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Journal of Mathematical Physics

Entanglement and the three-dimensionality of the Bloch ball

Lluís Masanes, Markus P. Müller, David Pérez-Garcı́a, Remigiusz Augusiak

Published Dec 1, 2014• 22 citations
Publisher

Abstract

We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.

DOI
10.1063/1.4903510

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We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.
We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this we impose two very natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories interacting dynamics is impossible. This result reveals that "existence of entanglement" is the requirement with minimal logical content which singles out quantum theory from our family of theories.
We explore the existence of entangling dynamics in a large family of theories which contains quantum theory as a special case. We classify all continuouslyreversible and locally-tomographic theories for bipartite systems where each subsystem has a state space with the geometry of a Euclidean ball (like the Bloch ball of a qubit but with dimension not necessarily equal to three). We show that the only theory in this family which has interacting dynamics is quantum theory, and all the other theories do not allow for entanglement nor violation of Bell inequalities. This suggests that, once dynamics is taken into account, entanglement and violation of Bell inequalities are very singular phenomena.
It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this paper, we suggest an information-theoretic analysis of this relationship, by proving a particular mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that carry 'minimal amounts of direction information', interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d = 3 and quantum theory on two qubits (including entanglement and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.
Inspired by its fundamental importance in quantum mechanics, we define and study the notion of entanglement for abstract physical theories, investigating its profound connection with the concept of superposition. We adopt the formalism of general probabilistic theories (GPTs), encompassing all physical models whose predictive power obeys minimal requirements. Examples include classical theories, which do not exhibit superposition and whose state space has the shape of a simplex, quantum mechanics, as well as more exotic models such as Popescu-Rohrlich boxes. We call two GPTs entangleable if their composite admits either entangled states or entangled measurements, and conjecture that any two non-classical theories are in fact entangleable. We present substantial evidence towards this conjecture by proving it (1) for the simplest case of 3-dimensional theories; (2) when the local state spaces are discrete, which covers foundationally relevant cases; (3) when one of the local theories is quantum mechanics. Furthermore, (4) we envision the existence of a quantitative relation between local non-classicality and global entangleability, explicitly describing it in the geometrically natural case where the local state spaces are centrally symmetric.
Abstract The computational efficiency of quantum mechanics can be characterized in terms of the qubit circuit model, which is defined by a few simple properties: each computational gate is a reversible transformation in a connected matrix group; single wires carry quantum bits, i.e. states of a three-dimensional Bloch ball; states on two or more wires are uniquely determined by local measurement statistics and their correlations. In this paper, we ask whether other types of computation are possible if we relax one of those characteristics (and keep all others), namely, if we allow wires to be described by d-dimensional Bloch balls, where d is different from three. Theories of this kind have previously been proposed as possible generalizations of quantum physics, and it has been conjectured that some of them allow for interesting multipartite reversible transformations that cannot be realized within quantum theory. However, here we show that all such potential beyond-quantum models of computation are trivial: if d is not three, then the set of reversible transformations consists entirely of single-bit gates, and not even classical computation is possible. In this sense, qubit quantum computation is an island in theoryspace.
Abstract Entanglement is a defining feature of quantum mechanics, producing correlations between the results of spacelike-separated measurements which cannot be explained by any local, single-world theory of nature. In this paper we employ a rigorous mathematical approach to explore the classification of bases for quantum systems by their entanglement properties. For bipartite systems, a basis is said to have type if it contains p entangled states and q separable states. Our main result is to prove a long-standing conjecture of Eakins and Jaroszkiewicz stating that no type orthonormal (o.n.) basis exists: an o.n. basis containing an entangled state is guaranteed to contain at least one other. Furthermore, we prove that o.n. bases of all other types do exist, before moving on to discuss the multipartite case. Our results show that there are fundamental constraints on the emergence of nonlocality in general orthogonal measurement.