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

TL;DR

This paper introduces a universal causality-based criterion for determining the separability of arbitrary quantum density matrices, linking physical causal properties to entanglement detection across diverse quantum systems.

Contribution

It proposes a new heuristic causal separability criterion based on physical causal considerations, applicable to all density matrices in complex quantum systems.

Findings

Derived general formulas for entanglement thresholds in EC-density matrices.

Reproduced known results for qubit pairs as special cases.

Identified features of entanglement thresholds dependent on system size and dimension.

Abstract

General physical background of Peres-Horodecki positive partial transpose (ppt-) separability criterion is revealed. Especially, the physical sense of partial transpose operation is shown to be equivalent to 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 the heuristic causal separability criterion has been proposed for arbitrary $ D^{N} \times D^{N}$ density matrices acting in $ \mathcal{H}_{D}^{\otimes 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 equivalent quantum subsystems being equally connected (EC-) with each other by means of a single entnaglement 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) $ which in the case of a qubit-pair density matrix in $ \mathcal{H}_{2} \otimes \mathcal{H}_{2} $ Hilbert space are shown to reduce to well-known results obtained earlier by Peres [5] and Horodecki [6]. 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 for a wide range of both interacting and non-interacting 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.