Arthur O. Pittenger

Classicality in discrete Wigner functions

Gibbons et al., [Phys. Rev. A 70, 062101 (2004)] have recently defined discrete Wigner functions W to represent quantum states in a Hilbert space with finite dimension. We show that such a class of Wigner functions W can be defined so that the only pure states having non-negative W for all such functions are stabilizer states, as conjectured by Galvao, [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of W for all definitions of W in the class form a subgroup of the Clifford group. This means pure states with non-negative W and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.

Convexity and the separability problem of quantum mechanical density matrices

Abstract A finite-dimensional quantum mechanical system is modelled by a density ρ, a trace one, positive semi-definite matrix on a suitable tensor product space H[N]. For the system to demonstrate experimentally certain non-classical behavior, ρ cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: 1. the determination whether a given ρ is in S, and 2. a measure of the “entanglement” of such a ρ in terms of its distance from S. In this paper we describe these two problems in detail for a linear algebra audience, discuss some recent results from the quantum computing literature, and prove some new results. We emphasize the roles of densities ρ as both operators on the Hilbert space H[N] and also as points in a real Hilbert space M. We are able to compute the nearest separable densities τ0 to ρ0 in particular classes of inseparable densities and we use the Euclidean distance between the two in M to quantify the entanglement of ρ0. We also show the role of τ0 in the construction of separating hyperplanes, so-called entanglement witnesses in the quantum computing literature.

Note on separability of the Werner states in arbitrary dimensions

Great progress has been made recently in establishing conditions for separability of a particular class of Werner densities on the tensor product space of n d-level systems (qudits). In this brief note we complete the process of establishing necessary and sufficient conditions for separability of these Werner densities by proving the sufficient condition for general n and d.

Unextendible product bases and the construction of inseparable states

Abstract Let H ( N ) denote the tensor product of n finite dimensional Hilbert spaces H ( r ) . A state ϕ of H ( N ) is separable if ϕ = α 1 ⊗⋯⊗ α n where the states α r are in H r . An orthogonal unextendible product basis is a finite set B of separable orthonormal states ϕ k ,1⩽k⩽m such that the non-empty space B ⊥ , the set of vectors orthogonal to B , contains no separable state. Examples of orthogonal UPB sets were first constructed by Bennett et al. 1 and other examples and references appear, for example, in 3 . If F=F B denotes the set of convex combinations of ϕ k ϕ k ,1⩽k⩽m , then F is a face in the set S of separable densities. In this note we show how to use F to construct families of positive partial transform states ( PPT ) which are not separable. We also show how to make an analogous construction when the condition of orthogonality is dropped. The analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable PPT states from S .

Order independence and factor convergence in iterative scaling

Abstract We prove two results concerning iterative scaling which have been claimed, but not proved, by other authors in the field. Iterative scaling is a procedure which begins with a probability measure Q on a finite set X and a finite set of partition constraints on X. It produces a sequence of iterates P0 = Q, P1,…,Pt,… in which each Pt (t #62; 0) is an adjustment (scaling) of Pt−1 to fit a single set of prescribed partition constraints. Under fairly general hypotheses, the iterates Pt converge to the unique probability P∗ which simultaneously satisfies the given constraints and is exponentially equivalent to the starting measure Q (or equivalently, which minimizes the I-divergence D(P‖Q) over all probabilities P on X which satisfy the constraints). The first main result (Theorem 3.1) states that the convergence of iterative scaling to the unique limit is independent of the order in which the adjustments for the prescribed partition constraints are made, provided each adjustment is made infinitely many times. The second main result (Theorem 4.2) is that, under certain commonly satisfied conditions, each factor of the exponential form being built up within the algorithm (one factor for each partition constraint) converges separately. A third group of results (Sections 5,6) provides an account of the relationship between the probability measures which satisfy all of the constraints and those which are exponentially equivalent to Q.

Complete separability and Fourier representations of n -qubit states

Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of n-qubit states whose densities all satisfy the Peres condition.

Generalized Circulant Densities and a Sufficient Condition for Separability

In a series of papers with Kossakowski, the first author has examined properties of densities for which the positive partial transpose (PPT) property can be readily checked. These densities were also investigated from a different perspective by Baumgartner, Hiesmayr and Narnhofer. In this paper, we show how the support of such densities can be expressed in terms of lines in a finite geometry and how the same structure lends itself to checking the necessary PPT condition and to a novel sufficient condition for separability.

Mappings of latin squares

Abstract Let L n denote the set of n by n Latin squares. We show that it is possible to map from one such square A to another square B using only the class of mappings defined by (1) symbol interchanges on cycles defined by two symbols and (2) a restricted class of mappings involving three symbols. The total number of mappings so defined is the smallest known class sufficient to connect L n . As an application of this result, we define a Markov chain on L n whose asymptotic distribution is uniform, thus providing a means of generating uniformly distributed Latin squares.

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

Abstract We provide an elementary proff of Fulkersons theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkersons theorem as a corollary.

Quantum Error-Correcting Codes

So far we have been dealing with a highly idealized situation in which physical states have been assumed to be stable and independent of time and in which unitary transformations on these states can be effected in a reliable fashion. The real situation is considerably more complicated and requires a model for the dynamics of a quantum system.