Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation

TL;DR

This paper introduces a quantum-control-assisted uncertainty relation that enables precise prediction of measurement outcomes and provides a novel method to classify multiparticle entanglement without full state knowledge.

Contribution

It develops a new variance-based uncertainty relation leveraging quantum control to analyze and distinguish multiparticle entanglement classes.

Findings

Defines 'multiparticle entanglement resolution lines' for classifying entangled states.

Allows identification of entanglement classes with limited measurements.

Does not require complete quantum state tomography.

Abstract

We construct a quantum-control-assisted multi-observable variance-based uncertainty relation, and the uncertainty relation obtained indicates that we can prepare a quantum state, in which the measurement results of any observables can be predicted precisely with the help of quantum control and entanglement resource. The new variance-based uncertainty relation provides a concept of "multiparticle entanglement resolution lines" for multiparticle entangled pure states, that can delineate different multiparticle entanglement classes, and thus it can be considered as an analyzer of multiparticle entanglement classes. The analyzer is used to identify different multiparticle entanglement classes with a suitable number of incompatible measurements, without a complete knowledge of the quantum state.

Equations

$$\mathrm{V}\left(R\right)\mathrm{V}\left(S\right)\ge \frac{1}{4}{\left|\mathrm{E}\left(\left[R,S\right]\right)\right|}^2,$$

$$\mathrm{V}\left(Q^{S_1}|O^{S_2}:={\lambda }_m\right)=\mathrm{Var}\left(Q^{S_1},{\rho }_{\left(O^{S_2}:={\lambda }_m\right)}^{S_1}\right),$$

$$\mathrm{E}\left[\mathrm{V}\left(Q^{S_1}|O^{S_2}\right)\right]=\sum _{j=1}\mathrm{P}\left(O^{S_2}:={\lambda }_j\right)\mathrm{V}\left(Q^{S_1}|O^{S_2}:={\lambda }_j\right),$$

$$\mathrm{E}\left[\mathrm{V}\left(Q^{S_1}|O^{S_2}\right)\right]\le \mathrm{V}\left(Q^{S_1}\right),$$

$$\sum _{k=1}^K\mathrm{E}[\mathrm{V}(Q_k^A|O_k^{C_1},\cdots ,O_k^{C_N})]\ge L_{tra}-\sum _{k=1}^K\mathrm{V}[\mathrm{E}(Q_k^A|O_k^{C_1})$$

$$-\sum _{k=1}^K\sum _{n=2}^N\mathrm{E}[\mathrm{V}(\mathrm{E}[Q_k^A|O_k^{C_n}]|O_k^{C_1},\cdots ,O_k^{C_{n-1}})]$$

$$|\Phi ⟩=\underset{l=1}{\overset{L}{\text{scalebox0.8⨂}}}|{\varphi }_l⟩,$$

$${\mathcal{L}}_0=$$

$${\mathcal{L}}_1=$$

$${\mathcal{L}}_2=$$

$$⋮$$

$${\mathcal{L}}_N=$$

$$\sqrt{1-2{\mu }^2}|2{⟩}_1|0{⟩}_2|0{⟩}_3+\mu |-1{⟩}_1|-1{⟩}_2|-1{⟩}_3-\mu |3{⟩}_1|1{⟩}_2|1{⟩}_3,$$

