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arXiv:1503.07103 · Quantum Information and Computation

Maximally coherent states

Zhaofang Bai, Shuanping Du

Published Nov 1, 2015• 19 citations
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Abstract

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.

DOI
10.26421/qic15.15-16-6

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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.
Abstract Quantum coherence and quantum correlation are two fundamental resources in quantum information and computation. Quantum correlated states in a bipartite system are the same as the states that are coherent with respect to any product basis, while a coherent state with respect to some given product basis may not be quantum correlated. In this work, we focus on the question of creating quantum correlation from a coherent state via incoherent quantum operations. We compare characterizations of incoherent operations (IOs), maximally incoherent operations, genuinely incoherent operations and coherence-breaking operations and then explicit interrelations among these operations. Moreover, we give the specific forms of states in two-qubit systems, called super-coherent ones, that can be mapped to quantum correlated ones via some genuinely incoherent operation. We find that there exist some coherent states that are not super-coherent. Furthermore, we prove that a state can be mapped to a quantum correlated state via an incoherent operation if and only if it is coherent. Lastly, we prove that any maximally entangled state can be mapped to an incoherent one via some one-side coherence-breaking operation.
In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.
We study partial coherence and its connection with entanglement. First, we provide a sufficient and necessary condition for bipartite pure-state transformation under partially incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by use of symmetric concave functions. Third, we establish the relationship between partial coherence and entanglement. We show that the minimal partial coherence under local unitary transformations is also a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof extension. Conversely, every entanglement measure can be constructed as a partial coherence measure, and we further show that partial coherence measures are induced through maximal entanglement under partially incoherent operations for bipartite pure states.
As an analogy of best separable approximation (BSA) in the framework of entanglement theory, here we concentrate on the notion of best incoherent approximation, with application to characterizing and quantifying quantum coherence. From both analytical and numerical perspectives, we have demonstrated that the weight-based coherence measure displays some unusual properties, in sharp contrast to other popular coherence quantifiers. First, by deriving a closed formula for single-qubit states, we have shown the exact value of weight-based coherence measure does not depend solely on the off-diagonal element but relies on the geometry of a given qubit state, thus exhibiting a rich structure even in this simplest case. Second, albeit the fact that almost all popular coherence measures only assign a maximal value to pure maximally coherent states, we have demonstrated the existence of mixed maximally coherent states (MMCS) with respect to this coherence measure and discussed the characteristic feature of MMCS in high-dimensional Hilbert spaces. Especially, we present several important families of MMCS by gaining insights from the numerical simulations. Moreover, it is pointed out that some considerations in this work can be generalized to general convex resource theories and a numerical method of improving the computational efficiency for finding the BSA is also discussed.
As an important quantum resource, quantum coherence play key role in quantum information processing. It is often concerned with manipulation of families of quantum states rather than individual states in isolation. Given two pairs of coherent states (ρ1,ρ2)(\rho_1,\rho_2)(ρ1​,ρ2​) and (σ1,σ2)(\sigma_1,\sigma_2)(σ1​,σ2​), we are aimed to study how can we determine if there exists a strictly incoherent operation Φ\PhiΦ such that Φ(ρi)=σi,i=1,2\Phi(\rho_i) =\sigma_i,i = 1,2Φ(ρi​)=σi​,i=1,2. This is also a classic question in quantum hypothesis testing. In this note, structural characterization of coherent preorder under strongly incoherent operations is provided. Basing on the characterization, we propose an approach to realize coherence distillation from rank-two mixed coherent states to qqq-level maximally coherent states. In addition, one scheme of coherence manipulation between rank-two mixed states is also presented.