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International Journal of Quantum Information

Mathematical framework for describing multipartite entanglement in terms of rows or columns of coefficient matrices

Yi Huang, Huapeng Yu, Fang Miao, Tianyong Han, Xiujun Zhang

Published Dec 13, 2021• 2 citations

Abstract

In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the [Formula: see text]-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

DOI: 10.1142/s0219749921500350

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In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the [Formula: see text]-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, assume that a state with six non-zero coefficients is not a trivially separable state. Then the state is separable if and only if its six basis states consist of the three partially complementary pairs and the corresponding coefficient matrix has proportional rows. The contrapositive of this result reads that the state is genuinely entangled if and only if its six basis states do not consist of the three partially complementary pairs or though the six basis states consist of the three partially complementary pairs, the corresponding coefficient matrix does not have proportional rows. We propose four corresponding coefficient 2 by 3 matrices and show that if the four coefficient matrices don't have proportional rows, then the state is genuinely entangled. It is trivial to know if two rows of a 2 by 3 coefficient matrix are proportional. The difference from the previous articles is that the structure of the basis states is used to detect entanglement in this paper. One can see that Osterloh and Siewert's states of five and six qubits are genuinely entangled because two rows for any one of the four corresponding coefficient 2 by 3 matrices are not proportional. These states were distinguished as the maximal entangled states by the complicated filters before. Keywords: entanglement, separability, entangled states, separable states, qubits.

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.

We present two sets of computable entanglement measures for multipartite systems where each subsystem can have different degrees of freedom (so-called qudits). One set, called ``separability'' measure, reveals which of the subsystems are separable or entangled. For that we have to extend the concept of

k

separability for multipartite systems to a unambiguous separability concept which we call ${\ensuremath{\gamma}}_{k}$ separability. The second set of entanglement measures reveals the ``kind'' of entanglement, i.e., if it is bipartite, tripartite, \dots{},

n

-partite entangled and is denoted as the ``physical'' measure. We show how lower bounds on both sets of measures can be obtained by the observation that any entropy may be rewritten via operational expressions known as

m

concurrences. Moreover, for different classes of bipartite or multipartite qudit systems we compute the bounds explicitly and discover that they are often tight or equivalent to positive partial transposition.

In this paper, we obtain an efficient separability criterion for bipartite quantum pure state systems, which is based on the two-order minors of the coefficient matrix corresponding to quantum state. Then, we generalize this criterion to multipartite arbitrarily dimensional pure states. Our criterion is directly built upon coefficient matrices, but not density matrices or observables, so it has the advantage of being computed easily. Indeed, to judge separability for an arbitrary n-partite pure state in a d-dimensional Hilbert space, it only needs at most O(d) times operations of multiplication and comparison. Our criterion can be extended to mixed states. Compared with Yu's criteria, our methods are faster, and can be applied to any quantum state.