Full text

The fundamental difference between quantum and classical mechanics is that the measurement results of the observables are unpredictable even if all the information of the state of the quantum system is known 1 ; 2 ; 3 ; 4 . This “unpredictability” is usually expressed by the quantum uncertainty relation 4D ; 5 ; 6 . The uncertainty of the measurement result of an observable can be quantified by its variance and entropy, and thus these uncertainty relations are divided into the variance-based uncertainty relations 2 ; 3 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 and the entropic uncertainty relations 4 ; 4D ; 5 ; 6 ; 13 ; 14 ; 15 ; 16 ; 17 . Both forms of uncertainty relations arise from the preparation of a quantum state in which the measurement results of incompatible observables cannot be predicted precisely at the same time 6 ; 8 ; 12 . In 2010, Berta et al. constructed an entropic uncertainty relation with the assistance of a quantum memory 13 . The uncertainty relation obtained indicates that, with the help of an entanglement resource and a quantum memory, Bob can prepare a quantum state for Alice, in which the measurement results of two given incompatible observables are predicted accurately by Bob 13C ; 13R . That is to say, the lower bound of the traditional uncertainty relation can be broken with the introduction of a quantum memory. Indeed, quantum-memory-assisted uncertainty relation is, essentially, a conditional uncertainty relation, and this conditional uncertainty relation opened up new directions for a deeper understanding of quantum uncertainty relations 18 ; 19 ; 20 ; 21 ; 22 ; 23 . Yet, conditional uncertainty relations, despite possible experimental realization, remain till today largely unexplored 8 .

Entanglement, a nonlocal correlation of two or more systems 25 ; 26 , is an indispensable ingredient in the quantum information science 25 ; 221 ; 222 ; 223 , with ramification to quantum simulation 223 and quantum metrology 0231 ; 0232 ; 0233 ; 0234 . Detection of entanglement is therefore an important tool 2320 ; 241 . In general, entanglement for two particles is easily characterized and identified 25 ; 26 , but the same question for the multiparticle system is difficult 2320 ; 241 ; 242 ; 243 ; 244 . Besides, multiparticle entanglement belongs to different inequivalent classes 24 ; 25 ; 26 ; 27 ; 28 , and there exists no effective method to distinguish these classes when we do not capture a complete knowledge of the density matrix 29 ; 30 ; 31 ; 32 ; 33 . The reconstruction of the full density matrix is usually done by the quantum state tomography 34 ; 243 , but the method becomes impossible for the states with a large number of particles 243 .

One of the most important applications of the uncertainty relation is entanglement witness 23 ; 231 ; 232 ; 234 ; 233 , and the advantage of the uncertainty relation over other methods is that it allows us to detect entanglement without a complete knowledge of the quantum states 231 ; 232 ; 233 ; 234 . Generally, the uncertainty relation can be used to identify bipartite and tripartite entanglement 231 ; 232 ; 233 ; 235 ; 236 ; 237 , but it becomes useless for four and more particles 234 ; 236 . Lots of effort has been made to overcome this problem, but none of them provides an effective solution.

In this Letter, we introduce multiparticle entanglement using variance-based uncertainty relation by constructing a multi-observable uncertainty relation through multiple quantum control systems. The new uncertainty relation shows that, even without the quantum memory, the lower bound of the traditional uncertainty relation can be broken with the help of quantum control. More importantly, based on the new uncertainty relation, we introduce the concept of “multiparticle entanglement resolution lines” (MERLs). The MERLs present different spectral for different multiparticle entanglement classes, and thus they can be considered as a form of analyzer of multiparticle entanglement classes. Remarkably, this new analyzer can identify the different multiparticle entanglement classes even when we do not have a complete knowledge of the quantum state.

*Quantum-Control-Assisted Uncertainty Relation.— *The most famous variance-based uncertainty relation is 2 ; 3 :

[TABLE]

where $R$ and $S$ are two incompatible observables, $\mathrm{E}([R,S])$ represents the expected value of the commutator $[R,S]$, and $\mathrm{V}(Y)$ is the variance of measurement result corresponding to the observable $Y$ with $Y\in\{R,S\}$.

In order to introduce multiparticle entanglement into variance-based uncertainty relation, we first consider the concept of the conditional variance. Traditional conditional variance is defined contingent on a measurement and its specific result; i.e.:

