Quantifying the resource content of quantum channels: An operational approach

TL;DR

This paper introduces an operational method to quantify the resourcefulness of quantum channels through their ability to be distinguished from free channels, linking success probability to resource generation.

Contribution

It establishes a universal framework connecting channel discrimination success with resource generation, applicable across various quantum resource theories.

Findings

Success probability equals the resource generating power of the channel.

Framework applies to coherence, entanglement, magic states, and asymmetry.

Provides a quantitative measure for resourcefulness in quantum channels.

Abstract

We propose a general method to operationally quantify the resourcefulness of quantum channels via channel discrimination, an important information processing task. A main result is that the maximum success probability of distinguishing a given channel from the set of free channels by free probe states is exactly characterized by the resource generating power, i.e. the maximum amount of resource produced by the action of the channel, given by the trace distance to the set of free states. We apply this framework to the resource theory of quantum coherence, as an informative example. The general results can also be easily applied to other resource theories such as entanglement, magic states, and asymmetry.

Equations

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

$$\tilde{\Omega }(\mathcal{N})$$

$${\omega }_D(\rho ):=\underset{\sigma \in \mathcal{F}}{\min }D(\rho ,\sigma ).$$

$${\Omega }_D(\mathcal{N})={\tilde{\Omega }}_D(\mathcal{N}).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathcal{M},\rho )$$

$$=$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho ):=\underset{\mathcal{M}\in \mathfrak{F}}{\min }{p}_{\mathrm{succ}}(\mathcal{N},\mathcal{M},\rho ),$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\mathcal{F}):=\underset{\rho \in \mathcal{F}}{\max }{p}_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho ),$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},Q):=\underset{\rho \in \mathcal{D}(\mathcal{H})}{\max }{p}_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho ).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\mathcal{F})=\frac{1}{2}+\frac{1}{2}{\tilde{\Omega }}_1(\mathcal{N})=\frac{1}{2}+\frac{1}{2}{\Omega }_1(\mathcal{N}).$$

$${\Omega }_1({\mathcal{M}}_1\circ \mathcal{N}\circ {\mathcal{M}}_2)\le {\Omega }_1(\mathcal{N}).$$

$${\Omega }_1(\sum _i{p}_i{\mathcal{N}}_i)\le \sum _i{p}_i{\Omega }_1({\mathcal{N}}_i).$$

$${\Omega }_1({\mathcal{N}}_1\otimes {\mathcal{N}}_2)\ge \max \{{\Omega }_1({\mathcal{N}}_1),{\Omega }_1({\mathcal{N}}_2)\}.$$

$${\Omega }_1({\mathcal{N}}_1\otimes {\mathcal{N}}_2)\le {\Omega }_1({\mathcal{N}}_1)+{\Omega }_1({\mathcal{N}}_2).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho )-p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\mathcal{F})\le \frac{1}{2}{\omega }_1(\rho ).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho )\le \frac{1}{2}+\frac{1}{2}{\Omega }_1(\mathcal{N})+\frac{1}{2}{\omega }_1(\rho ).$$

$$C_{l_1}(\rho ):$$

$$C_1(\rho ):$$

$$C_R(\rho )$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I})=\frac{1}{2}+\frac{1}{2}{\tilde{\mathcal{C}}}_1(\mathcal{N})=\frac{1}{2}+\frac{1}{2}{\mathcal{C}}_1(\mathcal{N}).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},Q)>p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I}).$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\rho )-p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I})\le \frac{1}{2}{C}_1(\rho ).$$

$$p_{succ}^I(\mathcal{N},\mathcal{M},\rho )$$

$$=$$

$$p_{\mathrm{succ}}^I(\mathcal{N},\mathfrak{I},\rho )=\frac{1}{2},$$

$$p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\rho )-p_{\mathrm{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I})\le \frac{1}{2}{E}_1({\rho }_{AB}),$$

$${\Omega }_D(\mathcal{N})=\underset{\rho \in \mathcal{F}}{\max }\underset{\mathcal{M}\in \mathfrak{F}}{\min }D(\mathcal{N}(\rho ),\mathcal{M}(\rho )).$$

$$\underset{\rho \in \mathcal{F}}{\max }\underset{\mathcal{M}\in \mathfrak{F}}{\min }D(\mathcal{N}(\rho ),\mathcal{M}(\rho ))$$

$$\ge$$

$$=$$

$$\underset{\rho \in \mathcal{F}}{\max }\underset{\mathcal{M}\in \mathfrak{F}}{\min }D(\mathcal{N}(\rho ),\mathcal{M}(\rho ))$$

$${\Omega }_D(\mathcal{N})={\tilde{\Omega }}_D(\mathcal{N}).$$

$${\omega }_D(\mathcal{N}(\rho ))-{\omega }_D(\rho )$$

$${\tilde{\Omega }}_1(\mathcal{N})={\Omega }_1(\mathcal{N})=\frac{1}{2}\underset{\rho \in \mathcal{F}}{\max }\underset{\mathcal{M}\in \mathfrak{F}}{\min }{\Vert \mathcal{N}(\rho )-\mathcal{M}(\rho )\Vert }_1.$$

