Quantum state space dimension as a quantum resource

Abstract

We argue that the dimensionality of the space of quantum systems' states should be considered as a legitimate resource for quantum information tasks. The assertion is supported by the fact that quantum states with discord-like capacities can be obtained from classically correlated states in spaces of dimension large enough. We illustrate things with some simple examples that justify our claim.Keywords: Hilbert space, quantum information, quantum discordThe Hilbert-space dimension has been related to physical resources for different physical systems, playing a fundamental role in quantum computation. Basically, the idea is that "if you want to avoid supplying an amount of some physical resource that grows exponentially with problem's size, the computer must be made up of parts whose number grows nearly linearly with the number of qubits required in an equivalent quantum computer. This thus becomes a fundamental requirement for a system to be a scalable quantum computer" [1]. Moreover, some recent results show that quantum dimensionality could be regarded as a physical entity. For example, Brunner et al. defined what they call 'dimension witnesses': observable quantities to estimate the minimum dimension of a given system state-space consistent with a number of measured correlations [2][3][4]. In the same spirit, Wehner et al. found a lower bound that gives a fundamental limit on the dimension of the state to implement certain measurement strategies [5]. Here, we propose to consider the dimension of the Hilbert space as a legitimate resource for quantum information processing. Our main argument lies in the observation, due to Li and Luo [6], that quantum separable states can be obtained from reductions of classically correlated ones. Although some authors have suggested the possibility of understanding the size of the Hilbert space as a resource by itself [7], the assertion that it is a quantum-better-than-classical resource was never technically analyzed, as far as we know.Under the discord paradigm, a classically correlated state (or simply, a classical state) is one that is information-wise accessible to local observers. Given a discord-like measure δ and a classical state σ AB of a composite system A + B, one knows that δ(σ AB ) = 0. The following theorem, due to Li and Luo, demonstrates a notable relation between separable states and classical states [6]. Figure 1. Every quantum state that is separable (within a given bipartition of the full system) is, in a formal sense, the reduction of a classical state of a system defined over a larger state-space (and preserves the original bipartition). Here, the state σ AB of the composite A + B should be regarded as a classical extension of the separable ρ ab .The proof is given by Li and Luo in Ref. [6]. The next result follows directly from the above theorem: Proposition. Any quantum task carried out using unentangled states can also be undertaken using classically correlated states.Indeed, if a quantum task needs appealing to a given un-entangled state ρ ab , then ...

Identifying nonconvexity in the sets of limited-dimension quantum correlations

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The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

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Repercusiones cuánticas de los estados clásicamente correlacionados

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“…¿Por qué esta extensión no es asequible por el método de Li y Luo? Para responderlo, basta con notar que = 4, de modo que el ALL demanda un sistema auxiliar en C 4…”

“…74, es decir, que incluso los estados CC pueden tener reducciones QC, fenómeno que no es capturado por las medidas usuales del tipo discord. Para un dado estado CC, sin embargo, toda reducción es separable para la misma bipartición [1,3,4,34]. Como consecuencia, la CP de un dado estado CC está acotado superiormente por aquel de los estados separables.…”