Continuous Tensor Products

Let H=X⊗Y be a tensor product of separable Hilbert spaces X and Y. We establish norm estimates for the resolvent and operator-valued functions of the operator A=∑k=0mBk⊗Sk, where Bk (k=0,…,m) are bounded operators acting in Y, and S is a self-adjoint operator acting in X. By these estimates we investigate spectrum perturbations of A. The abstract results are applied to the nonself-adjoint differential operators in Hilbert and Euclidean spaces. Our main tool is a combined use of some properties of operators on tensor products of Hilbert spaces and the recent estimates for the norm of the resolvent of a nonself-adjoint operator.

We study fusion frame in tensor product of Hilbert spaces and discuss some of its properties.\,The resolution of the identity operator on a tensor product of Hilbert spaces is being discussed.\,An alternative dual of a fusion frame in tensor product of Hilbert spaces is also presented.

A partial classification into unitary equivalence classes of complete Boolean algebras of type I factors is given. Any complete atomic Boolean algebra of type I factors is unitarily equivalent to a discrete tensor product of type I factors. We establish a one-to-one correspondence between the unitary equivalence classes of complete nonatomic Boolean algebras of type I factors satisfying a certain condition, and the unitary equivalence classes of complete nonatomic Boolean algebras of projections. A continuous tensor product of Hilbert spaces is defined which is a generalization of the discrete infinite incomplete tensor product space defined by von Neumann. On a separable Hilbert space, any complete nonatomic Boolean algebra of type I factors satisfying the previously mentioned condition is unitarily equivalent to a continuous tensor product of type I factors. An application to the representations of the canonical commutation relations of quantum field theory is made.

Frame vectors in the tensor product of Hilbert spaces that accomplish phase retrieval can be characterized. In this article, we determine the conditions under which the tensor product of vectors may do phase retrieval. Given that tensor product of two frames always implies a frame in the tensor product of Hilbert spaces, we particularly concentrate on finding conditions for phase retrieval in the tensor product of Hilbert spaces.

Uncountably many mutually non-isomorphic product systems (that is, continuous tensor products of Hilbert spaces) of types II-0 and III are constructed by probabilistic means (random sets and off-white noises), answering four questions of W. Arveson. Results of math.FA/0001070, math.FA/0006165 are improved, and proofs are more readable.

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.

‎‎The tensor product is the fundemental ingredient for extending one-dimensional techniques of filtering and compression in signal preprocessing to higher dimensions‎. ‎Woven frames play ‎a crucial role in signal preprocessing and distributed data processing‎. Motivated by these facts, we have investigated the tensor product of woven frames and presented some of their properties. Besides, we have studied some effects of operators on woven frames in the tensor products of Hilbert spaces.

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.

Infinite tensor products of Banach spaces I

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Generalized Aizerman–Myshkis problem for abstract differential-delay equations

We classify all continuous tensor product systems of Hilbert spaces which are “infinitely divisible” in the sense that they have an associated logarithmic structure. These results are applied to the theory of E 0-semigroups to deduce that every E 0-semigroup possessing sufficiently many “decomposable” operators must be cocycle conjugate to a CCR flow (Theorem 12.3).A path space is an abstraction of the set of paths in a topological space, on which there is given an associative rule of concatenation. A metric path space is a pair (P, g) consisting of a path space P and a function g : P 2 → ℂ which behaves as if it were the logarithm of a multiplicative inner product. The logarithmic structures associated with infinitely divisible product systems are such objects. The preceding results are based on a classification of metric path spaces.Key words and phrasesvon Neumann algebrassemigroupsautomorphism groups

We propose a generalization of the RT and HRT holographic entanglement entropy formulas to spacetimes with asymptotically Minkowski as well as asymptotically AdS regions. We postulate that such spacetimes represent entangled states in a tensor product of Hilbert spaces, each corresponding to one asymptotic region. We show that our conjectured formula has the same general properties and passes the same general tests as the standard HRT formula. We provide further evidence for it by showing that in many cases the Minkowski asymptotic regions can be replaced by AdS ones using a domain wall. We illustrate the use of our formula by calculating entanglement entropies between asymptotic regions in Brill-Lindquist spacetimes, finding phase transitions similar to those known to occur in AdS. We construct networks of universes by gluing together Brill-Lindquist spaces along minimal surfaces. Finally, we discuss a variety of possible extensions and generalizations, including to universes with asymptotically de Sitter regions; in the latter case, we identify an ambiguity in the homology condition, leading to two different versions of the HRT formula which we call “orthodox” and “heterodox”, with different physical interpretations.

Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ be a tensor product of Hilbert spaces and let $\tau_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state $\rho_{0}$. Let $\tilde{\tau}_{0}$ denote the closest separable state to $\rho_{0}$ along the line segment from $I/N$ to $\rho_{0}$ where $I$ is the identity matrix. Following [pitrubmat] a witness $W_{0}$ detecting the entanglement of $\rho_{0}$ can be constructed in terms of $I, \tau_{0}$ and $\tilde{\tau}_{0}$. If representations of $\tau_{0}$ and $\tilde{\tau}_{0}$ as convex combinations of separable projections are known, then the entanglement of $\rho_{0}$ can be detected by local measurements. G\"{u}hne \textit{et. al.} in [bruss1] obtain the minimum number of measurement settings required for a class of two qubit states. We use our geometric approach to generalize their result to the corresponding two qudit case when $d$ is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, $\tau_{0}=\tilde{\tau}_{0}$. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which $\tau_{0} \neq \tilde{\tau}_{0}$ and we show our approach works for $n$ qubits. In [pitt] we elaborated on the role of a ``far face'' of the separable states relative to a bound entangled state $\rho_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times $I$ and a separable density $\mu_{0}$ on the far face from $\rho_{0}$. Up to a normalization this coincides with the witness obtained in [bruss1] for the particular example analyzed there.

Natural one-to-one and two-to-one homomorphisms from SO(3) into SU(2) are built conventionally, and the collection of qubits, is identified with a subgroup of SU(2). This construction is suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilbert spaces reflect entanglement properties as well, and in this way a notion of entanglement is realised in the tensor product of symmetry groups.