Full text

I Introduction

Understanding and utilizing various forms of quantum resources represents a main theme of quantum information science. To this end, a powerful framework known as the quantum resource theory is being actively developed in recent years to systematically study the quantification and manipulation of quantum resources (see Chitambar and Gour for a recent review). In fact, the resource features of certain quantum effects, in particular quantum entanglement, have already been carefully studied earlier Plenio and Virmani (2007); Horodecki et al. (2009); Nielsen and Chuang (2010), but a key observation underlying the recent interests in the resource theory framework is that the theories of different kinds of resource properties (stemming from different physical constraints) can share a largely common structure and a wide range of general approaches and results Horodecki and Oppenheim (2013a); Brandão and Gour (2015); Liu et al. (2017); Regula (2018); Anshu et al. (2018); Takagi et al. (2019); Skrzypczyk and Linden (2019); Liu et al. (2019). Indeed, this idea has been successfully applied to the study of various other key quantum resources, such as coherence Baumgratz et al. (2014); Winter and Yang (2016); Streltsov et al. (2017), superposition Theurer et al. (2017), magic states Veitch et al. (2014); Howard and Campbell (2017), thermal non-equilibrium Brandão et al. (2013); Horodecki and Oppenheim (2013b), asymmetry Gour et al. (2009); Marvian and Spekkens (2014), etc.

The well-established schemes of resource theory (at a non-abstract level; see e.g. Coecke et al. (2016); Fritz (2017) for abstract, category-theoretic formulations that do not rely on the explicit mathematical structures of the object space) mostly handle in particular static resources encoded in quantum states (density operators). However, certain quantum processes or channels can represent dynamical quantum resources which play natural and fundamental roles in broad scenarios. The systematic study of channel resource theories is blueprinted recently by Liu and Winter (2019), but we are still at an early stage of developing the complete theory.

The quantification of resource is a central topic of all kinds of resource theories. In particular, one is interested in the operational interpretation of certain resource measures, i.e. how they correspond to the value of the resource in achieving some operational task. In state resource theories, general operational resource measures can be induced by several tasks, e.g. resource interconversion Horodecki and Oppenheim (2013a); Brandão and Gour (2015); Liu et al. (2019), resource erasure Anshu et al. (2018). However, for quantum channels, we only know that the smooth log-robustness characterizes the randomness cost of the task of one-shot resource erasure Liu and Winter (2019) at the general level. (Note that the quantification of channel resources have been previously considered in various specific contexts, such as entanglement Bennett et al. (2003), coherence Ben Dana et al. (2017); Theurer et al. , non-Gaussianity Zhuang et al. (2018), and magic Wang et al. (2019)).

In this work, we suggest a simple and general scheme of quantify the resourcefulness of quantum channels based on quantum channel discrimination, a fundamental problem in quantum information Acín (2001); Wang and Ying (2006); Pirandola et al. . (Note that channel discrimination is already known to play key roles in the characterization of state resources Napoli et al. (2016); Bu et al. (2017a); Takagi et al. (2019); Skrzypczyk and Linden (2019); Bae et al. (2019).) The core question here is how well one can distinguish a quantum channel from another by optimizing over input probe states and output measurements. We find that the maximum success probability of distinguishing the given channel from the set of free operations by all free probe states is exactly characterized by the maximum amount of resource that can be generated by the channel, i.e. the resource generating power, as measured by the trace-norm distance of resource. This resource generating power satisfies several desirable properties, such as faithfulness, convexity, sub-multiplicity and monotonicity. Besides, the advantage of using a resource state as the probe state, compared with free probe states, is upper-bounded by the trace-norm measure of resource. As a prominent example, we analyze in depth the widely-studied resource theory of coherence, the structure of which allows for further results. Our study leads to several new understandings of the coherence theory. This approach can be easily generalized to many other important resource theories. As an example, we state a basic result for entanglement theory.

II Main results

Given a finite dimensional Hilbert space $\mathcal{H}$, let $\mathcal{D}(\mathcal{H})$ denote the set of all quantum states on $\mathcal{H}$. Assume the set of free states $\mathcal{F}$ to be a non-empty, convex and closed subset of $\mathcal{D}(\mathcal{H})$. Let $\mathfrak{F}$ be the set of free quantum channels, or completely positive and trace preserving (CPTP) maps. Channels in $\mathfrak{F}$ must map all free states to free states.

