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arXiv:2010.15321 · Quantum Information and Computation

Coherent preorder of quantum states

Zhaofang Bai, Shuanping Du

Published Nov 1, 2020• 0 citations
arXivHTMLPDF

Abstract

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 $(\rho_1,\rho_2)$ and $(\sigma_1,\sigma_2)$, we are aimed to study how can we determine if there exists a strictly incoherent operation $\Phi$ such that $\Phi(\rho_i) =\sigma_i,i = 1,2$. 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 $q$-level maximally coherent states. In addition, one scheme of coherence manipulation between rank-two mixed states is also presented.

DOI
10.26421/qic20.13-14-3

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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.
We characterize the distillation of quantum coherence in the one-shot setting, that is, the conversion of general quantum states into maximally coherent states under different classes of quantum operations. We show that the maximally incoherent operations (MIO) and the dephasing-covariant incoherent operations (DIO) have the same power in the task of one-shot coherence distillation. We establish that the one-shot distillable coherence under MIO and DIO is efficiently computable with a semidefinite program, which we show to correspond to a quantum hypothesis testing problem. Further, we introduce a family of coherence monotones generalizing the robustness of coherence as well as the modified trace distance of coherence, and show that they admit an operational interpretation in characterizing the fidelity of distillation under different classes of operations. By providing an explicit formula for these quantities for pure states, we show that the one-shot distillable coherence under MIO, DIO, strictly incoherent operations, and incoherent operations is equal for all pure states.
Abstract State convertibility is fundamental in the study of resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, we give a complete characterization of state convertibility under genuinely incoherent operations. It is found that convexity of the robustness of coherence plays a central role. Based on this, the majorization condition of determining convertibility from pure states to mixed states under strictly incoherent operations is provided. Moreover, maximally coherent states in the set of all states with fixed diagonal elements are determined. It is somewhat surprising that convexity of the robustness of coherence can also decide conversion between off-diagonal parts of coherent states. This might be a big step to answer completely the question of state convertibility for mixed states under incoherent operations.
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 known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on k bases using Matrix Theory Method. First, by defining a correlation function m(e,f) of two orthonormal bases e and f, we investigate the relationships between sets I(e) and I(f) of incoherent states with respect to e and f. We prove that I(e)=I(f) if and only if the rank-one projective measurements generated by e and f are identical. We give a necessary and sufficient condition for the intersection I(e)⋂I(f) to include a state except the maximally mixed state. Especially, if two bases e and f are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set B of k bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to B when k=2 and it is not the case for k=3.
Coherence manipulation is fundamental in the study of the resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, it is shown that a finite number of measure conditions are not sufficient to characterize coherence manipulation on general mixed states. Further, a finite number of measure conditions to classify coherence manipulation on subspace-independent states are found. For the input pure state, we also furnish a structural characterization of coherence manipulation in terms of a finite number of measure conditions.
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.