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arXiv:1703.01067 · Physical Review Letters

Quantifying the Coherence between Coherent States

Kok Chuan Tan, Tyler J. Volkoff, Hyukjoon Kwon, Hyunseok Jeong

Published Nov 10, 2017• 77 citations
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Abstract

In this Letter, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present in an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz, Cramer, and Plenio and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan P distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum light. We show that our construction belongs to a family of resource monotones within the framework of a resource theory of linear optics, thus establishing deeper connections between the class of incoherent operations in the finite dimensional regime and linear optical operations in the continuous variable regime.

DOI
10.1103/physrevlett.119.190405

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In this Letter, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present in an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz, Cramer, and Plenio and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan P distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum light. We show that our construction belongs to a family of resource monotones within the framework of a resource theory of linear optics, thus establishing deeper connections between the class of incoherent operations in the finite dimensional regime and linear optical operations in the continuous variable regime.
We discuss the relationship between quantum coherence of finite dimensional systems and nonclassical quantum light. We demonstrate that quantifying the quantum coherence between coherent states always leads to a quantifier of nonclassicality that monotonically decreases under linear optical operations. This allows us to introduce a resource theory of nonclassicality based on linear optics that closely parallels the resource theory of coherence. Finally, we discuss the metrological power of a quantum state, which quantifies how sensitive a quantum state is to displacement operations. It is then shown that the metrological power leads to a nonclassicality monotone.
The superposition principle lies at the heart of many nonclassical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialized to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.
The resource theory of quantum superposition is an extension of the quantum coherent theory, in which linear independence relaxes the requirement of orthogonality. It can be used to quantify the nonclassical in superposition of finite number of optical coherent states. Based on convex roof extended, state transformation and weight, we give three methods of constructing superposition measures of quantum states, respectively. We also generalize the superposition resource theory from two perspectives.
The diverse range of resources which underlie the utility of quantum states in practical tasks motivates the development of universally applicable methods to measure and compare resources of different types. However, many of such approaches were hitherto limited to the finite-dimensional setting or were not connected with operational tasks. We overcome this by introducing a general method of quantifying resources for continuous-variable quantum systems based on the robustness measure, applicable to a plethora of physically relevant resources such as optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. We demonstrate in particular that the measure has a direct operational interpretation as the advantage enabled by a given state in a class of channel discrimination tasks. We show that the robustness constitutes a well-behaved, bona fide resource quantifier in any convex resource theory, contrary to a related negativity-based measure known as the standard robustness. Furthermore, we show the robustness to be directly observable-it can be computed as the expectation value of a single witness operator-and establish general methods for evaluating the measure. Explicitly applying our results to the relevant resources, we demonstrate the exact computability of the robustness for several classes of states.
Coherent superposition and entanglement are two fundamental aspects of nonclassicality. Here we provide a quantitative connection between the two on the level of operations by showing that the dynamical coherence of an operation upper bounds the dynamical entanglement that can be generated from it with the help of additional incoherent operations. In case a particular choice of monotones based on the relative entropy is used for the quantification of these dynamical resources, this bound can be achieved. In addition, we show that an analog to the entanglement potential exists on the level of operations and serves as a valid quantifier for dynamical coherence.
One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistically indistinguishable set of quantum state vectors forms a linearly dependent set. In quantum optics, the coherent states form an ``overcomplete basis'' of linearly dependent states and are useful in dealing with states that can be prepared in optical systems. Also, the resource theory of magic can be looked upon as a resource theory of quantum coherence with respect to a set of basis vectors that are probabilistically indistinguishable. These motivate us to consider a resource theory of coherence with respect to probabilistically indistinguishable pointers. We find the free states of the resource theory, and analyze the corresponding free operations, obtaining a necessary condition for an arbitrary quantum operation to be free. We identify a class of measures of the quantum coherence and, in particular, establish the monotonicity property of the measures. We find a connection of an arbitrary set of quantum state vectors with positive operator-valued measurements with respect to the resource theory being considered, which paves the way for an alternate definition of the free states. We subsequently examine the wave-particle duality in a double-slit setup in which superposition of probabilistically indistinguishable quantum state vectors is possible. Specifically, we report a complementary relation between quantum coherence and path distinguishability in such a setup.