Define the resource generating/increasing power ($\Omega/\widetilde{\Omega}$) of channel $\mathcal{N}:\mathcal{D}(\mathcal{H})\longrightarrow\mathcal{D}(\mathcal{H})$ as follows. Given some resource monotone of states $\omega$ and the set of free states $\mathcal{F}$:

[TABLE]

Note that the complete versions of resource generating/increasing power can also be defined which, in addition, optimize over any ancilla space (see Liu and Winter (2019) for extended discussions).

A representative type of resource monotones is the distance to $\mathcal{F}$. More explicitly, given some distance measure $D$, one can define resource measure $\omega_{D}$ for any quantum state $\rho$ as follows:

[TABLE]

The resource generating/increasing power given by $\omega_{D}$ is denoted $\Omega_{D}$/$\widetilde{\Omega}_{D}$. It can be shown that they are actually equivalent for contractive distance metrics (see the proof in Appendix A of Supplemental Material):

Proposition 1**.**

If the distance measure $D$ satisfies the triangle inequality and the data processing inequality ( i.e., non-increasing under CPTP maps), then we have

[TABLE]

Of particular importance to this work is the trace distance $\frac{1}{2}\left\lVert\rho-\sigma\right\rVert_{1}:=\frac{1}{2}\mathrm{Tr}{|\rho-\sigma|}$ , which we denote by subscript “1”.

Here, we aim at establishing connections between the resource generating power of a channel and its non-free feature in the task of channel discrimination. Given two channels $\mathcal{N}$ and $\mathcal{M}$, and the same probe state $\rho$ going through the channels $\mathcal{N},\mathcal{M}$ respectively, then the success probability of distinguishing $\mathcal{N}$ and $\mathcal{M}$ by the probe state $\rho$ is the success probability of distinguishing $\mathcal{N}(\rho)$ and $\mathcal{M}(\rho)$ as follows

[TABLE]

where the maximization is taken over all POVM $\{\Pi,\mathbb{I}-\Pi\}$. By the Holevo-Helstrom Theorem Helstrom (1976), $p_{\rm succ}(\mathcal{N},\mathcal{M},\rho)=\frac{1}{2}+\frac{1}{4}\left\lVert\mathcal{N}(\rho)-\mathcal{M}(\rho)\right\rVert_{1}$.

The success probability of distinguishing $\mathcal{N}$ from the set of channels $\mathfrak{F}$ by the probe state $\rho$ is defined as

[TABLE]

and the maximum success probability of distinguishing $\mathcal{N}$ from $\mathfrak{F}$ by using any free state or any quantum state (denoted by $Q$) as the probe state are respectively given by

[TABLE]

The following result provides an exact characterization of the success probability $p_{\rm succ}(\mathcal{N},\mathfrak{F},\mathcal{F})$:

Theorem 2**.**

Given a quantum channel $\mathcal{N}$ and the set of free channels $\mathfrak{F}$. The maximum success probability of discriminating $\mathcal{N}$ from $\mathfrak{F}$ by the set of free states $\mathcal{F}$ is only directly related to the resource increasing power given by trace distance (which equals the generating power due to Proposition 1) of $\mathcal{N}$ as follows:

[TABLE]

The proof of this theorem is provided in Appendix A of Supplemental Material. We now show that $\Omega_{1}(\mathcal{N})$ satisfies the basic conditions for resource quantifiers of quantum channels, e.g. normalized, and monotone under left and right compositions with free channels Liu and Winter (2019). More specifically,

Proposition 3**.**

The trace-norm resource generating power $\Omega_{1}(\mathcal{N})$ satisfies the following properties:

(i) $\Omega_{1}(\mathcal{N})\geq 0$, and $\Omega_{1}(\mathcal{N})=0$ if $\mathcal{N}\in\mathfrak{F}$. Moreover, if $\mathfrak{F}$ includes all CPTP maps which maps all free states to free states (resource non-generating maps), then $\Omega_{1}(\mathcal{N})=0$ iff $\mathcal{N}\in\mathfrak{F}$.

(ii) For any $\mathcal{M}_{1},\mathcal{M}_{2}\in\mathfrak{F}$, we have

[TABLE]

(iii) Given a set of quantum channels $\set{\mathcal{N}_{i},p_{i}}_{i}$ with $\sum_{i}p_{i}=1$,

[TABLE]

Moreover, if the free states on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is defined as convex combination of the tensor product of free states on $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, i.e., $\mathcal{F}_{AB}=Conv\set{\mathcal{F}_{A}\otimes\mathcal{F}_{B}}$, then resource generating power $\Omega_{1}(\mathcal{N})$ also satisfies the following properties,

(iv) Given two channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, it holds that

[TABLE]

(v) Given two channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, it holds that

[TABLE]

In fact, each of the above properties holds under weaker assumptions. The proof for more general distance measures is provided in Appendix B of Supplemental Material. Due to property (i), Theorem 2 also indicates that resource non-generating channels are effectively indistinguishable from each other by free probe states. Due to property (iv), it is easy to define a regularized version of $\Omega_{1}(\mathcal{N})$ by $\Omega^{\infty}_{1}(\mathcal{N})=\lim_{n\to\infty}\frac{1}{n}\Omega_{1}(\mathcal{N}^{\otimes n})$, which is invariant under tensoring, i.e., $\Omega^{\infty}_{1}(\mathcal{N}^{\otimes 2})=\Omega^{\infty}_{1}(\mathcal{N})$. However, this is not the focus of this work.

