Non-local Quantum Systems with Infinite Entanglement

Abstract

The study of quantum entanglement under quantum information has mostly been done in the mathematical model of tensor product of Hilbert spaces. In infinite dimensions, this model cannot capture all cases of non-local systems, and a more general model needs to be adopted; the most commonly used model here is the commuting operator model. We introduce another model, the C*-model, to describe non-local quantum systems in infinite dimensions. Instead of using Hilbert spaces to describe the states of a quantum system, a C*-algebra is used to describe the operators of a quantum system. The combination of two local quantum systems under this model is achieved by taking the tensor product of the two C*-algebras. The C*-model can be converted into the commuting operator model using the GNS representation theorem, and it is a generalization of the tensor product of the Hilbert spaces model.

One of the applications of infinite dimensional entanglement is the so-called linear system games. Linear system games are non-local games derived from linear systems of equations. We show that a linear system game has a perfect commuting operator strategy if and only if the equations have a potentially infinite-dimensional operator solution, and this is related to the properties of the representations of a certain group called the solution group of the system of equations. This understanding of linear system games is used as the foundation for the work by William Slofstra in which the author showed significant progress toward Tsirelson's problem.

Another application of infinite entanglement is the problem of embezzlement and self-embezzlement. Embezzlement is the task of locally creating an EPR pair using a shared state without changing the shared state; whereas self-embezzlement is the task of creating a copy of a shared entangled state locally without changing the original shared state. These tasks can be achieved if there exist infinitely many EPR pairs in the shared state, and thus would require a stronger model such as the commuting operator model to be used. We show the protocol for embezzlement and self-embezzlement using both the commuting operator model and the C*-model. An interesting property of embezzlement and self-embezzlement is that under the tensor product of the Hilbert spaces model, embezzlement is impossible to achieve, and self-embezzlement is impossible to approximate. These two problems show that the C*-model is indeed more powerful than the tensor product of the Hilbert spaces model in handling infinite-dimensional quantum systems.