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arXiv:2201.00526 · Laser Physics Letters

On coherence of quantum operations by using Choi–Jamiołkowski isomorphism

Xiaorong Wang, Ting Gao, Fengli Yan

Published Feb 10, 2022• 3 citations
arXivHTMLPDF

Abstract

Abstract In quantum information, most information processing processes involve quantum channels. One manifestation of a quantum channel is quantum operation acting on quantum states. The coherence of quantum operations can be considered as a quantum resource, which can be exploited to perform certain quantum tasks. From the viewpoint of Choi–Jamiołkowski isomorphism, we study the coherence of quantum operations in the framework of resource theory. We study the phase-out superoperation and give the operation which transforms the Choi–Jamiołkowski state of a quantum operation to the Choi–Jamiołkowski state of the another quantum operation obtained by using the phase-out superoperation to act on the quantum operation. The set of maximally incoherent superoperations, the set of nonactivating coherent superoperations and the set of de-phase incoherent superoperations are investigated and we prove that these sets are closed to compound operation and convex combination of quantum superoperations. Further, we introduce the fidelity coherence measure of quantum operations and obtain the exact form of the fidelity coherence measure of the unitary operations on the single qubit.

DOI
10.1088/1612-202x/ac50ad

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Abstract In quantum information, most information processing processes involve quantum channels. One manifestation of a quantum channel is quantum operation acting on quantum states. The coherence of quantum operations can be considered as a quantum resource, which can be exploited to perform certain quantum tasks. From the viewpoint of Choi–Jamiołkowski isomorphism, we study the coherence of quantum operations in the framework of resource theory. We study the phase-out superoperation and give the operation which transforms the Choi–Jamiołkowski state of a quantum operation to the Choi–Jamiołkowski state of the another quantum operation obtained by using the phase-out superoperation to act on the quantum operation. The set of maximally incoherent superoperations, the set of nonactivating coherent superoperations and the set of de-phase incoherent superoperations are investigated and we prove that these sets are closed to compound operation and convex combination of quantum superoperations. Further, we introduce the fidelity coherence measure of quantum operations and obtain the exact form of the fidelity coherence measure of the unitary operations on the single qubit.
We generalize the recently proposed resource theory of coherence (or superposition) [T. Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014); A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] to the setting where not just the free (``incoherent'') resources, but also the manipulated objects, are quantum operations rather than states. In particular, we discuss an information theoretic notion of the coherence capacity of a quantum channel and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel and prove achievability results for unitaries and general channels acting on a ddd-dimensional system; we show that a maximally coherent state of rank ddd is always sufficient as a resource if incoherent operations are allowed, and one of rank d2{d}^{2}d2 for ``strictly incoherent'' operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e., maps with nonzero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.
In this article, I use an operational formulation of the Choi-Jamiołkowski isomorphism to explore an approach to quantum mechanics in which the state is not the fundamental object. I first situate this project in the context of generalized probabilistic theories and argue that this framework may be understood as a means of drawing conclusions about the intratheoretic causal structure of quantum mechanics which are independent of any specific ontological picture. I then give an operational formulation of the Choi-Jamiołkowski isomorphism and show that, in an operational theory which exhibits this isomorphism, several features of the theory which are usually regarded as properties of the quantum state can be derived from constraints on non-local correlations. This demonstrates that there is no need to postulate states to be the bearers of these properties, since they can be understood as consequences of a fundamental equivalence between multipartite and temporal correlations.
In contrast to unitary evolutions, which are reversible, generic quantum processes (operations and quantum channels) are often irreversible. However, the degree of irreversibility is different for different channels, and it is desirable to have a quantitative characterization of irreversibility. In this paper, by exploiting the channel–state duality implemented by the Jamiołkowski–Choi isomorphism, we quantify the irreversibility of channels via entropy of the Jamiołkowski–Choi states of the corresponding channels and compare it with the notions of entanglement fidelity and entropy exchange. General properties of a reasonable measure of irreversibility are discussed from an intuitive perspective, and entropic measures of irreversibility are introduced. Several relations between irreversibility, entanglement fidelity, the degree of nonunitality, and decorrelating power are established. Some measures of irreversibility for a variety of prototypical channels are evaluated explicitly, revealing some information-theoretic aspects of the structure of channels from the perspective of irreversibility.
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.
Quantum coherence has recently emerged as a key candidate for use as a resource in various quantum information processing tasks. One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work, we ask if coherence, already present in a reference system, can be broadcast to the auxiliary system via incoherent operations. We prove that it is impossible for a weaker form of broadcasting via incoherent operations. This shows that broadcasting of coherence is forbidden via incoherent operations in every finite-dimensional system. We also prove that a generalized form of broadcasting via incoherent quantum operations can occur if and only if the state of system is coherent. In this case, we are able to give a method for constructing an incoherent quantum operation to broadcast coherence.
The concept of block coherence encompasses the case where experimental capabilities are not so delicate to perform arbitrary refined measurements on individual atoms. We develop a framework which facilitates further investigation of this resource theory in several respects. Using this framework, we investigate the problem of state conversion by incoherent operations and show that a majorization condition is the necessary and sufficient condition for state transformation by block-incoherent operations. We also determine the form of the maximally coherent state from which all other states and all unitary gates can be constructed by incoherent operations. Thereafter, we define the concept of block-cohering and block-decohering powers of quantum channels and determine these powers for several types of channels. Finally, we explore the relation between block coherence and a previous extension of coherence, known as kkk coherence.