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Scientific Reports

Coherence resource power of isocoherent states

Marcelo Losada, G. M. Bosyk, Héctor Freytes, Giuseppe Sergioli

Published May 5, 2022• 0 citations
Publisher

Abstract

We address the problem of comparing quantum states with the same amount of coherence in terms of their coherence resource power given by the preorder of incoherent operations. For any coherence measure, two states with null or maximum value of coherence are equivalent with respect to that preorder. This is no longer true for intermediate values of coherence when pure states of quantum systems with dimension greater than two are considered. In particular, we show that, for any value of coherence (except the extreme values, zero and the maximum), there are infinite incomparable pure states with that value of coherence. These results are not peculiarities of a given coherence measure, but common properties of every well-behaved coherence measure. Furthermore, we show that for qubit mixed states there exist coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states.

DOI
10.1038/s41598-022-11300-x

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We address the problem of comparing quantum states with the same amount of coherence in terms of their coherence resource power given by the preorder of incoherent operations. For any coherence measure, two states with null or maximum value of coherence are equivalent with respect to that preorder. This is no longer true for intermediate values of coherence when pure states of quantum systems with dimension greater than two are considered. In particular, we show that, for any value of coherence (except the extreme values, zero and the maximum), there are infinite incomparable pure states with that value of coherence. These results are not peculiarities of a given coherence measure, but common properties of every well-behaved coherence measure. Furthermore, we show that for qubit mixed states there exist coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states.
The coherence power of a quantum channel, that is, its maximum ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power and coherence rate and their basic properties. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails in the general case, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence in the case of the l1-coherence and dimension N > 2, the relative entropy of coherence and the geometric measure of coherence.
The relative entropy measure quantifying coherence, a key property of quantum system, is proposed recently. In this note, we firstly investigate structural characterization of maximally coherent states with respect to the relative entropy measure. It is shown that mixed maximally coherent states do not exist and every pure maximally coherent state has the form U|ψihψ|U† , |ψi = √1 d Pd k=1 |ki, U is diagonal unitary. Based on the characterization of pure maximally coherent states, for a bipartite maximally coherent state with dA = dB, we obtain that the super-additivity equality of relative entropy measure holds if and only if the state is a product state of its reduced states. From the viewpoint of resource in quantum information, we find there exists a maximally coherent state with maximal entanglement. Originated from the behaviour of quantum correlation under the influence of quantum operations, we further classify the incoherent operations which send maximally coherent states to themselves.
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.
The class of incoherent operations induces a preorder on the set of quantum pure states, defined by the possibility of converting one state into the other by transformations within the class. We prove that if two nnn-dimensional pure states are chosen independently according to the natural uniform distribution, then the probability that they are comparable vanishes as $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also study the maximal success probability of incoherent conversions and find an explicit formula for its large-nnn asymptotic distribution. Our analysis is based on the observation that the extreme values (largest and smallest components) of a random point uniformly sampled from the unit simplex are distributed asymptotically as certain explicit homogeneous Markov chains.
Abstract Quantum coherence is a fundamental property of quantum systems that can be studied within the framework of a quantum resource theory. From this resource-theoretic approach, many similarities emerge between coherence and quantum entanglement, the latter being another prominent quantum resource. In this paper, we study the relationship between coherence and entanglement from the perspective of resource loss/gain by unitary dynamics. Specifically we consider how much coherence can be either generated or destroyed on a bipartite system when the action is restricted to local unitary (LU) operations. For pure states, we find that the relative entropy of entanglement and the robustness of entanglement provide tight lower bounds on the amount of coherence that can be destroyed by LUs, as measured by the relative entropy and measures of coherence, respectively. This provides new operational interpretations of the entanglement entropy and robustness of entanglement in pure states as the minimal amount of coherence that persists when the system is subjected to LU transformations. We then study the amount of bipartite pure entanglement that can be either maximized or minimized when the action is restricted to a global incoherent operation. For two-qubit pure states, maximum and minimum are shown in terms of coefficients of given state with incoherent basis. Finally we consider a generalized version of the recently introduced CCP of a quantum channel.
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.