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Register a new userSeparability of Hermitian Tensors and PSD Decompositions
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Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect separability for much larger sized Hermitian tensors. This method is a good start on determining psd ranks of separable Hermitian tensors.
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Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.
We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: how closely can we approximate the set of unit-trace $n times n$ PSD matrices, denoted by $D$, using at most $N$ number of $k times k$ PSD constraints? In this paper, we prove lower bounds on $N$ to achieve a good approximation of $D$ by considering two constructions of an approximating set. First, we consider the unit-trace $n times n$ symmetric matrices that are PSD when restricted to a fixed set of $k$-dimensional subspaces in $mathbb{RR}^n$. We prove that if this set is a good approximation of $D$, then the number of subspaces must be at least exponentially large in $n$ for any $k = o(n)$. % Second, we show that any set $S$ that approximates $D$ within a constant approximation ratio must have superpolynomial $mathbf{S}_+^k$-extension complexity. To be more precise, if $S$ is a constant factor approximation of $D$, then $S$ must have $mathbf{S}_+^k$-extension complexity at least $exp( C cdot min { sqrt{n}, n/k })$ where $C$ is some absolute constant. In addition, we show that any set $S$ such that $D subseteq S$ and the Gaussian width of $D$ is at most a constant times larger than the Gaussian width of $D$ must have $mathbf{S}_+^k$-extension complexity at least $exp( C cdot min { n^{1/3}, sqrt{n/k} })$. These results imply that the cone of $n times n$ PSD matrices cannot be approximated by a polynomial number of $k times k$ PSD constraints for any $k = o(n / log^2 n)$. These results generalize the recent work of Fawzi on the hardness of polyhedral approximations of $mathbf{S}_+^n$, which corresponds to the special case with $k=1$.
In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose, firstly, we present that each B tensor is strictly semi-positive and each B$_0$ tensor is semi-positive. Subsequencely, the strictly lower and upper bounds of different operator norms are given for two positively homogeneous operators defined by B tensor. Finally, with the help of the upper bounds of different operator norms, we show the strcitly lower bound of solution set of tensor complementarity problem with B tensor. Furthermore, the upper bounds of spectral radius and $E$-spectral radius of B (B$_0$) tensor are obtained, respectively, which achieves our another objective. In particular, such the upper bounds only depend on the principal diagonal entries of tensors.
We combine stochastic control methods, white noise analysis and Hida-Malliavin calculus applied to the Donsker delta functional to obtain new representations of semimartingale decompositions under enlargement of filtrations. The results are illustrated by explicit examples.
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density matrices rho(MDS) for which tracing over any subsystem gives the unit density matrix). A sufficient condition that rho(MDS) is not separable is that it has an eigenvalue larger than 1/d for a qubit and a qudit, and larger than 1/2^(n-1) for n-qubits system. The PTU transformation does not change the eigenvalues of the n-qubits MDS density matrices for odd n. Thus the Peres-Horodecki criterion does not give any information about entanglement of these density matrices, but this criterion is useful for indicating inseparability for even n. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the W state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8 GHZ density matrices with probabilities p(i), is analyzed as function of these probabilities. In some cases we show that the Peres-Horodecki criterion is both sufficient and necessary.
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