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Physical Review A

Quantum coherence measures based on Fisher information with applications

Lei Li, Qing‐Wen Wang, Shu‐Qian Shen, Ming Li

Published Jan 5, 2021• 38 citations
Publisher

Abstract

Recently various ways to quantify coherence of a quantum state have been proposed. In this paper, several reliable quantum coherence measures based on quantum Fisher information (QFI) are presented. For pure states, this coherence measure is equivalent to that based on skew information, while for mixed states, we first define two natural coherence measures via QFI and find that the two definitions are exactly identical. By virtual of the special structure of qubit, we show that the mutual unbiased bases' (MUBs) qubit coherence is constrained only by the length of the Bloch vector. Then we provide upper bounds of the coherence measure with respect to any given basis and lower bounds in the case of a set of MUBs for both pure states and mixed states of arbitrary dimension. Finally, we investigate the connection between our coherence measures and quantum metrology, thereby providing an operational meaning for these coherence measures.

DOI
10.1103/physreva.103.012401

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Recently various ways to quantify coherence of a quantum state have been proposed. In this paper, several reliable quantum coherence measures based on quantum Fisher information (QFI) are presented. For pure states, this coherence measure is equivalent to that based on skew information, while for mixed states, we first define two natural coherence measures via QFI and find that the two definitions are exactly identical. By virtual of the special structure of qubit, we show that the mutual unbiased bases' (MUBs) qubit coherence is constrained only by the length of the Bloch vector. Then we provide upper bounds of the coherence measure with respect to any given basis and lower bounds in the case of a set of MUBs for both pure states and mixed states of arbitrary dimension. Finally, we investigate the connection between our coherence measures and quantum metrology, thereby providing an operational meaning for these coherence measures.
Quantum coherence is one of the old but always important concepts in quantum mechanics, and now it has been regarded as a necessary resource for quantum information processing and quantum metrology. However, the question of how to quantify the quantum coherence has just been paid the attention recently (see, e.g., Baumgratz et al. PRL, 113. 140401 (2014)). In this paper we verify that the well-known quantum Fisher information (QFI) can be utilized to quantify the quantum coherence, as it satisfies the monotonicity under the typical incoherent operations and the convexity under the mixing of the quantum states. Differing from most of the pure axiomatic methods, quantifying quantum coherence by QFI could be experimentally testable, as the bound of the QFI is practically measurable. The validity of our proposal is specifically demonstrated with the typical phase-damping and depolarizing evolution processes of a generic single-qubit state, and also by comparing it with the other quantifying methods proposed previously.
Quantum Fisher information (QFI) quantifies the sensitivity of a quantum state to a parameter change and plays a key role in quantum metrology. Meanwhile, quantum coherence is a crucial resource for quantum information processing. However, despite extensive studies on various aspects of quantum metrology, the interplay between the dynamics of the QFI and quantum coherence remains unexplored. Here, we explore the factorization relationship between the QFI and quantum coherence, both theoretically and experimentally. We prepare pure states in qubit and qutrit systems and apply unitary evolution to investigate this relationship. Our results confirm the factorization law, offering strong evidence for this relationship. Furthermore, we establish a lower bound on the variance of the estimated parameter that does not require the final state information. This approach reveals a deep connection between the QFI and quantum coherence, providing insights into quantum parameter estimation. These findings have potential applications in quantum sensing and quantum metrology.
Abstract Quantum coherence stands as a core property of quantum mechanics, its origin rooted in the superposition principle that governs quantum states. Leveraging mutually unbiased bases (MUBs), this paper delves into an in-depth exploration of quantum coherence characteristics through the lens of the l p -norm and s k e w Information. To begin with, we establish rigorous equality relations for the l p -norm coherence of qubit states under Pauli MUBs. Following this, we derive the complementary relations (upper bounds) and uncertainty relations (lower bounds) associated with l p -norm coherence under MUBs within arbitrary d-dimensional systems. Then, by virtue of deducing the connection between the classical purity and quantum purity of the square root of the quantum state, i.e. ρ , we further elaborate on the strict equality relations of skew-information coherence as well as several corresponding entropy-based coherences under MUBs in d-dimensional systems. Additionally, we provide illustrative examples to instantiate each of these relations, enhancing their clarity and practical relevance.
The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.
In quantum metrology, the parameter estimation accuracy is bounded by quantum Fisher information. In this paper we present coherence measures in terms of (quantum) Fisher information by directly considering the postselective nonunitary parametrization process. This coherence measure demonstrates the apparent operational meaning by the exact connection between coherence and parameter estimation accuracy. We also discuss the distinction between our coherence measure and the quantum Fisher information subject to unitary parametrization. The analytic coherence measure is given for qubit states.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.