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, 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.
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.
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. Chapters 1 to 3 motivate a new interior-point algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of observables. When all the expected values are known, the algorithm solves the separability problem. In Chapter 4, I give the motivation for the algorithm and show how it can be used in a particular physical scenario to detect entanglement (or decide separability) of an unknown quantum state using as few quantum resources as possible. I then explain the intuitive idea behind the algorithm and relate it to the standard algorithms of its kind. I end the chapter with a comparison of the complexities of the algorithms surveyed in Chapter 3. Finally, in Chapter 5, I present the details of the algorithm and discuss its performance relative to standard methods.
We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.
We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.