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arXiv:1711.10889 · Physical Review A

Optimization-free measures of quantum resources

Nikolaos K. Kollas

Published Jun 27, 2018• 6 citations
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Abstract

A closed expression is derived for the amount of resource present in a quantum state as quantified by a distance measure based on the Tsallis relative entropy introduced recently [H. Zhao and C.-s. Yu, Sci. Rep. 8, 299 (2018)], for any resource theory whose set of free states is described by the image of a unital resource-destroying map. By applying it to the resource theory of coherence it is demonstrated that the correct definition of a projective coarse-grained measurement needs to be modified in order for it to be correctly interpreted as a measurement which provides less information about a system's true state than the one obtained from a finer-grained measurement.

DOI
10.1103/physreva.97.062344

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A closed expression is derived for the amount of resource present in a quantum state as quantified by a distance measure based on the Tsallis relative entropy introduced recently [H. Zhao and C.-s. Yu, Sci. Rep. 8, 299 (2018)], for any resource theory whose set of free states is described by the image of a unital resource-destroying map. By applying it to the resource theory of coherence it is demonstrated that the correct definition of a projective coarse-grained measurement needs to be modified in order for it to be correctly interpreted as a measurement which provides less information about a system's true state than the one obtained from a finer-grained measurement.
The ability of quantum states to be in superposition is one of the key features that sets them apart from the classical world. This “coherence” is rigorously quantified by resource theories, which aim to understand how such properties may be exploited in quantum technologies. There has been much research on what the resource theory of coherence can reveal about quantum metrology, almost all of which has been from the viewpoint of Fisher information. We prove, however, that the relative entropy of coherence, and its recent generalization to positive operator-valued measures (POVMs), naturally quantify the performance of Bayesian metrology. In particular, we show how a coherence measure can be applied to an ensemble of states. We then prove that during parameter estimation, the ensemble relative entropy of coherence (C) is equal to the difference between the optimal Holevo information (X), and the mutual information attained by a measurement (I). We call this relation the CXI equality. The ensemble coherence lets us visualize how much information is locked away in superposition and hence is inaccessible with a given measurement scheme and quantifies the advantage that would be gained by using a joint measurement on multiple states. Our results hold regardless of how the parameter is encoded in the state, encompassing unitary, dissipative, and discrete settings. We consider both projective measurements and general POVMs. This work suggests new directions for research in coherence, provides a novel operation interpretation for the relative entropy of coherence and its POVM generalization, and introduces a new tool to study the role of quantum features in metrology. Published by the American Physical Society 2024
The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.
The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.
We propose a general method to operationally quantify the ``resourcefulness'' of quantum channels via channel discrimination, an important information processing task. A main result is that the maximum success probability of distinguishing a given channel from the set of free channels by free probe states is exactly characterized by the resource generating power, i.e., the maximum amount of resource produced by the action of the channel, given by the trace distance to the set of free states. We apply this framework to the resource theory of quantum coherence, as an informative example. The general results can also be easily applied to other resource theories such as entanglement, magic states, and asymmetry.
By studying generalizations of entanglement to quantum resource theories,we will provide a continuous family of optimization free measures to quantify the resource in a system, for any theory whose set of free states is given by the image of an idempotent and unital quantum operation. With the help of these measures we will argue for the correct definition of a coarse grained projective quantum measurement. The question of extraction of quantum coherence from one system to another in a faithful way is also studied. We construct two novel protocols, one able to extract only a small amount of coherence each time and the other a larger amount and apply these to a reservoir in a coherent and vacuum squeezed state of a harmonic oscillator. By studying extraction to qubits it is shown that for a single extraction from both reservoirs the stronger protocol outperforms a previous method known as catalytic coherence.
A question that is commonly asked in all areas of physics is how a certain property of a physical system can be used to achieve useful tasks and how to quantify the amount of such a property in a meaningful way. We answer this question by showing that, in a general resource-theoretic framework that allows the use of free states as catalysts, the amount of "resources" contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of a resource of that state. While we need to place a few assumptions on our resource-theoretical framework, it is still sufficiently general, and its special cases include quantum resource theories of entanglement, coherence, asymmetry, athermality, nonuniformity, and purity. As a by-product, our result also implies that the amount of noise one has to inject locally to erase all the entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement.