\otimes n})=nD_{\max}(\mathcal{N})$.

VII Experiment demonstrations on the IBM cloud

VII.1 Robustness of three example type of channels

Here we consider the robustness of the dephasing channels $\Delta_{p}(\rho)=p\rho+(1-p)Z\rho Z$, stochastic damping channels $\mathcal{D}_{p}(\rho)=p\rho+(1-p)|{0}\rangle\langle{0}|$, and erasure channels $\mathcal{E}_{p}(\rho)=p\rho+(1-p)|{2}\rangle\langle{2}|$ with $\ket{2}$ orthogonal to the computational basis $\{\ket{0},\ket{1}\}$. We implement the dephasing channel, erasure channel, and the stochastic damping channel in Fig. VII.1, VII.1, and VII.1, respectively. For these three different channels, we choose four different noise levels as $p=1,3/4,1/2,1/4$ and implement it on the IBM cloud. The game for the dephasing channel is the same one for the depolarising channel defined in Eq. (129) and Eq. (130). The game for the erasure channel is defined in Eq. (139) and Eq. (141). We use two qubits to encode $\ket{0}$, $\ket{1}$ and $\ket{2}$, via $\ket{0}\otimes\ket{0}\rightarrow\ket{0}$, $\ket{0}\otimes\ket{1}\rightarrow\ket{1}$, $\ket{1}\otimes I\rightarrow\ket{2}$. The game for the stochastic damping channel is defined in Eq. (153) and Eq. (154). The first qubit is used to choose the original state or $\ket{0}$. We use the second qubit to present the input state and use the third one to replace the original state with $\ket{0}$. We simulate this channel via collecting the post-processing statistics. When the outcome of first qubit is $\ket{0}$, we only care about the outcomes of second qubit. Otherwise, we focus on the outcomes of third one. The experiment results of the three channels are shown in Table 1, 2, and 3.