[TABLE]

where $Q^{S_{i}}$ $(O^{S_{i}})$ represents the observable $Q$ $(O)$ of the subsystem $S_{i}$, $S_{i}$ stands for a subsystem of the whole system with $i\in\{1,2\}$, $O^{S_{i}}:=\lambda_{j}$ means that the result of the measurement $O$ on the subsystem $S_{i}$ is $\lambda_{j}$ with $\lambda_{j}$ being an eigenvalue of $O$, and $\rho_{\left(O^{S_{i}}:=\lambda_{j}\right)}$ represents the corresponding state of the whole system after the measurement with $\rho$ being the state before the measurement. \mathrm{Var}\left(Q^{S_{1}},\rho^{S_{1}}_{\left(O^{S_{2}}:=\lambda_{j}\right)}\right)$$=\mathrm{Tr}\left(\rho^{S_{1}}_{\left(O^{S_{2}}:=\lambda_{j}\right)}{Q^{S_{1}}}^{2}\right)-\left[\mathrm{Tr}\left(\rho^{S_{1}}_{\left(O^{S_{2}}:=\lambda_{j}\right)}Q^{S_{1}}\right)\right]^{2} stands for the variance of $Q^{S_{1}}$ on the state $\rho^{S_{1}}_{\left(O^{S_{2}}:=\lambda_{j}\right)}$, with $\rho^{S_{1}}_{\left(O^{S_{2}}:=\lambda_{j}\right)}=\mathrm{Tr}^{S_{2}}[\rho_{\left(O^{S_{2}}:=\lambda_{j}\right)}]$ being the partial trace of $\rho_{\left(O^{S_{2}}:=\lambda_{j}\right)}$ over the basis of the subsystem $S_{2}$. Thus, $\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}:=\lambda_{j}\right)$ represents the variance of $Q^{S_{1}}$ after the measurement $O^{S_{2}}$ has been performed and the result is obtained as $\lambda_{j}$.

However, there exists a limitation in the definition, which greatly restricts the application of the conditional variance in the quantum uncertainty relation. That is, the variance of a measurement result is not reduced with the information gained from the given condition, and therefore the given condition does not reduce the uncertainty of a measurement result. For instance, considering two qubit subsystems, we have $\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}:=0\right)\geq\mathrm{V}\left(Q^{S_{1}}\right)$ when choosing $Q=O=\sigma_{z}$ and taking $\left(|0\rangle_{1}+|1\rangle_{1}\right)|0\rangle_{2}/2+|1\rangle_{1}|1\rangle_{2}/\sqrt{2}$ as the state of the whole system, with $|0\rangle$ and $|1\rangle$ being the eigenstates of $\sigma_{z}$, and $|\cdots\rangle_{i}$ being the state of subsystem $S_{i}$.

In order to fix this problem, we redefine the conditional variance as:

[TABLE]

where $\mathrm{P}\left(O^{S_{i}}:=\lambda_{j}\right)$ represents the probability that the measurement result is $\lambda_{j}$ when we perform the measurement $O$ on the subsystem $S_{i}$. The new definition of the conditional variance is actually the expectation of $\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}:=\lambda_{j}\right)$ in terms of the probability $\mathrm{P}\left(O^{S_{2}}:=\lambda_{j}\right)$. Based on (2), $\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}:=\lambda_{j}\right)$ is used to quantify the variance of $Q^{S_{1}}$ if the measurement $O^{S_{2}}$ has been performed and the corresponding result is obtained as $\lambda_{j}$. Hence, $\mathrm{E}\left[\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}\right)\right]$, the expectation of $\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}:=\lambda_{j}\right)$ , is actually the expectation of the remaining variance of $Q^{S_{1}}$ when the measurement $O^{S_{2}}$ has been performed. That is to say, the new definition of the conditional variance $\mathrm{E}\left[\mathrm{V}\left(Q^{S_{1}}|O^{S_{2}}\right)\right]$ is used to quantify how much variance of $Q^{S_{1}}$ remains after the measurement $O^{S_{2}}$ has been performed. Remarkably, we can obtain (please see the Section I in the Supplemental Material 52 ):

[TABLE]

which means, under the new definition of conditional variance, the uncertainty of the measurement is reduced subject to the given condition.

