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Rank in Matrix Analysis: On the Preservers of Maximally Entangled States and Fractional Minimal Rank

Benjamin Großmann

Published Jan 1, 2019• 0 citations
Publisher

Abstract

For Hilbert spaces X , Y, the set of maximally entangled states, MES X ,Y , is a set of rank-1 positive semidefinite operators over the space X ⊗ Y.In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X , Y.After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES X ,Y , which is a set of low-rank positive semidefinite operators over X ⊗ Y.We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices.We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the "minimal rank" and "triangular minimal ranks" differ in general.

DOI
10.17918/etd-9532

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For Hilbert spaces X , Y, the set of maximally entangled states, MES X ,Y , is a set of rank-1 positive semidefinite operators over the space X ⊗ Y.In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X , Y.After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES X ,Y , which is a set of low-rank positive semidefinite operators over X ⊗ Y.We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices.We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the "minimal rank" and "triangular minimal ranks" differ in general.
For Hilbert spaces X,Y, the set of maximally entangled states, MES_{X,Y}, is a set of rank-1 positive semidefinite operators over the space X [o times] Y. In this thesis, we consider the problem of classifying the linear maps that take maximally entangled states to maximally entangled states in the case of finite dimensional spaces X,Y. After classifying these linear maps in the case where dim X divides dim Y, we consider possible avenues of extending these results and consider the set WMES_{X,Y}, which is a set of low-rank positive semidefinite operators over X [o times] Y. We then discuss the "fractional minimal rank", a fractional parameter assigned to partial matrices. We compute the fractional minimal rank for partial matrices whose pattern of knowns is a "minimal cycle", which is the family of the smallest cases for which it is known that the ''minimal rank'' and ''triangular minimal ranks'' differ in general.
We present solutions to a set of problems that arise in quantum entanglement theory, whose common trait is the use of algebraic methods. The backbone of the thesis consists of two general theorems, pertaining to specific convex sets of quantum maps. They are complemented by solutions of many more specific problems of rather technical character. These problems concerned questions such as the higher rank numerical ranges, the lengths of separable states, the solutions of compression equations, the completely entangled subspaces, and maximally entangled states. The solutions were obtained partly by hand, and partly by using the Groebner basis toolset available in computer algebra systems. The main results, which are those of general nature, consist of a characterization of mapping cones, which is a full analogue of the positive maps criterion for separability, as well as of a full characterization of positive-partial-transpose entangled states of minimum rank. The thesis also contains an introduction to the theory of quantum entanglement and to basic algebraic geometry, whose methods have been used
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
Abstract. We consider a family of vector and operator norms which we refer to as Schmidt norms. We show that these norms have several uses in quantum information theory – they can be used to help classify k-positive linear maps (and hence entanglement witnesses), they are useful for approaching the problem of finding non-positive partial transpose bound entangled Werner states, they are related to the quantum fidelity and trace distance measures, and they are connected to the recently-defined local numerical range. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.
Abstract We study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗Hk. The rank of u is the minimal number p of decomposable states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C 2 )⊗3 and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for (C 2 )⊗4; in particular, we show that the maximal rank of a tensor in (C 2 )⊗4 is equal to 4.