Since $\mathcal{F}\subset\mathcal{D}(\mathcal{H})$, we have $p_{\rm succ}(\mathcal{N},\mathfrak{F},Q)\geq p_{\rm succ}(\mathcal{N},\mathfrak{F},\mathcal{F})$. If the probe state $\rho$ is not a free state, then the resource in $\rho$ may help improve the success probability of discriminating the given channel $\mathcal{N}$ from the set of free channels. Here we provide an upper bound on the advantage of using a resource probe state:

Theorem 4**.**

Given a quantum channel $\mathcal{N}$, a quantum state $\rho$ and the set of free channels $\mathfrak{F}$. The advantage provided by the state $\rho$ compared with all free states to distinguish any given channel $\mathcal{N}$ from $\mathfrak{F}$ is upper bounded by the trace-norm distance of resource:

[TABLE]

The proof is presented in Appendix C of Supplemental Material. A direct corollary is the following bound on the success probability of discriminating $\mathcal{N}$ from free channels by any probe state $\rho$:

Corollary 5**.**

Given a quantum channel $\mathcal{N}$, a quantum state $\rho$ and the set of free channels $\mathfrak{F}$, the success probability $p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho)$ is upper bounded by

[TABLE]

III Example

As an application of the above general framework, we now focus on quantum coherence, a prominent quantum feature emerging from the superposition principle of quantum mechanics. Coherence represents a key quantum resource which has a variety of applications in quantum information science, including quantum metrology Giovannetti et al. (2011), thermodynamics Lostaglio et al. (2015a, b) and biology Plenio and Huelga (2008); Levi and Mintert (2014). In recent years, the resource theory of coherence has drawn a lot of attention, where the manipulation and characterization of coherence in quantum states are thoroughly investigated (see Streltsov et al. (2017); Hu et al. (2018) for a review). Now we extend the study to quantum channels following the idea in the last section, that is, to characterize the coherence value of a channel by its distinguishability from the typical sets of coherence-free channels.

Given a fixed basis $\set{\ket{i}}^{d-1}_{i=0}$ for a $d$-dimensional system, any quantum state which is diagonal in the reference basis is called an incoherent state and is a free state in the resource theory of coherence. The set of incoherent states is denoted by $\mathcal{I}$. Let $\Delta$ denote the fully dephasing channel in the given basis, which is defined as $\Delta(\rho)=\sum_{i}\left\langle i\left|\rho\right|i\right\rangle|i\rangle\!\langle i|$. $\Delta$ is a prominent example of the resource destroying map Liu et al. (2017).

There are several individually motivated choices of free operations in the resource theory of coherence. The following four, which collectively emerge from the relations with $\Delta$ and can be broadly generalized via the theory of resource destroying map Liu et al. (2017), are considered most important: (1) maximally incoherent operations (MIO) Chitambar and Gour (2016a), the maximum possible set of coherence-free operations that contains all quantum operations $\mathcal{M}$ that maps incoherent states to incoherent states, i.e., $\mathcal{M}(\mathcal{I})\subset\mathcal{I}$; (2) incoherent operations (IO) Baumgratz et al. (2014), containing $\mathcal{M}$ that admit a set of Kraus operators $\set{K_{i}}$ such that $\mathcal{M}(\cdot)=\sum_{i}K_{i}(\cdot)K^{\dagger}_{i}$ and $K_{i}\mathcal{I}K^{\dagger}_{i}\subset\mathcal{I}$ for any $i$; (3) dephasing-covariant operations (DIO) Chitambar and Gour (2016a); Liu et al. (2017), containing $\mathcal{M}$ such that $[\Delta,\mathcal{M}]=0$ (4) strictly incoherent operations (SIO) Chitambar and Gour (2016b, a), containing all $\mathcal{M}$ admitting a set of Kraus operators $\set{K_{i}}$ such that $\Delta(K_{i}\rho K^{\dagger}_{i})=K_{i}\Delta(\rho)K^{\dagger}_{i}$ for any $i$ and any quantum state $\rho$.