Taking advantage of the new definition of conditional variance, we introduce multiparticle entanglement into variance-based uncertainty relation. Consider the following game, as shown in Fig.1. (i) Bob prepares $N+1$ particles, denoted by $A$, $C_{1},\cdots,C_{N-1}$ and $C_{N}$, which are entangled with each other, and sends the $A$-particle to Alice. (ii) Alice tells Bob about the measurement to be made. (iii) Based on the information provided by Alice and Bob’s knowledge about the quantum state of the whole system, Bob chooses a appropriate measurement and performs it on $C_{1},C_{2},\cdots,C_{N}$, respectively. Here we should mention that the measurements performed by Bob are not necessary the same as the measurement chosen by Alice. (iv) Finally, Alice performs the measurement on the particle $A$.

Based on the classical information captured by Bob, the measurements performed on the subsystems $C_{1},C_{2},\cdots,C_{N}$ are mainly used by Bob to manipulate the state of the whole system so as to minimize the uncertainty of the local measurement performed on the subsystem $A$ as much as possible. Thus, we label the particle $A$ as the measured system, the particles $C_{1},C_{2},\cdots,C_{N}$ as the control systems, and the measurements on the corresponding control systems as the quantum control. Obviously, from the perspective of Bob who possesses the control systems, the uncertainty of the measurement on the measured system is reduced with the entanglement between the measured system and control systems. In particular, by choosing a appropriate quantum control, the measurement result observed by Alice is precisely predicted by Bob when the control systems and measured system are in the maximum entangled state. The corresponding uncertainty relation can be expressed by the following inequality (please see the Section II in the Supplemental Material 52 ):

[TABLE]

where $Q_{1},Q_{2},\cdots,Q_{K}$ represent $K$ arbitrary incompatible observables, $O_{1},O_{2},\cdots,O_{K}$ stand for $K$ observables, and $\mathrm{E}\left[\mathrm{V}\left(Q_{k}^{A}|O_{k}^{C_{1}},\cdots,O_{k}^{C_{N}}\right)\right]$ is the conditional variance of $Q_{k}^{A}$ on the condition that we have performed the measurements $O_{k}^{C_{1}},O_{k}^{C_{2}},\cdots,O_{k}^{C_{N}}$. $\mathrm{V}\left[\mathrm{E}\left(Q_{k}^{A}|O_{k}^{C_{1}}\right)\right]$ represents the variance of $\mathrm{E}\left(Q_{k}^{A}|O_{k}^{C_{1}}:=\lambda^{(k)}_{j}\right)$ in terms of the possibility $\mathrm{P}\left(O_{k}^{C_{1}}:=\lambda^{(k)}_{j}\right)$ with $\lambda^{(k)}_{j}$ being an eigenvalue of $O_{k}$, and $\mathrm{E}[\mathrm{V}(\mathrm{E}[Q_{k}^{A}|O_{k}^{C_{n}}]|O_{k}^{C_{1}},\cdots,O_{k}^{C_{n-1}})]$ stands for the conditional variance of $\mathrm{E}\left(Q_{k}^{A}|O_{k}^{C_{n}}:=\lambda^{(k)}_{j}\right)$ on the condition that the measurements $O_{k}^{C_{1}},O_{k}^{C_{2}},\cdots,O_{k}^{C_{n-1}}$ have been performed 54 . $L_{tra}$ represents the lower bound of the traditional sum form variance-based uncertainty relation, namely $\sum_{k=1}^{K}\mathrm{V}(Q_{k}^{A})\geq L_{tra}$. Remarkably, based on Ref.12 , the lower bound $L_{tra}$ can be exactly equal to $\sum_{k=1}^{K}\mathrm{V}(Q_{k}^{A})$ by introducing auxiliary operators, and the uncertainty relation (Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation) will become an equality when $\sum_{k=1}^{K}\mathrm{V}(Q_{k}^{A})=L_{tra}$.

