For finite-dimensional operator systems $mathcal{S}_{mathsf T}$, ${mathsf T} in B({mathcal H})^d$, we show that the local lifting property and $1$-exactness of $mathcal{S}_{mathsf T}$ may be characterized by measurements of the disparity between the
matrix range $mathcal{W}({mathsf T})$ and the minimal/maximal matrix convex sets over its individual levels. We then examine these concepts from the point of view of free spectrahedra, direct sums of operator systems, and products of matrix convex sets.
We introduce a new class of non-local games, and corresponding densities, which we call bisynchronous. Bisynchronous games are a subclass of synchronous games and exhibit many interesting symmetries when the algebra of the game is considered. We deve
lop a close connection between these non-local games and the theory of quantum groups which recently surfaced in studies of graph isomorphism games. When the number of inputs is equal to the number of outputs, we prove that a bisynchronous density arises from a trace on the quantum permutation group. Each bisynchronous density gives rise to a completely positive map and we prove that these maps are factorizable maps.
We consider the tensor product of the completely depolarising channel on $dtimes d$ matrices with the map of Schur multiplication by a $k times k$ correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) uni
tary channel. When $d=1$, this recovers a result of OMeara and Pereira, and for larger $d$ is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras.
Let $A = (A_1, dots, A_m)$ be an $m$-tuple of elements of a unital $C$*-algebra ${cal A}$ and let $M_q$ denote the set of $q times q$ complex matrices. The joint $q$-matricial range $W^q(A)$ is the set of $(B_1, dots, B_m) in M_q^m$ such that $B_j =
Phi(A_j)$ for some unital completely positive linear map $Phi: {cal A} rightarrow M_q$. When ${cal A}= B(H)$, where $B(H)$ is the algebra of bounded linear operators on the Hilbert space $H$, the {bf joint spatial $q$-matricial range} $W^q_s(A)$ of $A$ is the set of $(B_1, dots, B_m) in M_q^m$ for which there is a $q$-dimensional $V$ of $H$ such that $B_j$ is a compression of $A_j$ to $V$ for $j=1,dots, m$. Suppose $K(H)$ is the set of compact operators in $B(H)$. The joint essential spatial $q$-matricial range is defined as $$W_{ess}^q(A) = cap { {bf cl}(W_s^q(A_1+K_1, dots, A_m+K_m)): K_1, dots, K_m in K(H) },$$ where ${bf cl}$ denotes the closure. Let $pi$ be the canonical surjection from $B(H)$ to the Calkin algebra $B(H)/K(H)$. We prove that $W_{ess}^q(A) =W^q(pi(A) $, where $pi(A) = (pi(A_1), dots, pi(A_m))$. Furthermore, for any positive integer $N$, we prove that there are self-adjoint compact operators $K_1, dots, K_m$ such that $${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A) quad hbox{ for all } q in {1, dots, N}.$$ These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the $m=1$ case, and also generalize a result of M{u}ller obtained in case $m ge 1$ and $q=1$. Furthermore, if $W_{ess}^1({bf A}) $ is a simplex in ${mathbb R}^m$, then we prove that there are self-adjoint $K_1, dots, K_m in K(H)$ such that ${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A)$ for all positive integers $q$.
We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel $mathfrak Z: M_d to M_d$ defined by begin{align*} mathfrak Z(X) = frac{1}{d+1}(X+text{Tr}(X)mathbb I
_d) end{align*} is $d^2$ if and only if there exists a symmetric informationally-complete POVM in dimension $d$.
We analyze certain class of linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the ten
sor product space. If a linear map is entanglement breaking after finite iterations, we say the map has a finite index of separability. In particular we show that every unital PPT-channel becomes entanglement breaking after a finite number of iterations. It turns out that the class of unital channels that have finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. This analysis is motivated by the PPT-squared conjecture made by M. Christandl that says every PPT channel, when composed with itself, becomes entanglement breaking.
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandls conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace pr
eserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandls conjecture holds for these families.
We generalise some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel require
s to realise a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalised minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lovasz theta number.
We introduce a game related to the $I_{3322}$ game and analyze a constrained value function for this game over various families of synchronous quantum probability densities.
Recently, W. Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum
approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes embedding problem in terms of synchronous correlations.