Several operational motived coherence measures have been introduced and here we consider the coherence measure defined by $l_{1}$-norm distance Baumgratz et al. (2014), trace-norm distance Shao et al. (2015) and robustness Piani et al. (2016),

[TABLE]

In fact, in single-qubit system $\mathbb{C}^{2}$, the trace-norm of coherence $C_{1}$ is equal to $l_{1}$-norm of coherence $C_{l_{1}}$ Rana et al. (2016); Shao et al. (2015) and the robustness of coherence $C_{R}$ Piani et al. (2016) up to a scalar 2.

In the resource theory of coherence, certain coherence generating power can also be used to characterize the cost of simulating the given channel by incoherent operations Bu et al. (2017b); Díaz et al. and the capacity of a channel to generate maximally coherent states Ben Dana et al. (2017). Besides, the ability of a quantum channel to detect non-classicality has also been introduced to quantify the resource of channels in terms of trace distance Theurer et al. and relative entropy Theurer et al. ; Yuan .

First, it follows from Theorem 2 that the success probability of distinguishing $\mathcal{N}$ from the set of free operations $\mathfrak{I}$, where $\mathfrak{I}$ can be any of $\{SIO,IO,DIO,MIO\}$, is universally determined by the trace-norm coherence generating power.

Proposition 6**.**

Given a quantum channel $\mathcal{N}$ and the set of coherence-free operations $\mathfrak{I}\in\{SIO,IO,DIO,MIO\}$, the maximum success probability of distinguishing $\mathcal{N}$ from $\mathfrak{I}$ by incoherent states is

[TABLE]

Again, the result indicates that channels in MIO are mutually indistinguishable by incoherent states since $C_{1}(\mathcal{N})=0$. Therefore, the task of discriminating a channel from coherence-free ones gives an operational interpretation for the coherence generating power. Compared with Bu et al. (2017a) and Napoli et al. (2016), which only consider the effect of coherence in the probe states in channel discrimination, the results here reveal the roles of coherence in quantum channels in this task.

Since the trace-norm of coherence $C_{1}\leq 1-1/d$ Chen et al. (2016); Yu et al. (2016), the success probability $p_{\text{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I})\leq 1-1/(2d)$. For example, for the Hadamard gate $H$ on single-qubit system $\mathbb{C}^{2}$, we have $p_{\rm succ}(H,\mathfrak{I},\mathcal{I})=3/4$, which follows from the fact that $\mathcal{C}_{1}(H)=1/2$ (see Appendix A of Supplemental Material for the calculation of $\mathcal{C}_{1}$ in single-qubit system). Due to the equivalence between trace-norm distance and robustness of coherence, it may be expected that this theorem can be experimentally testified in a future work, as the robustness of coherence can be measured in experiment Wang et al. (2017); Zheng et al. (2018).

Obviously, $p_{\text{succ}}(\mathcal{N},\mathfrak{I},Q)\geq p_{\text{succ}}(\mathcal{N},\mathfrak{I},\mathcal{I})$ for any quantum channel. There exists some quantum channel $\mathcal{N}$ such that the inequality is strict, which shows that the resource of probe states is useful for distinguishing the given channel from the set of free operations.

Proposition 7**.**

For $\mathfrak{I}\in\set{SIO,IO}$, there exists some quantum channel $\mathcal{N}$ such that

[TABLE]

The proof is presented in Appendix D of Supplemental Material. The above result shows that resource of probe states is useful for improving the success probability of distinguishing the given channel from the set of free operations $\mathfrak{I}\in\set{SIO,IO}$. However, whether the similar result holds for MIO or DIO is unknown.

By applying Theorem 4 to the resource theory of coherence, we obtain the following upper bound on the success probability when we choose a coherent state as the probe state.

Proposition 8**.**

Given a set of free operations $\mathfrak{I}\in\{SIO,IO,DIO,MIO\}$ and a probe state $\rho$. For any quantum channel $\mathcal{N}$, we have

[TABLE]

If we restrict the measurement in the channel discrimination to be an incoherent POVM, i.e., diagonal in the given basis $\set{\ket{i}}_{i}$, then the success probability to distinguish the given two channels by a probe state $\rho$ is

[TABLE]

In this case, the success probability of distinguishing the given channel $\mathcal{N}$ from the set of free operation $\mathfrak{I}\in\{SIO,IO,DIO,MIO\}$ is equal to the probability of random guessing.

Theorem 9**.**

Given a quantum channel $\mathcal{N}$ and the set of free operations $\mathfrak{I}\in\{SIO,IO,DIO,MIO\}$, then the success probability by incoherent POVM is

[TABLE]

for any $\rho\in\mathcal{D}(\mathcal{H})$.

The proof is provided in Appendix E of Supplemental Material. Therefore, the restriction of incoherent POVM will eliminate the advantage provided by the coherence of state and channel in the task of channel discrimination. Note that Ref. Theurer et al. considers a slightly different scenario (for example, the order of taking minimization over channels and maximization over states is different and the set of free operations there is consisted of detection-incoherent operations, which is different from those we consider), where, in contrast, it is possible to distinguish a channel from free ones with probability greater than $1/2$ even by free measurements. Moreover, the coherence feature of channels and its quantification that Ref. Theurer et al. studies rely on the resource destroying map (the fully dephasing channel), but our approach does not.