We then show that, using quantum entanglement resources, the lower bound of the traditional uncertainty relation is broken by the means of quantum control. Assuming that one of the control systems, says $C_{1}$, and measured system are in the maximum entangled state, and the measurements $Q_{k}$ and $O_{k}$ are the same, we deduce that the result of measurement $Q_{k}^{A}$ is predicted precisely with the result of the measurement $O_{k}^{C_{1}}$, and thus the lower bound (Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation) turns into $L_{tra}-\sum_{k=1}^{K}\mathrm{V}(O_{k}^{A})\leq 0$. That is to say, the measurement results of the incompatible observables $Q_{1}^{A},Q_{2}^{A},\cdots,Q_{K}^{A}$ are predicted precisely by Bob with the help of the control system. Alternatively, the lower bound (Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation) reduces to $L_{tra}$ when the entanglement between the control systems and measured system is equal to zero. In such case, the measurements on the control systems, namely the quantum control, have no effect on the measured system, and the uncertainty relation (Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation) turns into the traditional sum form one.

*Multiparticle Entanglement Resolution Analyzer.— *Entanglement for two particles can be easily characterized and identified, but the same question for $M$ particles is difficult with $M>2$. Unlike two-particle entanglement, entanglement for $M$ particles is divided into several classes 242 ; 243 ; 244 . Consider a fixed number $L$ with $M\geq L\geq 2$, a pure state $|\Phi\rangle$ of the whole system is $L$-separable when $|\Phi\rangle$ can be written as a tensor product of $L$ local states 24 :

[TABLE]

where the local state $|\phi_{l}\rangle$ is the state on the subsets of the $M$ parties 24 . There exists entanglement when the state is not an $L$-separable state, and the state is a genuinely multiparticle entangled state when it is not 2-separable. In general, it is difficult to delineate the multiparticle entanglement classes when we do not possess a full knowledge of the state.

We now show that the new conditional uncertainty relation can be used to identify the different classes of the multiparticle entangled pure state without a complete knowledge of the state. As shown in Fig.1, considering $N+1$ particles $A,C_{1},\cdots,C_{N}$, and using $\mathcal{L}_{m}$ to represent the lower bound conditioned on $m$ control systems with $0\leq m\leq N$, then we have:

[TABLE]

The reason why the uncertainty relation can be used to detect entanglement is that the local uncertainty relation is violated whenever entanglement exists. Thus, a violation of the local uncertainty relation is considered as a sign of entanglement. Following this scheme, we deduce that the control system $C_{1}$ is entangled with the measured system when the traditional lower bound is broken with the introduction of $C_{1}$, namely $\mathcal{L}_{0}>\mathcal{L}_{1}$. Similarly, we can deduce that the state of $N+1$ particles is a genuinely multiparticle entangled state if $\mathcal{L}_{0}>\mathcal{L}_{1}>\dots>\mathcal{L}_{N}$.

Here, the lower bounds $\left\{\mathcal{L}_{0},\mathcal{L}_{1},\dots,\mathcal{L}_{N}\right\}$ are named as “multiparticle entanglement resolution lines” (MERLs), and in general the MERLs are coincide with each other, because we have $\mathcal{L}_{0}\geq\mathcal{L}_{1}\geq\dots\geq\mathcal{L}_{N}$ and at least one equal sign is established for non-genuinely multiparticle entangled state. That is to say, the MERLs completely split from each other only for a genuinely multiparticle entangled state. Thus, the split MERLs is used to identify genuinely multiparticle entangled pure state; i.e., a pure state is genuinely multiparticle entangled when the number of the split MERLs is exactly equal to the number of the subsystems, as shown in Fig.2. Moreover, it can be seen from Fig.2 that the split MERLs are also used to identify the $L$-separable state. We deduce that a state is $(N+2-m)$-separable, $(N+1-m)$-separable, $\cdots$ , $2$-separable or genuinely multiparticle entangled when the $m$ split MERLs have been detected for the system which consists of $N+1$ subsystems. Here, we should mention that the conclusions obtained above only apply for the case of pure states, and the related proofs are presented in the Section III of the Supplemental Material 52 ).

