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arXiv:2008.09592 · Physical Review A

Restrictions on shareability of classical correlations for random multipartite quantum states

Saptarshi Roy, Shiladitya Mal, Aditi Sen

Published May 3, 2021• 2 citations
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Abstract

Unlike quantum correlations, the shareability of classical correlations (CCs) between two parties of a multipartite state is assumed to be free since there exist states for which CCs for each of the reduced states can simultaneously reach their algebraic maximum value. However, when one randomly picks out states from the state space, we find that the probability of obtaining those states possessing the algebraic maximum value is vanishingly small. Therefore, the possibility of a nontrivial upper bound on the distribution of CCs that is less than the algebraic maxima emerges. We explore this possibility by Haar uniformly generating random multipartite states and computing the frequency distribution for various CC measures, conventional classical correlators, and two axiomatic measures of classical correlations, namely, the classical part of quantum discord and local work of work-deficit. We find that the distributions are typically Gaussian-like and their standard deviations decrease with the increase in number of parties. It also reveals that, among the multiqubit random states, most of the reduced density matrices possess a low amount of CCs which can also be confirmed by the mean of the distributions, thereby showing a kind of restrictions on the shareability of classical correlations for random states. Furthermore, we also notice that the maximal value for random states is much lower than the algebraic maxima obtained for a set of states, and the gap between the two increases further for states with a higher number of parties. We report that, for a higher number of parties, the classical part of quantum discord and local work can follow a monogamy-based upper bound on shareability while classical correlators have a different upper bound. The trends of shareability for classical correlation measures in random states clearly demarcate between the axiomatic definition of classical correlations and the conventional ones.

DOI
10.1103/physreva.103.052401

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Unlike quantum correlations, the shareability of classical correlations (CCs) between two parties of a multipartite state is assumed to be free since there exist states for which CCs for each of the reduced states can simultaneously reach their algebraic maximum value. However, when one randomly picks out states from the state space, we find that the probability of obtaining those states possessing the algebraic maximum value is vanishingly small. Therefore, the possibility of a nontrivial upper bound on the distribution of CCs that is less than the algebraic maxima emerges. We explore this possibility by Haar uniformly generating random multipartite states and computing the frequency distribution for various CC measures, conventional classical correlators, and two axiomatic measures of classical correlations, namely, the classical part of quantum discord and local work of work-deficit. We find that the distributions are typically Gaussian-like and their standard deviations decrease with the increase in number of parties. It also reveals that, among the multiqubit random states, most of the reduced density matrices possess a low amount of CCs which can also be confirmed by the mean of the distributions, thereby showing a kind of restrictions on the shareability of classical correlations for random states. Furthermore, we also notice that the maximal value for random states is much lower than the algebraic maxima obtained for a set of states, and the gap between the two increases further for states with a higher number of parties. We report that, for a higher number of parties, the classical part of quantum discord and local work can follow a monogamy-based upper bound on shareability while classical correlators have a different upper bound. The trends of shareability for classical correlation measures in random states clearly demarcate between the axiomatic definition of classical correlations and the conventional ones.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
Monogamy of quantum correlations provides a way to study restrictions on their sharability in multiparty systems. We find the critical exponent of these measures, above which randomly generated multiparty pure states satisfy the usual monogamy relation, and show that the critical power decreases with the increase in the number of parties. For three-qubit pure states, we detect that W-class states are more prone to being nonmonogamous as compared to the GHZ-class states. We also observe a different criticality in monogamy power up to which random pure states remain nonmonogamous. We prove that the ``average monogamy'' score asymptotically approaches its maximal value on increasing the number of parties. Analyzing the monogamy scores of random three-, four-, five-, and six-qubit pure states, we also report that almost all random pure six-qubit states possess maximal monogamy score, which we confirm by evaluating statistical quantities like mean, variance, and skewness of the distributions. In particular, with the variation of number of qubits, means of the distributions of monogamy scores for random pure states approach to unity---which is the algebraic maximum---thereby conforming to the known results of random states having maximal multipartite entanglement in terms of geometric measures.
Quantum correlations typically decrease with increasing noise, although classical correlators (CCors) may rise for a particular class of states with noise. To analyse the behavior of classical correlation (CC) in the presence of local noise, we scrutinize the set of classical correlators, axiomatic CC measures like classical discord, and local work for Haar uniformly generated states. Like quantum correlation measures, we illustrate that when noise levels rise, the average value of the CC measures for noisy output states obtained from random input states decreases for most of the channels. We also demonstrate a connection between the CCors of the noise-affected multipartite states that are produced and the CCors of the initial states that exhibit exponential, polynomial, and constant behavior as the noise level changes. Moreover, based on CCors of the generalised N-qubit W state as input, we determine a method to discriminate between the quantum channels, namely phase damping, depolarizing, and amplitude damping channels. We also relate classical, quantum, and total correlation measures that exhibit a comparable reaction to decoherence for generic states.
In multiparty quantum systems, the monogamy inequality proposes an upper bound on the distribution of bipartite quantum correlation between a single party and each of the remaining parties in the system, in terms of the amount of quantum correlation shared by that party with the rest of the system taken as a whole. However, it is well known that not all quantum correlation measures universally satisfy the monogamy inequality. In this work, we aim at determining the nontrivial value by which the monogamy inequality can be violated by a quantum correlation measure. Using an information-theoretic complementarity relation between the normalized purity and quantum correlation in any given multiparty state, we obtain a nontrivial lower bound on the negative monogamy score for the quantum correlation measure. In particular, for the three-qubit states the lower bound is equal to the negative von Neumann entropy of the single qubit reduced density matrix. We analytically examine the tightness of the derived lower bound for certain nnn-qubit quantum states. Further, we report numerical results of the same for monogamy violating correlation measures using Haar uniformly generated three-qubit states.
Characterizing multipartite quantum correlations beyond two parties is of utmost importance for building cutting edge quantum technologies, although the comprehensive picture is still missing. Here we investigate quantum correlations (QCs) present in a multipartite system by exploring connections between magentathree different frameworks, namely, the monogamy score (MS), localizable quantum correlations (LQC), and genuine multipartite entanglement (GME) content of the state. We find that the frequency distribution of GME for Dicke states with higher excitations resembles that of random states. We show that there is a critical value of GME beyond which all states become monogamous and it is investigated by considering different powers of MS which provides various layers of monogamy relations. Interestingly, such a relationship between LQC and MS as well as GME does not hold. States having a very low GME (low monogamy score, both positive and negative) can localize a high amount of QCs in two parties. We also provide an upper bound to the sum of bipartite QC measures including LQC for random states and establish a gap between the actual upper bound and the algebraic maximum.
Monogamy is a nonclassical property that restricts the sharability of quantum correlation among the constituents of a multipartite quantum system. Quantum correlations may satisfy or violate monogamy for quantum states, which was tested mainly for three-qubit states. Here we establish a sufficient condition for monogamy of arbitrary quantum correlation measures of states of an arbitrary number of parties, using which and further numerical results, we obtain evidence for monogamy of measures such as distillable entanglement and relative entropy of entanglement, which are physically important but mathematically intractable, for almost all quantum states of a moderate number of parties. The result is generic and holds for a large class of quantum correlation measures. Nonetheless, we identify important zero Haar measure classes of pure states that remain nonmonogamous with respect to quantum discord and quantum work deficit, irrespective of the number of qubits.