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arXiv:1802.00160 · Physical Review A

Bipartite discrimination of independently prepared quantum states as a counterexample to a parallel repetition conjecture

Seiseki Akibue, Go Kato

Published Apr 6, 2018• 4 citations
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Abstract

For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to 1 as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ if the entropy of the probability distribution associated with the ensemble is less than 1, even if the success probability is less than 1 for any finite $N$. In other words, the nonlocality of the $N$-fold ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the disappearance of the nonlocality can be regarded as a remarkable counterexample of a fundamental open question in theoretical computer science, called a parallel repetition conjecture of interactive games with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.

DOI
10.1103/physreva.97.042309

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For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of NNN subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to 1 as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ if the entropy of the probability distribution associated with the ensemble is less than 1, even if the success probability is less than 1 for any finite NNN. In other words, the nonlocality of the NNN-fold ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the disappearance of the nonlocality can be regarded as a remarkable counterexample of a fundamental open question in theoretical computer science, called a parallel repetition conjecture of interactive games with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of NNN subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to 111 as N→∞N\rightarrow\inftyN→∞ if the entropy of the probability distribution associated with the ensemble is less than 1/21/21/2. In other words, the nonlocality of the NNN-fold ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the discrimination task is a remarkable counterexample of a fundamental open question in theoretical computer science, called a parallel repetition conjecture of interactive games with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.
The uncertainty principle may be considered as giving rise to the notion of incompatibility of observables, a property that has been carefully analyzed in the literature for single systems. A pack of quantum measurements that cannot be measured simultaneously is said to form a set of incompatible measurements. Every set of incompatible measurements has an advantage over the compatible ones in a quantum state discrimination task where one prepares a state from an ensemble and sends it to another party, and the latter tries to detect the state using available measurements. Comparison between global and local quantum state discriminations is known to lead to a phenomenon of ``nonlocality.'' In this work, we seal a connection between the domains of local quantum state discrimination and incompatible quantum measurements. We consider the local quantum state discrimination task where a sender prepares a bipartite state and sends the subsystems to two receivers. The receivers try to detect the sent state using locally incompatible measurements. We analyze the ratio of the probability of successfully guessing the state using incompatible measurements and the maximum probability of successfully guessing the state using compatible measurements. We find that this ratio is upper bounded by a simple function of robustnesses of incompatibilities of the local measurements. Interestingly, corresponding to every pair of sets of incompatible measurements, there exists at least one local state discrimination task where this bound can be achieved. We argue that the optimal local quantum state discrimination task does not present any ``nonlocality,'' where the term is used in the sense of a difference between the ratios, of probabilities of successful detection via incompatible and compatible measurements, in global and local state discriminations. The results can be generalized to the regime of multipartite local quantum state distinguishing tasks.
Abstract In the case of a multi-party system, through local operations and classical communication (LOCC), each party may not perform perfect discrimination of quantum states that are separable and orthogonal. This property of quantum ensemble is called “nonlocality without entanglement” since each local party has a limit to full information of given quantum states. When this property is extended to the case of minimum-error discrimination, one can see that it is revealed when a nonlocal measurement provides more information about the unentangled states than LOCC does. One may infer the fact that the property depends on quantum states composing the quantum ensemble. However, an essential but unsettled question about the property is whether an explicit dependence on prior probabilities in terms of minimum-error discrimination could be shown in nonlocality without entanglement. In a simple term, one can ask whether different quantum ensembles made of the same separable quantum states could exhibit explicitly different behavior of the nonlocality. We answer this question in the positive, and we furthermore provide the explicit functional dependence of guessing probability on prior probabilities for the mirror-symmetric ensemble.
We show that a pair of two-qubit product measurements can be perfectly distinguished when accessed jointly, yet cannot be perfectly distinguished when their local components are distributed between distant parties restricted to local quantum operations and classical communication (LOCC). This establishes a measurement analogue of nonlocality without entanglement that, unlike its state-based counterpart, emerges in a remarkably simple setting. We further uncover a hierarchy of LOCC discrimination strategies: adaptive protocols, in which the probe used at one site may depend on classical information obtained from another, can strictly outperform restricted non-adaptive protocols, as demonstrated by a set of six bipartite qutrit product measurements. We also show that LOCC distinguishability of bipartite measurements can be intrinsically directional, depending on which party initiates the protocol. More generally, we construct pairs of nnn-qubit product measurements that are globally perfectly distinguishable but cannot be perfectly distinguished across any bipartition, even when all parties on either side of the bipartition are allowed arbitrary joint quantum operations. Finally, we show how quantum data hiding based on quantum-process nonlocality without entanglement can be realized using elementary optical components.
Nonlocality exhibited by ensembles of composite quantum states, wherein local operations and classical communication (LOCC) yield suboptimal discrimination probabilities compared to global strategies, is one of the striking nonclassical features of quantum theory. A variant of this phenomenon arises in conclusive local state discrimination, where the correct state must be identified with zero error, albeit allowing for inconclusive outcomes. More recently, the notion of local state marking has been introduced, with the focus shifted to correctly identifying the permutation of a subset of states randomly chosen from a given set of multipartite states under LOCC. In this work, we unify these two approaches by introducing the task of conclusive local state marking, which reveals a finer hierarchy of nonlocality in multipartite quantum state ensembles. Notably, we demonstrate that certain ensembles of product states can exhibit a stronger form of nonlocality than entangled ensembles traditionally considered highly nonlocal.
Quantum state discrimination is a fundamental problem in quantum information science. We investigate the optimal distinguishability of orthogonal two-qubit (bipartite) quantum states. The scenario consists of three parties: Alice, Bob, and Charlie. Charlie prepares one of two orthogonal states and sends one qubit to Alice and the other to Bob. Their goal is to correctly identify which state Charlie sent. In most state discrimination scenarios, it is assumed that Alice and Bob can freely communicate with one another so as to collectively agree on the best guess. In this research, we consider a more restricted setting where only one-way classical communication is possible from Alice to Bob. Under this setting, we study two figures of merit (i) Alice's optimal probability, PPP, of identifying the state , and (ii) Alice's optimal probability, P⊥P^\perpP⊥, of identifying the state along with helping Bob identify the state perfectly. We show that in general P≠P⊥P\neq P^\perpP=P⊥ and we prove a theorem for when P=P⊥P=P^\perpP=P⊥. We also found that the maximum of P−P⊥P-P^\perpP−P⊥ can arbitrarily approach 1/21/21/2.