Hidden Tensor Structures

TL;DR

This paper reveals that even simple systems have hidden tensor structures that can be used for quantum computation and other applications.

Contribution

The discovery of infinitely many hidden tensor-like structures in separable Hilbert spaces is novel and unifies quantum and classical systems.

Findings

Single systems like a harmonic oscillator can be decomposed into arbitrary subsystems.

Hidden tensor structures enable quantum computation and violation of Bell’s inequalities.

These structures explain how classical devices can emulate quantum computers.

Abstract

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and classical signal analysis. Accordingly, systems as simple as a single one-dimensional harmonic oscillator, an infinite potential well, or a classical finite-amplitude signal of finite duration can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell’s inequalities, and formulation of universal quantum gates. Less standard quantum applications involve a distinction between position and hidden position. The hidden position can be accompanied by a hidden spin, even if the particle is spinless. Hidden degrees of freedom are, in many respects, analogous to modular variables. Moreover, it is shown that these hidden structures are at the roots of some well-known theoretical constructions, such as the Brandt–Greenberg multi-boson representation of creation–annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In the context of classical signal analysis, the discussed structures explain why it is possible to emulate a quantum computer by classical analog circuit devices.