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arXiv:2003.03372 · Journal of the Optical Society of America B

Intrinsic degree of coherence of two-qubit states and measures of two-particle quantum correlations

Nilakantha Meher, Abu Musa Patoary, Girish Ashok Kulkarni, Anand K. Jha

Published Mar 5, 2020• 10 citations
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Abstract

Recently, a basis-invariant measure of coherence known as the intrinsic degree of coherence has been established for classical and single-particle quantum states [ J. Opt. Soc. Am. B 36 , 2765 ( 2019 ) JOBPDE 0740-3224 10.1364/JOSAB.36.002765 ]. In this article, we use the same mathematical construction to define the intrinsic degree of coherence of two-qubit states and demonstrate its usefulness in quantifying two-particle quantum correlations and entanglement. In this context, we first show that the intrinsic degree of coherence of a two-qubit state puts an upper bound on the violations of Bell inequalities that can be achieved with the state and that a two-qubit state with an intrinsic degree of coherence less than 1 / 3 cannot violate Bell inequalities. We then show that the quantum discord of a two-qubit state, which quantifies the amount of quantum correlations available in the two-qubit state for certain tasks, is bounded from above by the intrinsic degree of coherence of the state. Next, in the context of two-particle entanglement, we show that the range of values that the concurrence of a two-qubit state can take is decided by the intrinsic degree of coherence of the two-qubit state together with that of the individual qubits. Finally, for the polarization two-qubit states generated by the parametric down-conversion of a pump photon, we propose an experimental scheme to measure the intrinsic degree of coherence of two-qubit states. We also present our theoretical study that shows how the intrinsic degree of coherence of a pump photon dictates the maximum intrinsic degree of coherence of the generated two-qubit state.

DOI
10.1364/josab.384936

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Recently, a basis-invariant measure of coherence known as the intrinsic degree of coherence has been established for classical and single-particle quantum states [ J. Opt. Soc. Am. B 36 , 2765 ( 2019 ) JOBPDE 0740-3224 10.1364/JOSAB.36.002765 ]. In this article, we use the same mathematical construction to define the intrinsic degree of coherence of two-qubit states and demonstrate its usefulness in quantifying two-particle quantum correlations and entanglement. In this context, we first show that the intrinsic degree of coherence of a two-qubit state puts an upper bound on the violations of Bell inequalities that can be achieved with the state and that a two-qubit state with an intrinsic degree of coherence less than 1 / 3 cannot violate Bell inequalities. We then show that the quantum discord of a two-qubit state, which quantifies the amount of quantum correlations available in the two-qubit state for certain tasks, is bounded from above by the intrinsic degree of coherence of the state. Next, in the context of two-particle entanglement, we show that the range of values that the concurrence of a two-qubit state can take is decided by the intrinsic degree of coherence of the two-qubit state together with that of the individual qubits. Finally, for the polarization two-qubit states generated by the parametric down-conversion of a pump photon, we propose an experimental scheme to measure the intrinsic degree of coherence of two-qubit states. We also present our theoretical study that shows how the intrinsic degree of coherence of a pump photon dictates the maximum intrinsic degree of coherence of the generated two-qubit state.
We study how one-particle correlations transfer to manifest as two-particle correlations in the context of parametric down-conversion (PDC), a process in which a pump photon is annihilated to produce two entangled photons. We work in the polarization degree of freedom and show that for any two-qubit generation process that is both trace-preserving and entropy-nondecreasing, the concurrence $C(\ensuremath{\rho})$ of the generated two-qubit state $\ensuremath{\rho}$ follows an intrinsic upper bound with $C(\ensuremath{\rho})\ensuremath{\le}(1+P)/2$, where PPP is the degree of polarization of the pump photon. We also find that for the class of two-qubit states that is restricted to have only two nonzero diagonal elements such that the effective dimensionality of the two-qubit state is the same as the dimensionality of the pump polarization state, the upper bound on concurrence is the degree of polarization itself, that is, $C(\ensuremath{\rho})\ensuremath{\le}P$. Our work shows that the maximum manifestation of two-particle correlations as entanglement is dictated by one-particle correlations. The formalism developed in this work can be extended to include multiparticle systems and can thus have important implications towards deducing the upper bounds on multiparticle entanglement, for which no universally accepted measure exists.
In quantum resource theories (QRTs), evidence of intrinsic connections between coherence and quantum correlations, including Bell nonlocality, entanglement, quantum steering, and so on, exists. However, building these relationships is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among coherence and different quantum correlations in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence is derived for tripartite entanglement states. Then we obtain the trade-off relation between the first-order coherence and the genuine tripartite entanglement by the genuinely multipartite concurrence and concurrence fill. Furthermore, the trade-off relation between the maximum steering inequality violation and concurrence fill for arbitrary three-qubit states is found. In addition, we investigate the close relation between the maximum steering inequality violation and the first-order coherence. The results show that coherence and quantum correlations are intrinsically related and can be converted to one another in the framework of QRTs.
Concurrence is viewed as the most commonly approach for quantifying entanglement of two-qubit states, while intrinsic concurrence contains concurrence of four pure states consisting of a special pure state ensemble concerning an arbitrary two-qubit state. Thus, a natural question arises: Whether there is a specified relation between them. We firstly examine the relation between concurrence and intrinsic concurrence for the maximally nonlocal mixed state under a special unitary operation, which is not yet rigorously proved. In order to obtain a general result, we investigate the relation between concurrence and intrinsic concurrence using randomly generated two-qubit states, and derive an inequality relation between them. Finally, we take into account the relation between concurrence and intrinsic concurrence in open systems, and reveal the ratio of the two quantum resources, which is only correlated with the experiencing channels.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.
By introducing a quantitative ``degree of commutativity'' in terms of the angle between spin observables we present two tight quantitative trade-off relations in the case of two qubits. First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to $\ensuremath{\pi}∕2$ (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach $\ensuremath{\pi}∕2$ the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anticommuting local observables where, respectively, the bounds of 222\sqrt{2}22​ and 2\sqrt{2}2​ hold for the expectation value of the Bell operator. The trade-off relations show that noncommmutativity gives ``a more than classical result'' for entangled states, whereas ``a less than classical result'' is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment.