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.
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.
We study the rank of a general tensor u in a tensor product $H_1\ot...\ot H_k$. The rank of u is the minimal number p of pure 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 $(\C^2)^{\ot 3}$ and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for $(\C^2)^{\ot 4}$; in particular, we show that the maximal rank of a tensor in $(\C^2)^{\ot 4}$ is equal to 4.
The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called entangled, if it cannot be written in the product form, v≠v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.
The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called {\sl entangled}, if it {\sl cannot} be written in the product form, v=v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..
Abstract. We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let M = Mm(C) and N = Mn(C) be full matrix algebras with m ≥ n, fix an arbitrary state ω of N, and let E(ω) be the set of all states of M ⊗N that extend ω. The space E(ω) contains states of rank r for every r = 1, 2,..., m · rank ω, and it has a filtration into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E m·rank ω = E(ω), where E r (ω) is the set of all states of E(ω) having rank ≤ r. We show first that for every r, there is a real-analytic manifold V r, homogeneous under a transitive action of a compact group G r, which parameterizes E r (ω). The unique G r-invariant probability measure on V r promotes to a probability measure P r,ω on E r (ω), and P r,ω assigns probability 1 to states of rank r. The resulting probability space (E r (ω),P r,ω) represents “choosing a rank r extension of ω at random”. Main result: For every r = 1, 2,..., [rank ω/2], states of (E r (ω), P r,ω) are almost surely entangled. 1.
We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rank-one approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.
Abstract In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: the vacuum in quantum field theory, Araki–Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.