Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPerfect Commuting-Operator Strategies for Linear System Games
104
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Linear system games are a generalization of Mermins magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bobs measurement operators act on a joint Hilbert space, and Alices operators must commute with Bobs operators. We show that perfect strategies in this model correspond to possibly-infinite-dimensional operator solutions of the non-commutative equations. The proof is based around a finitely-presented group associated to the linear system which arises from the non-commutative equations.
rate research
Read More
We unify and consolidate various results about non-signall-ing games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are {it reflexive games,} which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce {it imitation games,} in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and {it unique} games. We associate a C*-algebra $C^*(mathcal{G})$ to any imitation game $mathcal{G}$, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of $mathcal{G}$ can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call {it mirror games,} and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.
There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p$ a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of $n$-simplices, i. e., pairs of $n$-simplices which are emph{mutually inscribed and circumscribed} to each other. For group elements representing an $n$-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension $n$ of the associated polar space in group theoretic terms. Any Moebius pair of $n$-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$.
Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosmans complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.
We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue distributions with the help of free probability theory. Similarly, we obtain the large system limit for several quantities of interest in quantum information theory, such as the sharpness, the noise content, and the probability range. Finally, we study different compatibility criteria, and we compare them for generic POVMs.
We consider the process of flux insertion for ground states of almost local commuting projector Hamiltonians in two spatial dimensions. In the case of finite dimensional local Hilbert spaces, we prove that this process cannot pump any charge and we conclude that the Hall conductance must vanish.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
