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Journal of Physics A Mathematical and Theoretical

Entanglement in orthonormal bases for quantum systems

Jacob Chevalier Drori, Crispin H W Barnes

Published Dec 20, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ab64a7

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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.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.
We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are local unitary transformations of the original state. We prove that for bipartite states with a local dimension that is either 2, 4, or 8, every state corresponds to a basis. Via numerics, we strongly evidence the same conclusion for two qutrits and three qubits also. However, for some states of four qubits, we are unable to find a basis, leading us to conjecture that not all quantum states admit a corresponding measurement. Furthermore, we investigate whether there can exist a set of local unitaries that transform any state into a basis. While we show that such a state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued nnn-qubit states if and only if n=2,3n=2,3n=2,3, and that such constructions are impossible for any multipartite system of an odd local dimension. Our results suggest a rich relationship between entangled states and iso-entangled measurements with a strong dependence on both particle numbers and dimension.
We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of this issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type (p,q) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space. The phenomenon of entanglement is of central importance in the interpretation of quantum mechanics. Historically, entanglement was the focus of the famous Einstein-Podolsky-Rosen (EPR) paper [1], which suggested that standard quantum mechanics is an incomplete theory of physical reality. The central argument of the EPR paper was that more information about incompatible variables such as momentum and position could in principle be deduced about an entangled two particle quantum state than quantum mechanics permits, effectively giving information about each particle
We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of the issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type ( p , q ) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space.
A set of orthogonal pure states is an unextendible biseparable basis (UBB), which means that its complementary subspace contains only genuinely entangled states. UBBs thus serve as an effective tool for constructing genuinely entangled subspaces. If every state within such a subspace exhibits distillable entanglement across all bipartitions, it becomes particularly advantageous for applications in quantum information. In this paper, we mainly conduct research on the 4-qudit quantum systems, where the local dimension ddd is not less than 3. We present an approach for constructing UBB and prove that the UBB established in this way is strongly nonlocal. We build several genuinely entangled subspaces and demonstrate the distillability of the genuinely entangled subspaces across all bipartitions. In addition, we also describe the specific orthonormal basis for some genuinely entangled subspaces. These results will not only contribute to the development of quantum nonlocality theory, but also provide a crucial theoretical foundation for practical quantum information processing tasks.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.