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arXiv:1711.02291 · Physical Review A

Coherence-generating power of quantum dephasing processes

Georgios Styliaris, Lorenzo Campos Venuti, Paolo Zanardi A

Published Mar 7, 2018• 28 citations
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Abstract

We provide a quantification of the capability of various quantum dephasing processes to generate coherence out of incoherent states. The measures defined, admitting computable expressions for any finite Hilbert-space dimension, are based on probabilistic averages and arise naturally from the viewpoint of coherence as a resource. We investigate how the capability of a dephasing process (e.g., a nonselective orthogonal measurement) to generate coherence depends on the relevant bases of the Hilbert space over which coherence is quantified and the dephasing process occurs, respectively. We extend our analysis to include those Lindblad time evolutions which, in the infinite-time limit, dephase the system under consideration and calculate their coherence-generating power as a function of time. We further identify specific families of such time evolutions that, although dephasing, have optimal (over all quantum processes) coherence-generating power for some intermediate time. Finally, we investigate the coherence-generating capability of random dephasing channels.

DOI
10.1103/physreva.97.032304

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Given a preferred orthonormal basis BBB in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to BBB. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.
Given a preferred orthonormal basis BBB in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to BBB. We give its explicit analytical form and provide an operational protocol to directly detect it. This measure vanishes if and only if the unitary map transforms all the incoherent states into themselves and is non-increasing under pre- or post-processing by incoherent operations. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital completely positive maps and the relation to the concept of asymmetry are discussed.
Given an orthonormal basis BBB in a ddd-dimensional Hilbert space and a unital quantum operation E\mathcal{E}E acting on it, one can define a nonlinear mapping that associates with E\mathcal{E}E a $d\ifmmode\times\else\texttimes\fi{}d$ real-valued matrix that we call the coherence matrix of E\mathcal{E}E with respect to BBB. This is the Gram matrix of the coherent part of the basis projections of BBB under E\mathcal{E}E. We show that one can use this coherence matrix to define vast families of measures of the coherence-generating power (CGP) of the operation. These measures have a natural geometrical interpretation as separation of E\mathcal{E}E from the set of incoherent unital operations. The probabilistic approach to CGP discussed in P. Zanardi et al. [Phys. Rev. A 95, 052306 (2017).] can be reformulated and generalized, introducing, alongside the coherence matrix, another $d\ifmmode\times\else\texttimes\fi{}d$ real-valued matrix, the simplex correlation matrix. This matrix describes the relevant statistical correlations in the input ensemble of incoherent states. Contracting these two matrices, one finds CGP measures associated with the process of preparing the given incoherent ensemble and processing it with the chosen unital operation. Finally, in the unitary case, we discuss how these concepts can be made compatible with an underlying tensor product structure by defining families of CGP measures that are additive.
Non-classical dynamics in open quantum systems can be hidden in temporal correlations that are invisible to standard state- and channel-based tests. We define coherence for multi-time quantum processes and show that access to intermediate times can reveal coherence absent from all corresponding pairwise channels. Incoherent processes are fixed points of dephasing on all input and output spaces, and their coherence is quantified by relative entropy. As a proof of principle, we use an Ising model in which all three pairwise channels are incoherent while the full three-time process remains coherent. We observe this multi-time coherence on a superconducting quantum processor using a coherence witness and full process tomography.
We study the ability of a quantum channel to generate quantum coherence when it applies to incoherent states. Based on probabilistic averages, we define a measure of such coherence generating power (CGP) for a generic quantum channel, based on the average coherence generated by the quantum channel acting on a uniform ensemble of incoherent states. Explicit analytical formula of the CGP for any unitary channels are presented in terms of subentropy. An upper bound for CGP of unital quantum channels has been also derived. Detailed examples are investigated.
Recently, there has been a renewed interest in the quantification of coherence or other coherencelike concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be two sides of the same coin. Therefore, a natural question is raised: How can the conventional decoherence processes, such as the von Neumann--L\"uders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretical framework? Here we show that the state collapse rules of the von Neumann or L\"uders-type measurements, as special cases of genuinely incoherent operations (GIOs), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"uders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed-point theory for C∗{C}^{*}C∗ algebra, we prove that GIOs indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIOs in fact constitute the core of strictly incoherent operations and generally incoherent operations and the unspeakable notion of coherence induced by GIOs can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.