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arXiv:2408.08574 · Physical Review A

Detection power of real entanglement witnesses under local unitary equivalence

Yi Shen, Lin Chen, Zhihao Bian

Published Jul 28, 2025• 1 citation
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Abstract

The imaginary unit $i$ has recently been experimentally proven to be indispensable for quantum mechanics. We study the differences in detection power between real and complex entanglement witnesses (EWs) distinguished by whether their matrix expressions incorporate imaginary parts. We show that a real EW (REW), denoted by a real Hermitian matrix, must detect one entangled state of a real density matrix, and conversely an entangled state of a real density matrix must be detected by one REW. We present a necessary and sufficient condition for the entangled states detected by REWs and give a specific example implying the detection limitations of REWs. From an operational perspective, we investigate whether all entangled states are detected by the EWs locally equivalent to some REWs. We prove the validity for all nonpositive partial transpose states. We also derive a necessary and sufficient condition of the validity for the positive partial transpose (PPT) entangled states of complex density matrices. By this condition we show the validity for a family of two-qutrit PPT entangled states of rank four. Another way to figure out the problem is to check whether a counterexample exists. We propose a method to examine the existence from a set-theoretic perspective and provide some supporting evidence of nonexistence. Finally, we derive some results on local projections of EWs with product projectors.

DOI
10.1103/py8r-jklj

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Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the matrix inertias of EWs with a focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.
Entanglement witnesses (EWs) are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we prove that for a given pair of entangled states ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ an EW, WWW, common to both of them exists if and only if for all $0\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}1$, $\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$ is an entangled state. Based on this result, we find a sufficient condition for the presence of entanglement. We then show that an EW which can detect all bell states simultaneously does not exist.
We present a systematic investigation of the spectral properties of entanglement witnesses (EWs). Specifically, we analyze the infimum and supremum of the largest eigenvalue, the smallest eigenvalue, the negativity (defined as the absolute value of the sum of negative eigenvalues), and the squared Frobenius norm of a unit-trace (normalized) entanglement witness, along with the conditions under which these values are attained. Our study provides distinct characterizations for decomposable (DEWs) and nondecomposable entanglement witnesses (NDEWs). While these two classes share many spectral similarities, we reveal a fundamental divergence by proving that the infimum of the smallest eigenvalue can be attained by DEWs, yet remains strictly unattainable for all NDEWs. We apply the results to provide necessary conditions for an EW to possess a mirrored EW. Furthermore, we demonstrate the superior detection capability of NDEWs by proving that any non-positive-partial-transpose state beyond the two-qubit and qubit-qutrit systems can be detected by an NDEW.
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From Certifying Rank kkk Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers
We all know that a density matrix ρρρ is pure if and only if Tr ρ2=1\mathrm{Tr} \,ρ^2 = 1Trρ2=1. But is this the only way to prove the purity of ρρρ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.