This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with non-negative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching. We show that the new notion inherits many 'classical' properties, but not all of them. This new notion goes somewhere beyound matroids. For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids. In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states. Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some 'classical' complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k^{4/5}n^{9/5}) algorithm as well as a lower bound of Omega(k^{1/2}n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.
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.