The optimal positive operator-valued measure for two nonorthogonal mixed states

Wei Li, Shengmei Zhao

Published Aug 24, 2021• 0 citations

Abstract

Evaluating the amount of information obtained from nonorthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides an upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for two nonorthogonal mixed states. We construct a generalized POVM measurement operation, and derive the optimal POVM measurement by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a lower bound for quantum state measurement, which is significantly lower than that predicted by Holevo bound. The derivation of optimal POVM measurement will play an important role in the security research of quantum key distribution.

Related papers

The optimal positive operator-valued measure for state discrimination

Wei Li, Shengmei Zhao

Evaluating the amount of information obtained from non-orthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides a loose upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for discriminating nonorthogonal states. We construct a generalized POVM measurement operation, and derive the optimal one for state discrimination by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a tight upper bound for state discrimination, which is significantly lower than that predicted by Holevo bound. The derivation of optimal POVM measurement will play an important role in the security research of quantum key distribution.

arXiv:2108.10474Aug 24, 20210 citations

Optimal individual and collective measurements for nonorthogonal quantum key distribution signals

Isabella Cerutti, Petra F. Scudo

We consider how the theory of optimal quantum measurements determines the maximum information available to the receiving party of a quantum key distribution (QKD) system employing linearly independent but nonorthogonal quantum states. Such a setting is characteristic of several practical QKD protocols. Due to nonorthogonality, the receiver is not able to discriminate unambiguously between the signals. To understand the fundamental limits that this imposes, the quantity of interest is the maximum mutual information between the transmitter (Alice) and the receiver, whether legitimate (Bob) or an eavesdropper (Eve). To find the optimal measurement—taken individually or collectively—we use a framework based on operator algebra and general results derived from singular-value decomposition, achieving optimal solutions for von Neumann measurements and positive operator-valued measures (POVMs). The formal proof and quantitative analysis elaborated for two signals allow us to conclude that optimal von Neumann measurements are uniquely defined and provide a higher information gain compared to POVMs. Interestingly, collective measurements not only do not provide additional information gain with respect to individual ones but also suffer from a gain reduction in the case of POVMs. Published by the American Physical Society 2024

Physical Review AMar 20, 20242 citations

Minimax Strategy of Optimal Unambiguous State Discrimination

Wen‐Hai Zhang, Long-Bao Yu, Zhuo‐Liang Cao +1 more

In this paper, we consider the minimax strategy to unambiguously discriminate two pure nonorthogonal quantum states without knowing a priori probability. By exploiting the positive-operator valued measure, we derive the upper bound of the minimax measurement of the optimal unambiguous state discrimination. Based on the linear optical devices, we propose an experimentally feasible scheme to implement a minimax measure of a general pair of two nonorthogonal quantum states.

Communications in Theoretical PhysicsAug 1, 20120 citations

Bounds on positive operator-valued measure based coherence of superposition

Meng‐Li Guo, Jin‐Min Liang, Bo Li +2 more

Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.

arXiv:2305.06705Chinese Physics BFeb 8, 20231 citation

Can optimal collective measurements outperform individual measurements for non-orthogonal QKD signals?

Isabella Cerutti, Petra F. Scudo

We consider how the theory of optimal quantum measurements determines the maximum information available to the receiving party of a quantum key distribution (QKD) system employing linearly independent but non-orthogonal quantum states. Such a setting is characteristic of several practical QKD protocols. Due to non-orthogonality, the receiver is not able to discriminate unambiguously between the signals. To understand the fundamental limits that this imposes, the quantity of interest is the maximum mutual information between the transmitter (Alice) and the receiver, whether legitimate (Bob) or an eavesdropper (Eve). To find the optimal measurement we use a framework based on operator algebra and general results derived from singular value decomposition, achieving optimal solutions for von Neumann measurements and positive operator-valued measures (POVMs). The formal proof and quantitative analysis elaborated for two signals allow one to conclude that optimal von Neumann measurements are uniquely defined and provide a higher information gain compared to POVMs. Interestingly, collective measurements not only do not provide additional information gain with respect to individual ones, but also suffer from a gain reduction in the case of POVMs.

arXiv:2401.01616Jan 3, 20240 citations

QestOptPOVM: An iterative algorithm to find optimal measurements for quantum parameter estimation

Jianchao Zhang, Suzuki, Jun

Quantum parameter estimation holds significant promise for achieving high precision through the utilization of the most informative measurements. While various lower bounds have been developed to assess the best accuracy for estimates, they are not tight, nor provide a construction of the optimal measurement in general. Thus, determining the explicit forms of optimal measurements has been challenging due to the non-trivial optimization. In this study, we introduce an algorithm, termed QestOptPOVM, designed to directly identify optimal positive operator-valued measure (POVM) using the steepest descent method. Through rigorous testing on several examples for multiple copies of qubit states (up to six copies), we demonstrate the efficiency and accuracy of our proposed algorithm. Moreover, a comparative analysis between numerical results and established lower bounds serves to validate the tightness of the Nagaoka-Hayashi bound in finite-sample quantum metrology for our examples. Concurrently, our algorithm functions as a tool for elucidating the explicit forms of optimal POVMs, thereby enhancing our understanding of quantum parameter estimation methodologies.

arXiv:2403.20131Mar 29, 20241 citation