Universal separability criterion for arbitrary density matrices from causal properties of separable and entangled quantum states

General physical background of famous Peres–Horodecki positive partial transpose (PH- or PPT-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to what one could call as the “local causality reversal” (LCR-) procedure for all separable quantum systems or to the uncertainty in a global time arrow direction in all entangled cases. Using these universal causal considerations brand new general relations for the heuristic causal separability criterion have been proposed for arbitrary $$ D^{N} \times D^{N}$$ D N × D N density matrices acting in $$ {\mathcal {H}}_{D}^{\otimes N} $$ H D ⊗ N Hilbert spaces which describe the ensembles of N quantum systems of D eigenstates each. Resulting general formulas have been then analyzed for the widest special type of one-parametric density matrices of arbitrary dimensionality, which model a number of equivalent quantum subsystems being equally connected (EC-) with each other to arbitrary degree by means of a single entanglement parameter p. In particular, for the family of such EC-density matrices it has been found that there exists a number of N- and D-dependent separability (or entanglement) thresholds$$ p_{th}(N,D) $$ p th ( N , D ) for the values of the corresponded entanglement parameter p, which in the simplest case of a qubit-pair density matrix in $$ {\mathcal {H}}_{2} \otimes {\mathcal {H}}_{2} $$ H 2 ⊗ H 2 Hilbert space are shown to reduce to well-known results obtained earlier independently by Peres (Phys Rev Lett 77:1413–1415, 1996) and Horodecki (Phys Lett A 223(1–2):1–8, 1996). As the result, a number of remarkable features of the entanglement thresholds for EC-density matrices has been described for the first time. All novel results being obtained for the family of arbitrary EC-density matrices are shown to be applicable to a wide range of both interacting and non-interacting (at the moment of measurement) multi-partite quantum systems, such as arrays of qubits, spin chains, ensembles of quantum oscillators, strongly correlated quantum many-body systems with the possibility of many-body localization, etc.

Robustness of the Gaussian interferometric power in two optomechanical systems

In two optomechanical systems, we study the robustness of the Gaussian interferometric power (GIP) against thermal noises. We use the Mancini et al. criterion to distinguish between entangled and separable states. Also, we employ the GIP to capture the nonclassical feature of the two studied systems beyond entanglement. We evaluate explicitly these two indicators of nonclassicality as functions of parameters characterizing the environment and the optomechanical systems (temperature, coupling, etc). We show that the inseparable quantum correlations decay strongly enough than those existing in separable states under thermal noises. On the other hand, when the separability criterion failed, the GIP remains almost constant and nonzero in the two systems even for high temperatures. This show that the GIP is an appropriate measure to capture the quantumness of correlations especially for systems coupled to environment strongly affected by thermal noises.

TESTING FOR WEAK SEPARABILITY USING STOCHASTIC SEMI-NONPARAMETRIC TESTS: AN EMPIRICAL STUDY ON US DATA

In this paper, we use the weak separability criterion to check for the existence of six different monetary aggregates reported by the Center of Financial Stability (CFS). We implement an extended version of the semi-nonparametric tests introduced by Barnett and de Peretti on US monthly data from January 1967 to December 2012. The test, first, checks for the necessary existence conditions of an overall utility function and a monetary subutility function, and then tests for the separability of the latter. On different subsamples, our results suggest that only the DM1 aggregate meets the separability criterion. Implemented on macroeconomic data, we have tested a joint assumption about separability and the existence of a representative agent. Thus, the rejection of the null could also be due to the rejection of stringent Gorman's conditions. More advanced tests for weak separability are clearly required to confirm the results found in this paper.

Exact algebraic separability criterion for two-qubit systems

A conceptually simpler proof of the separability criterion for two-qubit systems, which is referred to as “Hefei inequality” in literature, is presented. This inequality gives a necessary and sufficient separability criterion for any mixed two-qubit system unlike the Bell-CHSH inequality that cannot test the mixed-states such as the Werner state when regarded as a separability criterion. The original derivation of this inequality emphasized the uncertainty relation of complementary observables, but we show that the uncertainty relation does not play any role in the actual derivation and the Peres–Hodrodecki condition is solely responsible for the inequality. Our derivation, which contains technically novel aspects such as an analogy to the Dirac equation, sheds light on this inequality and on the fundamental issue to what extent the uncertainty relation can provide a test of entanglement. This separability criterion is illustrated for an exact treatment of the Werner state.