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Bipartite discrimination of independently prepared quantum states: revival of distinguishability with entanglement

Seiseki Akibue, Go Kato

Published Feb 1, 2018• 0 citations
OpenAlex

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 $N\rightarrow\infty$ if the entropy of the probability distribution associated with the ensemble is less than $1/2$. 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 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.

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

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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 $N$ 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.

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Donghoon Ha, Younghun Kwon

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We relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".

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TRADE-OFFS IN DISTINGUISHING TWO-QUBIT STATE PREPARATIONS USING ONE-WAY LOCC

Alvin Gonzales

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, $P$, of identifying the state , and (ii) Alice's optimal probability, $P^\perp$, of identifying the state along with helping Bob identify the state perfectly. We show that in general $P\neq P^\perp$ and we prove a theorem for when $P=P^\perp$. We also found that the maximum of $P-P^\perp$ can arbitrarily approach $1/2$.

•Jan 1, 2017•0 citations
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.
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 $N\ensuremath{\rightarrow}\ensuremath{\infty}$ 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 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.
Two pure orthogonal quantum states can be perfectly distinguished by sequential local action of multiple pairs of parties. However, this process typically leads to the complete dissolution of entanglement in the states being discriminated. We propose a protocol that allows an arbitrary number of pairs of parties to distinguish between any two orthogonal, entangled, two-qubit pure states using local quantum operations and classical communication, with a success probability greater than that of random guessing, while ensuring that at each step, the individual ensemble states retain a finite amount of entanglement. Our protocol employs the minimum-error state discrimination approach. For demonstrating the retention of entanglement in the ensemble states at each step, we use logarithmic negativity as well as the concept of entanglement witnessing. For a large family of sets of the two states, the success probability of discrimination can be as close as required to unity, while sustaining a finite amount of entanglement in each step.
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.
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 relate the the distinguishability of quantum states with their robustness of the entanglement, where the robustness of any resource quantifies how tolerant it is to noise. In particular, we identify upper and lower bounds on the probability of discriminating the states, appearing in an arbitrary multiparty ensemble, in terms of their robustness of entanglement and the probability of discriminating states of the closest separable ensemble. These bounds hold true, irrespective of the dimension of the constituent systems the number of parties involved, the size of the ensemble, and whether the measurement strategies are local or global. Additional lower bounds on the same quantity is determined by considering two special cases of two-state multiparty ensembles, either having equal entanglement or at least one of them being separable. The case of equal entanglement reveals that it is always easier to discriminate the entangled states than the ones in the corresponding closest separable ensemble, a phenomenon which we refer as "More indistinguishability with less entanglement". Furthermore, we numerically explore how tight the bounds are by examining the global discrimination probability of states selected from Haar-uniformly generated ensembles of two two-qubit states. We find that for two-element ensembles of unequal entanglements, the minimum of the two entanglements must possess a threshold value for the ensemble to exhibit "More indistinguishability with less entanglement".
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.