In this paper, we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.
In this paper, we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.
We consider random bipartite quantum states obtained by tracing out one subsystem from a random, uniformly distributed, tripartite pure quantum state. We compute thresholds for the dimension of the system being traced out, so that the resulting bipartite quantum state satisfies the reduction criterion in different asymptotic regimes. We consider as well the basis-independent version of the reduction criterion (the absolute reduction criterion), computing thresholds for the corresponding eigenvalue sets. We do the same for other sets relevant in the study of absolute separability, using techniques from random matrix theory. Finally, we gather and compare the known values for the thresholds corresponding to different entanglement criteria, and conclude with a list of open questions.
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.
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
In quantum information theory,quantum entangled state is a class very important resource.Detecting entanglement for states is a extremely important problem.In 2001,Nielsen M A proposed a reduction criterion,then William Hall proposed a different reduction criterion for multipartite finite dimensional systems in 2005.In this paper,we generalized above two reduction criteria to infinite dimensional multipartite quantum systems.Two class reduction criteria axe obtained for states in infinite dimensional multipartite quantum systems.
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.
In this paper, we combine the k -reduction map, the moment method, and randomized measurements into a practical protocol for certifying the entanglement dimensionality. Our approach is based on the observation that a state with entanglement dimensionality at most k must stay positive under the action of the k -reduction map. The core of our protocol utilizes the moment method to determine whether the k -reduced operator, i.e., the operator obtained after applying the k -reduction map on a quantum state, contains a negative eigenvalue or not. Notably, we propose a systematic method for constructing reduction moment criteria that apply to a broader range of states, including k -unfaithful states, compared with fidelity-based methods. The performance of our approach gets better and better with the moment order employed, which is corroborated by extensive numerical simulations. To apply our approach, it suffices to implement a unitary 3-design instead of a 4-design, which is more feasible in practice than the correlation matrix method. In the course of study, we show that the k -reduction negativity, the absolute sum of the negative eigenvalues of the k -reduced operator, is monotonic under local operations and classical communication for pure states.