The general results Theorem 2 and 4 can also be applied to other resource theories, such as entanglement, magic states and so on. For instance, in the resource theory of bipartite entanglement, the free states are separable states, and the free operations are typically chosen to be Local Operations and Classical Communication (LOCC), or Separable operations (SEP)—the maximal set of entanglement non-generating operations. Then we have

Proposition 10**.**

Given the set of free operations $\mathfrak{I}\in\{LOCC,SEP\}$ and a probe state $\rho_{AB}\in\mathcal{D}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$. For any quantum channel $\mathcal{N}$, we have

[TABLE]

where $E_{1}(\rho_{AB}):=\min_{\sigma\in Sep(A:B)}\left\lVert\rho_{AB}-\sigma\right\rVert_{1}$ and $Sep(A:B)$ denotes the set of separable states on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$.

As for the free measurement case, in general, we can also define the free measurement $\set{\Pi,\mathbb{I}-\Pi}$, where $\Pi$ and $\mathbb{I}-\Pi$ are proportional to some free states. If a resource theory has resource destroying channel $\lambda$ and $\lambda^{{\dagger}}$ is a resource destroying channel as well, then Theorem 9 is still true (see Appendix E of Supplemental Material). However, whether Theorem 9 can be applied to other convex resource theories is unknown.

IV Conclusion

This work considers the fundamental task of channel discrimination from a resource theory perspective, which leads to an intuitive and general framework of operationally quantifying the resource value of quantum channels by how efficiently they can be distinguished from the resource-free ones. The key observation is that the maximum success probability of distinguishing a channel from the set of free operations by all free states is characterized by the trace-norm resource generating power of the channel. As the resource generating power satisfies the properties like positivity, convexity, sub-multiplicity and the monotonicity under free operations, it establishes an operational framework of quantifying resource in quantum channels. We demonstrate the power of this framework in the resource theory of quantum coherence. In addition to the de-generalized results, we also show that restricting to incoherent POVMs in this task will eliminate any advantage over random guessing. Our results shed new light on the operational resource theory of quantum channels and in particular the resource theory of coherence. We hope that the framework will lead to more interesting results for a variety of resource theories and information processing tasks.

Note added. During the revision of this paper, we became aware of a recent work by Liu and Yuan Liu and Yuan (2019), which establishes general connections between the resource generating/increasing power and channel distillation/dilution tasks.

Acknowledgements.

This research was supported in part by the Templeton Religion Trust under grant TRT 0159. L. Li and K. Bu acknowledge Arthur Jaffe for the support and help. K. Bu also thanks the support of Academic Awards for Outstanding Doctoral Candidates from Zhejiang University. Z.-W. Liu is supported by AFOSR, ARO, and Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Appendix A Connections between channel discrimination and resource

generating/increasing power

Given a distance measure $D:\mathcal{D}(\mathcal{H})\times\mathcal{D}(\mathcal{H})\to\mathbb{R}_{+}$, we consider the following conditions:

(1) Positivity: $D(\rho,\sigma)\geq 0$, $D(\rho,\sigma)=0$ iff $\rho=\sigma$.

(2) Pseudo joint convexity: $D(\sum_{i}p_{i}\rho_{i},\sum_{i}p_{i}\sigma_{i})\leq\max_{i}D(\rho_{i},\sigma_{i})$ with $\sum_{i}p_{i}=1$.

(2’) Joint convexity: $D(\sum_{i}p_{i}\rho_{i},\sum_{i}p_{i}\sigma_{i})\leq\sum_{i}p_{i}D(\rho_{i},\sigma_{i})$ with $\sum_{i}p_{i}=1$.

(3) Data processing inequality: $D(\mathcal{N}(\rho),\mathcal{N}(\sigma))\leq D(\rho,\sigma)$ for any CPTP map $\mathcal{N}$.

(4) Triangle inequality: $D(\rho,\sigma)\leq D(\rho,\tau)+D(\tau,\sigma)$ for any $\tau\in\mathcal{D}(\mathcal{H})$.

Here, we assume the distance measure always satisfies the condition (1) , i.e., positivity.

Lemma 11**.**

For any given distance measure $D$ and quantum channel $\mathcal{N}$, it holds that

[TABLE]

Proof.

First, we have

[TABLE]

where the inequality comes from the fact that $\mathcal{M}(\rho)\in\mathcal{F}$ for any $\rho\in\mathcal{F}$.

