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arXiv:1507.00213 · Physical Review Letters

Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

Jamie Sikora, Antonios Varvitsiotis, Zhaohui Wei

Published Aug 4, 2016• 44 citations
arXivHTMLPDF

Abstract

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.

DOI
10.1103/physrevlett.117.060401

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Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.
We introduce novel upper bounds on the Hilbert space dimensions required to realize quantum correlations in Bell scenarios. We start by considering bipartite cases wherein one of the two parties has two settings and two outcomes. Regardless of the number of measurements and outcomes of the other party, the Hilbert space dimension of the first party can be limited to two while still achieving all convexly extremal quantum correlations. We then leverage Schmidt decomposition to show that the remaining party can losslessly also be restricted to a qubit Hilbert space. We then extend this idea to multipartite scenarios. We also adapt our results to provide upper bounds of local Hilbert space dimensions to achieve any quantum correlation, including convexly non-extremal correlations, by utilizing Caratheodory's theorem. Finally, we generalize our results to nonstandard Bell scenarios with communication. Taken together, our results fill in several previously unresolved aspects of the problem of determining sufficient Hilbert space dimensionality, expanding the collection of scenarios for which finite-dimensional quantum systems are known to be sufficient to reproduce any quantum correlation.
Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two different ways by constructing explicit examples for every dimension ddd, which demonstrates that there exist bipartite correlations that necessitate ddd-dimensional local quantum systems in order to generate them. We also show that at least ten two-outcome measurements must be carried out by the two parties altogether so as to generate bipartite joint correlations not achievable by two-dimensional local systems. The smallest explicit example we found involves 11 settings.
Resource theory is applied to quantify the quantum correlation of a bipartite state and a computable measure is proposed. Since this measure is based on quantum coherence, we present another possible physical meaning for quantum correlation, i.e., the minimum quantum coherence achieved under local unitary transformations. This measure satisfies the basic requirements for quantifying quantum correlation and coincides with concurrence for pure states. Since no optimization is involved in the final definition, this measure is easy to compute irrespective of the Hilbert space dimension of the bipartite state.
We analyze the structure of the space of temporal correlations generated by quantum systems. We show that the temporal correlation space under dimension constraints can be nonconvex. For the general case, we provide the necessary and sufficient dimension of a quantum system needed to generate a convex correlation space for a given scenario. We further prove that this dimension coincides with the dimension necessary to generate any point in the temporal correlation polytope. As an application of our results, we derive nonlinear inequalities to witness the nonconvexity for qubits and qutrits in the simplest scenario, and present an algorithm which can help to find the minimum for a certain type of nonlinear expressions under dimension constraints.
We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.
We describe a simple method to derive high performance semidefinite programing relaxations for optimizations over complex and real operator algebras in finite dimensional Hilbert spaces. The method is very flexible, easy to program, and allows the user to assess the behavior of finite dimensional quantum systems in a number of interesting setups. We use this method to bound the strength of quantum nonlocality in Bell scenarios where the dimension of the parties is bounded from above. We derive new results in quantum communication complexity and prove the soundness of the prepare-and-measure dimension witnesses introduced in Gallego et al., Phys. Rev. Lett. 105, 230501 (2010). Finally, we propose a new dimension witness that can distinguish between classical, real, and complex two-level systems.