Related papers: On quantum Strassen's theorem
We consider a partial trace transformation which maps a multipartite quantum state to collection of local density matrices. We call this collection a mean field state. The necessary and sufficient conditions under which a mean field state…
Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of…
The main purpose of this paper is pedagogical. Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations…
In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x…
Given a semiring with a preorder subject to certain conditions, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is a compact Hausdorff space together with a map from the semiring to the ring of continuous…
Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if there are two different ways to compute an amplitude the two answers must agree. This…
Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this article, we present a thorough description of this function and analyze the…
In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result…
In the context of the Oppenheim-Horodecki paradigm of nonclassical correlation, a bipartite quantum state is (properly) classically correlated if and only if it is represented by a density matrix having a product eigenbasis. On the basis of…
Analytic expressions for the probability density distribution of the linear entropy and the purity are derived for bipartite pure random quantum states. The explicit distributions for a state belonging to a product of Hilbert spaces of…
The semi-tensor product (STP) of matrices which was proposed by Daizhan Cheng in 2001 [2], is a natural generalization of the standard matrix product and well defined at every two finite-dimensional matrices. In 2016, Cheng proposed a new…
Density matrices are the most general descriptions of quantum states, covering both pure and mixed states. Positive semidefiniteness is a physical requirement of density matrices, imposing nonnegative probabilities of measuring physical…
Quantum mechanics can emerge from classical statistics. A typical quantum system describes an isolated subsystem of a classical statistical ensemble with infinitely many classical states. The state of this subsystem can be characterized by…
We generalize, improve and unify theorems of Rumin, and Maassen--Uffink about classical entropies associated to quantum density matrices. These theorems refer to the classical entropies of the diagonals of a density matrix in two different…
A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schr\"odinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the…
We derive upper bounds for Hilbert-Schmidt's quantum coherence of general states of a $d$-level quantum system, a qudit, in terms of its incoherent uncertainty, with the latter quantified using the linear and von Neumann's entropies of the…
We develop two general approaches to characterising the manipulation of quantum states by means of probabilistic protocols constrained by the limitations of some quantum resource theory. First, we give a general necessary condition for the…
Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fr\'{e}chet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a…
For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain…
By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent $2$ between a quantum density operator and a classical…