Besides, for any $\rho\in\mathcal{F}$, we can define the quantum channel $\mathcal{N}_{\rho}$ as $\mathcal{N}_{\rho}(\tau)=\sigma^{\star}_{\mathcal{N}(\rho)}$ for any quantum state $\tau\in\mathcal{D}(\mathcal{H})$ with $\sigma^{\star}_{\mathcal{N}(\rho)}\in\mathcal{F}$ and $\omega_{D}(\mathcal{N}(\rho))=D(\mathcal{N}(\rho),\sigma^{\star}_{\mathcal{N}(\rho)})$. It is easy to verify that $\mathcal{N}_{\rho}$ is a free operation, i.e., $\mathcal{N}_{\rho}\in\mathfrak{F}$. Thus,

[TABLE]

where the inequality comes from the fact that $\mathcal{N}_{\rho}\in\mathfrak{F}$ and $\mathcal{N}_{\rho}$ maps any quantum state to the free state $\sigma^{\star}_{\mathcal{N}(\rho)}$.

∎

Lemma 12**.**

If the distance measure $D$ satisfies the triangle inequality and the data processing inequality ( i.e., non-increasing under CPTP maps), then we have

[TABLE]

Proof.

It is obvious that $\Omega_{D}\leq\widetilde{\Omega}_{D}$, thus we only need to prove $\widetilde{\Omega}_{D}\leq\Omega_{D}$.

For any quantum state $\rho\in\mathcal{D}(\mathcal{H})$, we have

[TABLE]

where the first inequality comes from the data processing inequality and the second inequality comes from the triangle inequality of $D$. Therefore, we have $\widetilde{\Omega}_{D}(\mathcal{N})\leq\Omega_{D}(\mathcal{N})$.

∎

Proof of Theorem 2.

It is easy to verify that trace-norm satisfies the data processing inequality and the triangle inequality. Thus, according to Lemma 11 and 12, we have

[TABLE]

Besides, the success probability $p_{\rm succ}(\mathcal{N},\mathfrak{F},\mathcal{F})$ can be expressed as

[TABLE]

∎

Corollary 13**.**

If we take the distance measure $D$ to be max-relative entropy $D_{\max}$ or fidelity $D_{F}$, then we have

[TABLE]

where $D_{F}(\rho,\sigma)=\sqrt{1-F^{2}(\rho,\sigma)}$ with $F(\rho,\sigma)=\operatorname{Tr}\left[|\sqrt{\rho}\sqrt{\sigma}|\right]$.

Proof.

It has been proved that $D_{\max}$ satisfies the data processing inequality Datta (2009) and the triangle inequality comes directly from the definitions. Besides, it has been proved that $D_{F}$ satisfies the data processing inequality Barnum et al. (1996) and the triangle inequality Gilchrist et al. (2005); Uhlmann (1976). ∎

Now, let us consider the example of coherence. In single-qubit system, it has been proved that trace-norm of coherence $C_{1}$ is equivalent to $l_{1}$ norm of coherence $C_{l_{1}}$ Rana et al. (2016); Shao et al. (2015) and the analytic form of coherence generating power for unitary operations has been obtained in Bu et al. (2017b). Therefore, we have the following corollary,

Corollary 14**.**

Given a single-qubit unitary $U=[U_{ij}]_{i,j=1,2}$, the coherence generating power by trace-norm is

[TABLE]

Specially, for the Hadamard gate $H$, $\mathcal{C}_{1}(H)=1/2$.

Appendix B Properties of $\Omega_{D}(\mathcal{N})$

Now, let us investigate the properties of $\Omega_{D}(\mathcal{N})$ for any distance measure $D$. We assume that the free states on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is defined as convex combination of the tensor product of free states on $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, i.e., $\mathcal{F}_{AB}=Conv\set{\mathcal{F}_{A}\otimes\mathcal{F}_{B}}$.

Lemma 15**.**

Given any distance measure $D$, $\Omega_{D}(\cdot)$ has the following properties:

(i) $\Omega_{D}(\mathcal{N})\geq 0$, and $\Omega_{D}(\mathcal{N})=0$ if $\mathcal{N}\in\mathfrak{F}$. Moreover, if $\mathfrak{F}$ includes all CPTP maps which maps all free states to free states, then $\Omega_{D}(\mathcal{N})=0$ iff $\mathcal{N}\in\mathfrak{F}$.

(ii) If the distance measure $D$ satisfies the data processing inequality: For any $\mathcal{M}_{1},\mathcal{M}_{2}\in\mathfrak{F}$,

[TABLE]

(iii) If the distance measure $D$ satisfies joint convexity: Given a set of quantum channels $\set{\mathcal{N}_{i},p_{i}}_{i}$ with $\sum_{i}p_{i}=1$,

[TABLE]

(iv) If the distance measure $D$ satisfies the pseudo joint convexity and data processing inequality: Given two channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, it holds that

[TABLE]

(v) If the distance measure $D$ satisfies the pseudo joint convexity, data processing inequality and triangle inequality: Given two channels $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$, it holds that

[TABLE]

Proof.

