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arXiv:0810.1923 · Physical Review Letters

Simulating Quantum Systems Using Real Hilbert Spaces

Matthew McKague, Michele Mosca, Nicolas Gisin

Published Jan 15, 2009• 89 citations
arXivHTMLPDF

Abstract

We develop a means of simulating the evolution and measurement of a multipartite quantum state under discrete or continuous evolution using another quantum system with states and operators lying in a real Hilbert space. This extends previous results which were unable to simulate local evolution and measurements with local operators and was limited to discrete evolution. We also detail applications to Bell inequalities and self-testing of the quantum apparatus.

DOI
10.1103/physrevlett.102.020505

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We develop a means of simulating the evolution and measurement of a multipartite quantum state under discrete or continuous evolution using another quantum system with states and operators lying in a real Hilbert space. This extends previous results which were unable to simulate local evolution and measurements with local operators and was limited to discrete evolution. We also detail applications to Bell inequalities and self-testing of the quantum apparatus.
A discrete quantum process is defined as a sequence of local states ρtρ_tρt​, t=0,1,2,...t=0,1,2,...t=0,1,2,..., satisfying certain conditions on an L2L_2L2​ Hilbert space HHH. If ρ=lim⁡ρtρ=\limρ_tρ=limρt​ exists, then ρρρ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure μμμ on the algebra of cylinder sets $\cscript$. We consider the problem of extending μμμ to other physically relevant sets in a systematic way. To this end we show that μμμ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable fff can be "quantized" to form a self-adjoint operator $\fhat$ on HHH. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of noncommuting projectors. We present local and nonlocal examples of violation of generalized Bell inequalities in phase space, which sense the negativity of the Wigner function.
Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces 1,2 . This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states 4 . Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
Bell nonlocality provides the foundation for device-independent (DI) certification of quantum devices. We introduce a Bell inequality capable of identifying genuine multipartite nonlocality (GMNL) in an arbitrary m-partite scenario with an arbitrary odd number of measurements per party. Since the multi-setting nature of this inequality precludes the use of Jordan's Lemma, we construct an analytical sum-of-squares (SOS) decomposition to obtain the optimal quantum violation without assuming any bound on the Hilbert space dimension. This, in turn, enables self-testing of the shared entangled state and the corresponding measurement observables, up to local isometries, whose existence we confirm using a swap-based certification scheme. In addition, we show that our framework enables the extraction of maximal global DI randomness (m bits) at the optimal quantum violation, thereby exceeding previous limitations in the GMNL regime. Finally, we demonstrate that the architecture of our inequality yields improved robustness to noise as the number of measurement settings grows, ensuring experimental feasibility.
Bell nonlocality provides a robust scalable route to the efficient certification of quantum states. Here, we introduce a general framework for constructing Bell inequalities tailored to the Zd\mathbb{Z}_dZd​ toric code for odd prime local dimensions. Selecting a suitable subset of stabilizer operators and mapping them to generalized measurement observables, we compute multipartite Bell expressions whose quantum maxima admit a sum-of-squares decomposition. We show that these inequalities are maximally violated by all states in the ground-state manifold of the Zd\mathbb{Z}_dZd​ toric code, and determine their classical (local) bounds through a combination of combinatorial tiling arguments and explicit optimization. As a concrete application, we analyze the case of d=3d=3d=3 and demonstrate that the maximal violation self-tests the full qutrit toric-code subspace, up to local isometries and complex conjugation. This constitutes, to our knowledge, the first-ever example of self-testing a qutrit subspace. Extending these constructions, we further present schemes to enhance the ratio of classical--quantum bounds and thus improve robustness to experimental imperfections. Our results establish a pathway toward device-independent certification of highly entangled topological quantum matter and provide new tools for validating qudit states in error-correcting codes and quantum simulation platforms.