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AIP conference proceedings

Separability condition for higher dimensional systems based on realignment operation

Naman Sharma, Gunjan Arora, Hemaank Mahajan, Satyabrata Adhikari

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We begin by describing our study of the singular value decomposition. We move on describing the separability/entanglement detection problem for a two-qubit system. After describing a few other criterions used for detection of entangled states, we divert our focus to higher dimensional systems, where we find a necessary condition on the minimum singular value for a d ⊗ d dimensional state to be separable. The condition is based on realignment criterion and structural physical approximation of partial transposition (SPA-PT) criterion. We also discuss the geometrical interpretation of the necessary condition. We have also included some examples that show the situations where other criterion might fail but our condition is useful to detect entangled states. The condition we found is useful in the fact that only the smallest singular value of the realigned matrix is required instead of all the singular values that are required in the other criterion. This makes our criterion have a small computational advantage. After this, we also discussed some ways to approximate the smallest singular value which may in turn help finding entangled states when used along with the above criterion. Index Terms—entanglement detection, quantum information theory, realignment criterion, singular value, structural physical approximation of partial transposition.

DOI
10.1063/5.0294153

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We begin by describing our study of the singular value decomposition. We move on describing the separability/entanglement detection problem for a two-qubit system. After describing a few other criterions used for detection of entangled states, we divert our focus to higher dimensional systems, where we find a necessary condition on the minimum singular value for a d ⊗ d dimensional state to be separable. The condition is based on realignment criterion and structural physical approximation of partial transposition (SPA-PT) criterion. We also discuss the geometrical interpretation of the necessary condition. We have also included some examples that show the situations where other criterion might fail but our condition is useful to detect entangled states. The condition we found is useful in the fact that only the smallest singular value of the realigned matrix is required instead of all the singular values that are required in the other criterion. This makes our criterion have a small computational advantage. After this, we also discussed some ways to approximate the smallest singular value which may in turn help finding entangled states when used along with the above criterion. Index Terms—entanglement detection, quantum information theory, realignment criterion, singular value, structural physical approximation of partial transposition.
It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher dimensional system and obtained the necessary condition on the minimum singular value of the realigned matrix of d⊗dd\otimes dd⊗d dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for d⊗dd\otimes dd⊗d dimensional system. Furthermore, we show that our criterion may also detect bound entangled state. The entanglement detection criterion studied here is beneficial in the sense that it requires to calculate only minimum singular value of the realigned matrix while on the other hand realignment criterion requires all singular values of the realigned matrix. Thus, our criterion has computational advantage over the realignment criterion.
Entanglement detection is an important problem in quantum information theory because quantum entanglement is a key resource in quantum information processing. Realignment criterion is a powerful tool for the detection of entangled states in bipartite and multipartite quantum systems. It works well not only for negative-partial-transpose entangled states (NPTESs) but also for positive-partial-transpose entangled states (PPTESs). Since the matrix corresponding to the realignment map is indefinite, the experimental implementation of the map is an obscure task. In this work, first, we approximate the realignment map to a positive map using the method of structural physical approximation, and then we show that the structural physical approximation of the realignment map (SPA-R) is completely positive. Positivity of the constructed map is characterized using moments which can be physically measured. Next, we develop a separability criterion based on our SPA-R map in the form of an inequality and show that the developed criterion not only detect NPTESs but also PPTESs. Further, we show that for a special class of states called Schmidt-symmetric states, the SPA-R separability criteria reduce to the original form of the realignment criteria. We provide some examples to support the results obtained. Moreover, we analyze the error that may occur because of approximating the realignment map.
The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.
Detection of entanglement through partial knowledge of the quantum state is a challenge to implement efficiently. Here we propose a separability criterion for detecting bipartite entanglement in arbitrary-dimensional quantum states using partial moments of the realigned density matrix. Our approach enables the detection of both distillable and bound entangled states through a common framework. We illustrate the significance of our method through examples of states belonging to both the above categories, which are not detectable using comparable other schemes relying on partial state information. The formalism of employing partial realigned moments proposed here is further shown to be effective for two-qubit systems too, with a slight modification of our separability criterion.
It is crucial to know whether the quantum state generated in the experiment is entangled. In the literature, this topic was studied extensively and researchers proposed different approaches for the detection of a mixed bipartite entangled state in arbitrary dimension. Proceeding in this line of research, we propose three different criteria for the detection of mixed bipartite negative partial transpose (NPT) entangled state in arbitrary dimension. Our criteria are based on the method of structural physical approximation (SPA) of partial transposition (PT). We show that the proposed criteria for the detection of a NPT entangled state can be realized experimentally. Two of the proposed criteria are given in terms of the concurrence of the given state in arbitrary dimension so it is essential to find out the concurrence. Thus, we provide lower and upper bounds of concurrence of the quantum state under investigation in terms of average fidelity of two quantum states, and hence these bounds can be realized experimentally. Moreover, we show how to perform a SPA map on a qutrit-qubit system and then explicitly calculate the matrix elements of the density matrix describing the SPA-PT of the qutrit-qubit system. We then illustrate our criteria for the detection of entanglement by considering a class of qubit-qubit system and a class of qutrit-qubit system.
Great advances have been achieved in studying characteristics of entanglement for fundamentals of quantum mechanics and quantum information processing. However, even for N-qubit systems, the problem of entanglement criterion has not been well solved. In this Letter, using the method of state decomposition and high order singular value decomposition (HOSVD), we propose a necessary and sufficient entanglement criterion for general N-qubit systems. As an example, we apply our method to study the multi-qubit W state with white noise. We not only obtain the separability critical point, which is tight and thus better than known results, but also the separate states ensemble for decomposition. More examples are presented to show our criterion is accurate, which is tighter than the well-known positive partial transpose criterion. For two-qubit case, we can provide an entanglement measurer which gives similar results with concurrence up to a factor. Our results pave the way to solve the entanglement-separability criterion for more general cases.