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arXiv:1507.02358 · Scientific Reports

Quantum coherence of steered states

Xueyuan Hu, Antony Milne, Boyang Zhang, Heng Fan

Published Jan 19, 2016• 72 citations
arXivHTMLPDF

Abstract

Lying at the heart of quantum mechanics, coherence has recently been studied as a key resource in quantum information theory. Quantum steering, a fundamental notion originally considered by Schödinger, has also recently received much attention. When Alice and Bob share a correlated quantum system, Alice can perform a local measurement to 'steer' Bob's reduced state. We introduce the maximal steered coherence as a measure describing the extent to which steering can remotely create coherence; more precisely, we find the maximal coherence of Bob's steered state in the eigenbasis of his original reduced state, where maximization is performed over all positive-operator valued measurements for Alice. We prove that maximal steered coherence vanishes for quantum-classical states whilst reaching a maximum for pure entangled states with full Schmidt rank. Although invariant under local unitary operations, maximal steered coherence may be increased when Bob performs a channel. For a two-qubit state we find that Bob's channel can increase maximal steered coherence if and only if it is neither unital nor semi-classical, which coincides with the condition for increasing discord. Our results show that the power of steering for coherence generation, though related to discord, is distinct from existing measures of quantum correlation.

DOI
10.1038/srep19365

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Lying at the heart of quantum mechanics, coherence has recently been studied as a key resource in quantum information theory. Quantum steering, a fundamental notion originally considered by Schödinger, has also recently received much attention. When Alice and Bob share a correlated quantum system, Alice can perform a local measurement to 'steer' Bob's reduced state. We introduce the maximal steered coherence as a measure describing the extent to which steering can remotely create coherence; more precisely, we find the maximal coherence of Bob's steered state in the eigenbasis of his original reduced state, where maximization is performed over all positive-operator valued measurements for Alice. We prove that maximal steered coherence vanishes for quantum-classical states whilst reaching a maximum for pure entangled states with full Schmidt rank. Although invariant under local unitary operations, maximal steered coherence may be increased when Bob performs a channel. For a two-qubit state we find that Bob's channel can increase maximal steered coherence if and only if it is neither unital nor semi-classical, which coincides with the condition for increasing discord. Our results show that the power of steering for coherence generation, though related to discord, is distinct from existing measures of quantum correlation.
We establish an operational connection between coherence and quantum correlation by showing that, Alice can create quantum coherence on Bob's side using local selective measurement as long as the previously shared bipartite state has non-vanishing quantum correlation on BBB, and the maximum average coherence she can extract can not surpass the BBB-side quantum correlation in the initial bipartite state. Steering-induced coherence is introduced to characterize Alice's ability to extract quantum coherence on Bob's side. For pure states and the maximally correlated states, the steering-induced relative entropy of coherence is proved to reach the initially shared quantum correlation which is measured by the relative entropy of BBB-side measurement-induced disturbance. The condition to reach the upper bound of quantum correlation varies for different measures of coherence. While the steering-induced l1l_1l1​-norm of coherence is shown reach the initially shared quantum correlation (measured by trace-norm of BBB-side measurement-induced disturbance) for any two-qubit states, an example is found whose steering-induced relative entropy of coherence is strictly less than the relative entropy of BBB-side measurement-induced disturbance.
Quantum steering refers to the possibility for Alice to remotely steer Bob's state by performing local measurements on her half of a bipartite system. Two necessary ingredients for steering are entanglement and incompatibility of Alice's measurements. In particular, it is known that for the case of pure states of maximal Schmidt rank the problem of steerability for Bob's assemblage is equivalent to the problem of joint measurability for Alice's observables. We show that such an equivalence holds in general; namely, the steerability of any assemblage can always be formulated as a joint measurability problem, and vice versa. We use this connection to introduce steering inequalities from joint measurability criteria and develop quantifiers for the incompatibility of measurements.
Coherence and nonlocal correlations are two fundamental features of quantum mechanics and the relations between them have received much attention recently. While the relations of coherence to Bell nonlocality and to entanglement have been established, an interesting question left open is how to quantitatively characterize the relation of coherence to steering, which is one kind of nonlocal correlation lying between Bell nonlocality and entanglement. In this paper, we address this question by considering the first-order coherence and the three-setting linear steering inequality for a two-qubit system. Interestingly, we find that the first-order coherence for both subsystems has a complementary relation with the maximum violation of the three-setting linear steering inequality. A weighted sum of their squares keeps invariant under unitary operations, although each of them changes in general. This suggests that the maximum violation of the three-setting linear steering inequality can be taken as a measure of hidden coherence unveiled by unitary operations.
In a recent work [Phys. Rev. Lett. 98, 140402 (2007)] we defined ``steering,'' a type of quantum nonlocality that is logically distinct from both nonseparability and Bell nonlocality. In the bipartite setting, it hinges on the question of whether Alice can affect Bob's state at a distance through her choice of measurement. More precisely and operationally, it hinges on the question of whether Alice, with classical communication, can convince Bob that they share an entangled state under the circumstances that Bob trusts nothing that Alice says. We argue that if she can, then this demonstrates the nonlocal effect first identified in the famous Einstein-Podolsky-Rosen paper [Phys. Rev. 47, 777 (1935)] as a universal effect for pure entangled states. This ability of Alice to remotely prepare Bob's state was subsequently called steering by Schr\"odinger, whose terminology we adopt. The phenomenon of steering has been largely overlooked, and prior to our work had not even been given a rigorous definition that is applicable to mixed states as well as pure states. Armed with our rigorous definition, we proved that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. In this work we expand on these results and provide further examples of steerable states. We also elaborate on the connection with the original EPR paradox.
Nonclassical correlations are a key resource to explore foundational quantum information tasks and find applications in device-independent protocols. Quantum steering was formalized in 2007 [1] and is distinct from other nonclassical correlations such as Bell nonlocality [2] and entanglement. Steering describes the effect of a local measurement on one system and affecting the measurement results on the other system. This can be visualized for two-qubit states using quantum steering ellipsoids (QSE), which is the set of Bloch vectors that Alice can collapse Bob's state to. They are theoretically well studied [3-6] and have been recently experimentally demonstrated [7].
As the precious resource for quantum information processing, quantum coherence can be created remotely if the involved two sites are quantum correlated. It can be expected that the amount of coherence created should depend on the quantity of the shared quantum correlation, which is also a resource. Here, we establish an operational connection between coherence induced by steering and the quantum correlation. We find that the steering-induced coherence quantified by such as relative entropy of coherence and trace-norm of coherence is bounded from above by a known quantum correlation measure defined as the one-side measurement-induced disturbance. The condition that the upper bound saturated by the induced coherence varies for different measures of coherence. The tripartite scenario is also studied and similar conclusion can be obtained. Our results provide the operational connections between local and non-local resources in quantum information processing.