A key concept in quantum information theory is the notion of separable states; states for which there is no quantum entanglement.Determining if a given state is entangled has been shown to be NP-hard, and so no deterministic and efficient algorithm will likely be created for this purpose.Current tests are either computationally intensive , return false positives or worsen as the dimension of the problem grows larger.These large dimensions arise naturally in quantum computing.It is therefore necessary to develop a reliable test that is naturally scalable to higher dimensions if one wishes to create efficient quantum computers.Using random matrix theory, a new algorithm has been developed that, with high probability, can determine if a quantum state is separable or entangled.The purpose of this research is to implement this algorithm, checking its efficiency through numerical tests, as well as testing the limitations of the theory behind it.
A key concept in quantum information theory is the notion of separable states; states for which there is no quantum entanglement.Determining if a given state is entangled has been shown to be NP-hard, and so no deterministic and efficient algorithm will likely be created for this purpose.Current tests are either computationally intensive , return false positives or worsen as the dimension of the problem grows larger.These large dimensions arise naturally in quantum computing.It is therefore necessary to develop a reliable test that is naturally scalable to higher dimensions if one wishes to create efficient quantum computers.Using random matrix theory, a new algorithm has been developed that, with high probability, can determine if a quantum state is separable or entangled.The purpose of this research is to implement this algorithm, checking its efficiency through numerical tests, as well as testing the limitations of the theory behind it.
Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. It has recently been shown that this problem is NP-hard, suggesting that an efficient, general solution does not exist. There is a highly inefficient ``basic algorithm'' for solving the quantum separability problem which follows from the definition of a separable state. By exploiting specific properties of the set of separable states, we introduce a classical algorithm that solves the problem significantly faster than the ``basic algorithm,'' allowing a feasible separability test where none previously existed, e.g., in $3\ifmmode\times\else\texttimes\fi{}3$-dimensional systems. Our algorithm also provides a unique tool in the experimental detection of entanglement.
Ever since entanglement was identified as a computational and cryptographic resource, effort has been made to find an efficient way to tell whether a given density matrix represents an unentangled, or separable, state. Essentially, this is the quantum separability problem. In Section 1, I begin with a brief introduction to separability and entanglement, and a basic formal definition of the quantum separability problem. I conclude with a summary of one-sided tests for separability, including those involving semidefinite programming. In Section 2, I treat the separability problem as a computational decision problem and motivate its approximate formulations. After a review of basic complexity-theoretic notions, I discuss the computational complexity of the separability problem. In Section 3, I give a comprehensive survey of deterministic algorithmic solutions to the (approximate) quantum separability problem. Finally, Section 4 contains a comparison of the complexities of the algorithms surveyed in Section 3. I have attempted to give a comprehensive treatment of the deterministic quantum separability problem from a computational perspective and identify relevant open problems. Features and new results include:
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper’s main focus. I then give a thorough treatment of the problem’s relationship with NP, NP-completeness, and co-NP. I also discuss extensions of Gurvits ’ NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem [1, 2, 3]; and the entanglement-witness search (via interior-point algorithms and global optimization) [4, 5]. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.
Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. However, there has not been a general, necessary and sufficient, and operational separability condition to determine whether an arbitrary quantum state is entangled or separable. In this paper, we show that whether a quantum state is entangled or not is determined by a threshold within the quantum state. We first introduce the concept of finer and optimal separable states based on the properties of separable states in the role of higher-level witnesses. Then we show that any bipartite quantum state can be decomposed into a convex mixture of its optimal entangled state and its optimal separable state. Furthermore, we show that whether an arbitrary quantum state is entangled or separable, as well as positive partial transposition (PPT) or not, is determined by the robustness of its optimal entangled state to its optimal separable state with reference to a crucial threshold. Moreover, for an arbitrary quantum state, we provide operational algorithms to obtain its optimal entangled state, its optimal separable state, its best separable approximation (BSA) decomposition, and its best PPT approximation decomposition. And the open question of how to calculate the BSA in high-dimension systems is partially done.
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or \emph{separable}, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper's main focus. I then give a thorough treatment of the problem's relationship with NP, NP-completeness, and co-NP. I also discuss extensions of Gurvits' NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem \cite{DPS02,DPS04,qphCKMR06}; and the entanglement-witness search (via interior-point algorithms and global optimization) \cite{ITCE04,IT06}. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper's main focus. I then give a thorough treatment of the problem's relationship with the complexity classes NP, NP-complete, and co-NP. I also discuss extensions of Gurvits' NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem; and the entanglement-witness search (via interior-point algorithms and global optimization). These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.