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arXiv:quant-ph/0504041

Towards efficient algorithm deciding separability of distributed quantum states

Piotr Badziąg, Paweł Horodecki, Ryszard Horodecki

Published Apr 6, 2005• 2 citations
arXivHTMLPDF

Abstract

It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly exponentially with the state's rank. Nevertheless, we argue that for generic states, analysis of concurrence matrices essentially reduces the task of solving separability problem in $m \times n$ dimensions to solving a set of linear equations in about $\binom{mn+D-1}{D}$ variables, where $D$ decreases with $mn$ and for large $mn$ it should not exceed 4. Moreover, the same method is also applicable to multipartite states where it is at least equally efficient.

DOI
10.48550/arxiv.quant-ph/0504041

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It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly exponentially with the state's rank. Nevertheless, we argue that for generic states, analysis of concurrence matrices essentially reduces the task of solving separability problem in m×nm \times nm×n dimensions to solving a set of linear equations in about (mn+D−1D)\binom{mn+D-1}{D}(Dmn+D−1​) variables, where DDD decreases with mnmnmn and for large mnmnmn it should not exceed 4. Moreover, the same method is also applicable to multipartite states where it is at least equally efficient.
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.
Abstract Determining whether an unknown quantum state is entangled can be approached through quantum tomography and related algorithms. However, these methods are generally inefficient due to the NP-hardness of the problem. An alternative strategy is to treat it as a decision (or estimation) problem with a quantum data structure (Horodecki and Ekert 2002 Phys. Rev. Lett. 89 127902). Over the years, significant progress has been made: the two-qubit case is now fully understood, but for higher-dimensional systems only entanglement detection via negativity has been established. In this work, we propose schemes for measuring upper bounds on bipartite biconcurrence and concurrence, expressed as functions of a fixed (but arbitrary) unitary operation acting on a Hilbert space that is quadratically larger than the original state space. Independently, we provide a scheme for measuring a lower bound for concurrence. In the case of the states proportional to the projectors the schemes are getting significantly simpler and the upper bound on the biconcurrence, as well as the lower bound on the concurrence, become exact (up to experimental and numerical errors). This framework opens the door to tackling the separability problem using methods inspired by the variational quantum eigensolver.
While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad class of structured dense matrices arise in potential theory, covariance modeling, and computational physics, namely, hierarchically block separable (HBS) matrices. We develop two distinct methods to make these systems amenable to quantum solvers. The first is a pre-processing approach that transforms the dense matrix into a larger but sparse format. The second is a direct block encoding scheme that recursively constructs the necessary oracles from the HBS structure. We provide a detailed complexity analysis and rigorous error bounds for both methods. Numerical experiments are presented to validate the effectiveness of our approaches.
The discretized path integral expression for the reduced density matrix (RDM) of a system interacting with a dissipative harmonic bath is fully entangled because of influence functional terms that couple the variables at different time points. The iterative decomposition of the path integral, which exploits the finite length of influence functional memory, involves a tensor propagator whose size grows exponentially with the memory length. The present Communication disentangles the path integral by recursively spreading the temporal entanglement over longer path segments, while decreasing its contribution. Eventually, the entangled term becomes sufficiently small and may be neglected, leading to iterative propagation of the RDM through simple multiplication of matrices whose size is equal to that of the bare system. It is found that the temporal entanglement length is practically equal to the bath-induced memory length. The small matrix decomposition of the path integral (SMatPI) is stable and very efficient, extending the applicability of numerically exact real-time path integral methods to multi-state systems.
Abstract We investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems’ dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N -partite case, we provide conditions on the subsystems’ size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Also, allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration-free parent Hamiltonians. We further argue that, to a high probability, pure quantum states sampled from a t -design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N -partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.