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Coherence generating power of quantum unitary maps

Paolo Zanardi A, Georgios Styliaris, Lorenzo Campos Venuti

Published Oct 2, 2016• 0 citations
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Abstract

Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to $B$. 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.

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•Jan 1, 2020•0 citations
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 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 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.
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 coherence power of a quantum channel, that is, its 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. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1l_1l1​-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails for dimensions N>2, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence.
We establish a direct connection between the power of a unitary map in d-dimensions (d < ∞) to generate quantum coherence and the geometry of the set Md of maximally abelian subalgebras (of the quantum system full operator algebra). This set can be seen as a topologically non-trivial subset of the Grassmannian over linear operators. The natural distance over the Grassmannian induces a metric structure on Md, which quantifies the lack of commutativity between the pairs of subalgebras. Given a maximally abelian subalgebra, one can define, on physical grounds, an associated measure of quantum coherence. We show that the average quantum coherence generated by a unitary map acting on a uniform ensemble of quantum states in the algebra (the so-called coherence generating power of the map) is proportional to the distance between a pair of maximally abelian subalgebras in Md connected by the unitary transformation itself. By embedding the Grassmannian into a projective space, one can pull-back the standard Fubini-Study metric on Md and define in this way novel geometrical measures of quantum coherence generating power. We also briefly discuss the associated differential metric structures.
Recently, how to measure quantum coherence is a hot topic in the quantum resource theory. In 2014, Baumgratz et al. proposed a rigorous framework to quantify coherence. In this framework, a particular basis of the Hilbert space is required to define the coherence measure, and hence, the proposed coherence measure of the same quantum state does not remain the same under the basis transformations. In this paper, a measure of coherence based on the normalized modulus of the coherent vector is proposed in coherent vector representation, where the property of the coherent vector under basis transformation is considered. The measure of coherence proposed here is independent of the choice of the basis. The incoherent states and incoherent operators based on this measure can also be well defined as the classical maximal mixed states and the unitary operators respectively. And three important properties can be obtained, namely, non-negativity, convexity and invariance on the incoherent operators.