In general, the traditional lower bound $L_{tra}$ is taken as $\sum_{k=1}^{K}\mathrm{V}(Q_{k}^{A})$ , and therefore the uncertainty relation (Multiparticle Entanglement Resolution Analyzer based on Quantum-Control-Assisted Uncertainty Relation) becomes an equality. Then, the MERLs are rewritten as $\mathcal{L}_{m}=\sum_{k=1}^{K}\mathrm{E}[\mathrm{V}(Q_{k}^{A}|O_{k}^{C_{1}},\cdots,O_{k}^{C_{m}})]$ for $m\geq 1$ and $\sum_{k=1}^{K}\mathrm{V}(Q_{k}^{A})$ for $m=0$. Based on the new definition of the conditional variance, we can see that the MERLs are obtained only through some suitable incompatible measurements, even without a complete knowledge of the quantum state.

*GHZ State Witness.— *Here, an example will be presented to demonstrate the superiority of the MERLs. In many experiments, due to technical limitation, the research on multiparticle entanglement has been confined mainly on qubits. However, entanglement beyond qubit, such as multiparticle entanglement based on the qutrit system, provides more information than the entangled qubit state, and thus a lot of work has been done to prepare the genuinely multiparticle entangled state beyond qubit system in experiment. In 2016, using the orbital angular momentum of the photon, Malik et al. prepared a high dimension multiparticle entangled state 57 , where two photons reside in a three-dimension space and the third one lives in a two-dimension space (denoted by (3,3,2)-type multiparticle entanglement). In 2018, they create a real three-particle GHZ state entangled in (3,3,3) way 58 , which reads:

[TABLE]

where $|\cdots\rangle_{h}$ stands the orbital angular momentum quanta of the $h-th$ photon, $\mu$ changes from 0 to $1/\sqrt{2}$ by adjusting the relevant experimental parameters, and the state becomes a real GHZ state when $\mu=1/\sqrt{3}$.

With the success in the preparation of high dimension multiparticle entangled state in experiment, another problem arises. That is “how does one verify these multiparticle entangled states experimentally”. Malik et al. certify the prepared multiparticle entangled state by measuring the fidelity of the prepared state to the ideal GHZ state, and they show that the prepared state possesses genuine multiparticle entanglement when the fidelity is less than a certain threshold value 58 . Indeed, this method still depends on the tomographic reconstruction of the quantum state, and thus the number of measurements increases dramatically with the increase in the dimension of the subsystem. For instance, as mentioned in Refs.57 and 58 , the verification of the (3,3,2)-type and (3,3,3)-type genuine entanglement needs 162 and 219 total measurements, respectively. We note that, based on our analyzer, the prepared GHZ state is verified with only polynomial number of incompatible measurements, which does not change with the increase of the dimension of the subsystem , as shown in Fig.3.

*Conclusion.— *In this Letter, we construct a multi-observable variance-based conditional uncertainty relation by introducing multiparticle entanglement and several quantum control systems. The uncertainty relation obtained indicates that we can prepare a quantum state, in which the measurement results of any observables are predicted precisely at the same time, when the prepared system and the control systems are in maximum entangled state. Multiparticle entanglement is divided into several classes, and, in general, it is difficult to characterize them when we cannot obtain a full knowledge of the quantum state. Here, using the lower bounds of the new conditional uncertainty relation, we introduce the concept of “multiparticle entanglement resolution lines” for multiparticle entangled pure states. The “multiparticle entanglement resolution lines” present different spectral for different multiparticle entanglement classes, and these lines are obtained only using some incompatible measurements. Finally, the “multiparticle entanglement resolution lines” can be used to identify multiparticle entangled pure state, even without a complete knowledge of the quantum state.

This work was supported by the NSFC under Grants No. 11574022, No. 61775242, No. 11434015, No. 61227902, No. 61835013; the National Key R$\&$D Program of China under Grants No. 2016YFA0301500; and the Strategic Priority Research Program of the Chinese Academy of Sciences under Grants No. XDB01020300 and No. XDB21030300, and L. C. Kwek was supported by the National Research Foundation Singapore and the Ministry of Education Singapore.

S. Q. M., X. Z. and G. F. Z. contributed equally to this work.