Minimum guesswork discrimination between quantum states

2015 ◽  
Vol 15 (9&10) ◽  
pp. 737-758
Author(s):  
Weien Chen ◽  
Yongzhi Cao ◽  
Hanpin Wang ◽  
Yuan Feng
Keyword(s):  
Error Probability ◽  
Sufficient Conditions ◽  
Optimization Criterion ◽  
Quantum States ◽  
Actual State ◽  
Information Theoretic ◽  
Minimum Error ◽  
State Discrimination ◽  
Semidefinite Programming Problem ◽  
Necessary And Sufficient

Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy.

scholarly journals Adaptive estimation and discrimination of Holevo-Werner channels

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas P. W. Cope ◽  
Stefano Pirandola
Keyword(s):  
Quantum Information ◽  
Error Probability ◽  
Adaptive Estimation ◽  
Analytical Formula ◽  
Quantum States ◽  
Minimum Error ◽  
Chernoff Bound ◽  
Werner States

AbstractThe class of quantum states known as Werner states have several interesting properties, which often serve to illuminate unusual properties of quantum information. Closely related to these states are the Holevo- Werner channels whose Choi matrices are Werner states. Exploiting the fact that these channels are teleportation covariant, and therefore simulable by teleportation, we compute the ultimate precision in the adaptive estimation of their channel-defining parameter. Similarly, we bound the minimum error probability affecting the adaptive discrimination of any two of these channels. In this case, we prove an analytical formula for the quantum Chernoff bound which also has a direct counterpart for the class of depolarizing channels. Our work exploits previous methods established in [Pirandola and Lupo, PRL

scholarly journals An information-theoretic treatment of quantum dichotomies

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 209 ◽  
Author(s):  
Francesco Buscemi ◽  
David Sutter ◽  
Marco Tomamichel
Keyword(s):  
Relative Entropy ◽  
Sufficient Conditions ◽  
The Other ◽  
State Transformation ◽  
Initial State ◽  
Information Theoretic ◽  
Target State ◽  
State Pair ◽  
Strong Converse ◽  
Necessary And Sufficient

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

scholarly journals Uniqueness of Minimax Strategy in View of Minimum Error Discrimination of Two Quantum States

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 671 ◽  
Author(s):  
Jihwan Kim ◽  
Donghoon Ha ◽  
Younghun Kwon
Keyword(s):  
Optimal Strategy ◽  
Quantum States ◽  
Minimax Strategy ◽  
Minimum Error ◽  
Optimal Measurement ◽  
Single Side ◽  
Quantum Ensemble ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Specific Quantum

This study considers the minimum error discrimination of two quantum states in terms of a two-party zero-sum game, whose optimal strategy is a minimax strategy. A minimax strategy is one in which a sender chooses a strategy for a receiver so that the receiver may obtain the minimum information about quantum states, but the receiver performs an optimal measurement to obtain guessing probability for the quantum ensemble prepared by the sender. Therefore, knowing whether the optimal strategy of the game is unique is essential. This is because there is no alternative if the optimal strategy is unique. This paper proposes the necessary and sufficient condition for an optimal strategy of the sender to be unique. Also, we investigate the quantum states that exhibit the minimum guessing probability when a sender’s minimax strategy is unique. Furthermore, we show that a sender’s minimax strategy and a receiver’s minimum error strategy cannot be unique if one can simultaneously diagonalize two quantum states, with the optimal measurement of the minimax strategy. This implies that a sender can confirm that the optimal strategy of only a single side (a sender or a receiver but not both of them) is unique by preparing specific quantum states.

scholarly journals On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-old Problem

2020 ◽  
Author(s):  
Lingyi Zhang ◽  
David Sidoti ◽  
Adam Bienkowski ◽  
Krishna Pattipati ◽  
Yaakov Bar-Shalom ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Optimization Criterion ◽  
Covariance Matrices ◽  
Gradient Algorithm ◽  
Measurement Noise Covariance ◽  
Noise Covariance ◽  
Noise Statistics ◽  
Necessary And Sufficient

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally <i>unknown</i>. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance <i>R</i> via <i>post-fit residuals</i>, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

scholarly journals On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-old Problem

2020 ◽  
Author(s):  
Lingyi Zhang ◽  
David Sidoti ◽  
Adam Bienkowski ◽  
Krishna Pattipati ◽  
Yaakov Bar-Shalom ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Optimization Criterion ◽  
Covariance Matrices ◽  
Gradient Algorithm ◽  
Measurement Noise Covariance ◽  
Noise Covariance ◽  
Noise Statistics ◽  
Necessary And Sufficient

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally <i>unknown</i>. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance <i>R </i>via <i>post-fit residuals</i>, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

scholarly journals Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

2019 ◽  
Vol 35 ◽  
pp. 156-180 ◽  
Author(s):  
Nathaniel Johnston ◽  
Olivia MacLean
Keyword(s):  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Quantum States ◽  
Completely Positive ◽  
Positive Partial Transpose ◽  
Completely Positive Matrices ◽  
Partial Transposition ◽  
Partial Transpose ◽  
Positive Matrices ◽  
Necessary And Sufficient

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.

scholarly journals On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-old Problem

2020 ◽  
Author(s):  
Lingyi Zhang ◽  
David Sidoti ◽  
Adam Bienkowski ◽  
Krishna Pattipati ◽  
Yaakov Bar-Shalom ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Optimization Criterion ◽  
Covariance Matrices ◽  
Gradient Algorithm ◽  
Measurement Noise Covariance ◽  
Noise Covariance ◽  
Noise Statistics ◽  
Necessary And Sufficient

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally <i>unknown</i>. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance <i>R </i>via <i>post-fit residuals</i>, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

scholarly journals On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-old Problem

2020 ◽  
Author(s):  
Lingyi Zhang ◽  
David Sidoti ◽  
Adam Bienkowski ◽  
Krishna Pattipati ◽  
Yaakov Bar-Shalom ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Sufficient Conditions ◽  
Positive Definiteness ◽  
Optimization Criterion ◽  
Covariance Matrices ◽  
Gradient Algorithm ◽  
Measurement Noise Covariance ◽  
Noise Covariance ◽  
Noise Statistics ◽  
Necessary And Sufficient

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally <i>unknown</i>. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance <i>R </i>via <i>post-fit residuals</i>, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

scholarly journals Locally Maximally Entangled States of Multipart Quantum Systems

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 115 ◽  
Author(s):  
Jim Bryan ◽  
Samuel Leutheusser ◽  
Zinovy Reichstein ◽  
Mark Van Raamsdonk
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Quantum States ◽  
Geometric Invariant ◽  
Henri Poincaré ◽  
System A ◽  
Unitary Transformations ◽  
Common Framework ◽  
Entangled Quantum States ◽  
Necessary And Sufficient

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions(d1,d2,…,dn), and computes the dimension of the spaceSLME/Kof LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions(2,A,B)and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points inSLME/Kcorrespond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroupS⊂SL⁡(d1,C)×⋯×SL⁡(dn,C)for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

scholarly journals Locally undetermined states, generalized Schmidt decomposition, and application in distributed computing

2009 ◽  
Vol 9 (11&12) ◽  
pp. 997-1012
Author(s):  
Y. Feng ◽  
R. Duan ◽  
M. Ying
Keyword(s):  
Distributed Computing ◽  
Quantum State ◽  
Fault Tolerant ◽  
A Priori ◽  
Sufficient Conditions ◽  
Quantum States ◽  
Consensus Problem ◽  
Schmidt Decomposition ◽  
Pure States ◽  
Necessary And Sufficient

Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computing. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state.

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