(i) This comes directly from the definition.

(ii) For any $\mathcal{M}\in\mathfrak{F}$,

[TABLE]

where the first inequality comes from the fact that $\mathcal{M}\circ\mathcal{M}\in\mathfrak{F}$ for any $\mathcal{M}\in\mathfrak{F}$ and the second inequality comes from the data processing inequality.

Besides,

[TABLE]

where the inequality comes from the fact $\mathcal{M}(\mathcal{F})\subset\mathcal{F}$. ∎

(iii) Since $D$ is jointly convex, then the corresponding resource monotone $\omega_{D}$ is convex, i.e., $\omega_{D}(\sum_{i}p_{i}\rho_{i})\leq\sum_{i}p_{i}\omega_{D}(\rho_{i})$. Thus,

[TABLE]

(iv) We only need to prove that $\max\set{\omega_{D}(\rho_{1}),\omega_{D}(\rho_{2})}\leq\omega_{D}(\rho_{1}\otimes\rho_{2})$.

First,

[TABLE]

where $\tau_{1}=\operatorname{Tr}_{2}\left[\tau_{12}\right]$ and the inequality comes from the data processing inequality. Hence, we have $\omega_{D}(\rho_{1}\otimes\rho_{2})\geq\omega_{D}(\rho_{1})$. Similarly, we have $\omega_{D}(\rho_{1}\otimes\rho_{2})\geq\omega_{D}(\rho_{2})$.

(v) We only need to prove that $\omega_{D}(\rho_{1}\otimes\rho_{2})\leq\omega_{D}(\rho_{1})+\omega_{D}(\rho_{2})$. Due to the data processing inequality, we have

[TABLE]

because both partial trace and tensoring with a quantum state are CPTP maps.

Therefore, we have

[TABLE]

where the free states $\tau_{1}$ and $\tau_{2}$ are chosen to satisfy the conditions $\omega_{D}(\rho_{1})=D(\rho_{1},\tau_{1})$ and $\omega_{D}(\rho_{2})=D(\rho_{2},\tau_{2})$.

Proof of Proposition 3.

Since the trace norm satisfies the joint convexity, data processing inequality and triangle inequality, then the Proposition 3 comes directly from the Lemma 15.

∎

Appendix C Upper bound for $p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho)$

Proof of Theorem 4.

Since

[TABLE]

then by Theorem 2 and the definition of $p_{\mathrm{succ}}(\mathcal{N},\mathfrak{F},\rho)$, we have

[TABLE]

where the second inequality comes from the fact that

[TABLE]

for any $\rho\in\mathcal{D}(\mathcal{H})$. Thus, we complete the proof.

∎

Appendix D Improvement from coherent states in channel discrimination

Proof of Proposition 7.

It has been shown that there exists some quantum channel $\mathcal{N}_{*}\in MIO$ but not $IO$, i.e, there exists some quantum state $\rho$ such that $\mathcal{N}_{*}(\rho)\neq\mathcal{M}(\rho)$ for any $\mathcal{M}\in IO$ Bu and Xiong (2017), which implies that

[TABLE]

Thus, we have $p_{\rm succ}(\mathcal{N}_{*},IO,Q)>1/2$. However, due to Proposition 6, we have

[TABLE]

as $\mathcal{N}_{*}\in MIO$. Thus, we have

[TABLE]

Besides, since $SIO\subset IO$, then $p_{\rm succ}(\mathcal{N}_{*},SIO,Q)\geq p_{\rm succ}(\mathcal{N}_{*},IO,Q)$. Therefore,

[TABLE]

∎

Appendix E Discrimination with incoherent measuresment

Proof of Theorem 9.

It is easy to see that

[TABLE]

Besides, $\Delta^{{\dagger}}$ satisfies the conditions that $\Delta^{{\dagger}}(\mathcal{D}(\mathcal{H}))\subset\mathcal{I}$ and $\Delta^{{\dagger}}(\rho)=\rho$ for any $\rho\in\mathcal{I}$, which implies that

[TABLE]

∎

Note that, in any other resource theory with resource destroying channel $\lambda$, we can also define the free measurement $\set{\Pi,\mathbb{I}-\Pi}$, where $\Pi$ and $\mathbb{I}-\Pi$ are propositional to some free states. Then it is to see that the above proof still works for the free measurement case if the resource destroying map satisfies that conditions that $\lambda^{{\dagger}}(\mathcal{D}(\mathcal{H}))\subset\mathcal{F}$ and $\lambda^{{\dagger}}(\rho)=\rho$ for any $\rho\in\mathcal{F}$, i.e, $\lambda^{{\dagger}}$ is a resource destroying map Liu et al. (2017).