Abstract Quantum entanglement plays a key role in quantum computation and quantum information processing. It is of great significance to find efficient and experimentally friend separability criteria to detect entanglement. In this paper, we firstly propose two easily used entanglement criteria based on matrix moments. The first entanglement criterion only uses the first two realignment moments of a density matrix. The second entanglement criterion is based on the moments related to the partially transposed matrix. By detailed examples we illustrate the effectiveness of these criteria in detecting entanglement. Moreover, we provide an experimentally measurable lower bound of concurrence based on these moments. Finally, we present both bipartite and genuine tripartite entanglement measures based on the moments of the reduced states. By detailed examples, we show that our entanglement measures characterize the quantum entanglement in a more fine ways than the existing measures.
Efficiently detecting entanglement based on measurable quantities is a basic problem for quantum information processing. Recently, the measurable quantities called partial-transpose (PT) moments have been proposed to detect and characterize entanglement. Recently [L. Zhang et al., Ann. Phys. (Berlin) 534, 2200289 (2022)], we have already identified the two-dimensional region, comprised of the second and third PT moments, corresponding to two-qubit entangled states, and described the whole region for all two-qubit states. In the present paper, we visualize the three-dimensional (3D) region corresponding to all two-qubit states by further involving the fourth PT moment (the last one for two-qubit states). The characterization of this 3D region can finally be achieved by optimizing some polynomials. Furthermore, we identify the dividing surface which separates the two parts of the whole 3D region corresponding to entangled and separable states, respectively. Due to the measurability of PT moments, we obtain a complete and operational criterion for the detection of two-qubit entanglement.
The
p3-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments
pk=Tr
[(ρΓ)k], where
ρΓ denotes the partially transposed density matrix of a quantum state
ρ. While most of these generalizations are strictly more powerful than the
p3-PPT criterion, their
m-th level versions usually rely on the availability of
pk for all moment orders
k≤m. Here, we show that one can alternatively compare any three PT moments of orders
k<l<m, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where
x=(m−l)/(m−k). Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to
pk for
k≤5 is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-
m criterion is as powerful as the PPT criterion whenever
ρΓ has no more than
(m+1)/2 distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of
pk under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.
Abstract A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.
Abstract We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k -th condition involves comparing moments of the partially transposed density operator up to order k . Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on
Cd⊗Cd, obtained as the marginal over an environment
Cs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level
m∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension
s=λmd2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to
1 as
d grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension
d is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.