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arXiv:2301.13285 · Physical Review A

Correspondence between entangled states and entangled bases under local transformations

Florian Pimpel, Martin J. Renner, Armin Tavakoli

Published Aug 25, 2023• 9 citations
arXivHTMLPDF

Abstract

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 $n$-qubit states if and only if $n=2,3$, 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.

DOI
10.1103/physreva.108.022220

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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.
Every two-qubit measurement basis composed of maximally entangled eigenstates can be transformed into the Bell basis via local unitary operations. For higher dimensions, in contrast, there exist inequivalent bases composed of maximally entangled eigenstates. Here, we provide a single-parameter family of two-qutrit maximally entangled measurement bases, and demonstrate that none of the bases are equivalent to each other under local unitaries. These bases are constructed from the continuous family of symmetric informationally complete sets of states in dimension three. By studying the local unitary bases that generate the family of two-qutrit maximally entangled measurement bases, we construct the first examples of wild error bases in the smallest dimension where these can exist. Finally, we discuss how distinct measurements in the family lead to differences in performance in several scenarios relevant in quantum information.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
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 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.
Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on Cd1⊗Cd2\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}Cd1​⊗Cd2​ for any nonprime dimension d=d1×d2d=d_1\times d_2d=d1​×d2​. Moreover, based on the structure of the orthogonal product basis in C2⊗Cn\mathbb{C}^2\otimes \mathbb{C}^nC2⊗Cn, we obtain another construction of absolutely entangled set with 2n+12n+12n+1 elements in C2⊗Cn\mathbb{C}^2\otimes \mathbb{C}^nC2⊗Cn.
It is known that there exist sets of pure orthogonal product states which cannot be perfectly distinguished by local operations and classical communication (LOCC). Such sets are nonlocal sets which exhibit nonlocality without entanglement. These nonlocal sets can be completable or uncompletable. In this work both completable and uncompletable small nonlocal sets of multipartite orthogonal product states are constructed. Apart from nonlocality, these sets have other interesting properties. In particular, the completable sets lead to the construction of a class of complete orthogonal product bases with the property that if such a basis is given then no state can be eliminated from that basis by performing orthogonality-preserving measurements. On the other hand, an uncompletable set of the present kind contains several Shifts unextendible product bases (UPBs) that belong to qubit subspaces. Identifying these subspace UPBs, it is possible to obtain a class of high-dimensional multipartite bound entangled states. Finally, it is shown that a two-qubit maximally entangled Bell state shared between any two parties is sufficient as a resource to distinguish the states of any completable set (of the above kind) perfectly by LOCC. This constitutes an example where the amount of entanglement, sufficient to accomplish the aforesaid task, depends neither on the dimension of the individual subsystems nor on the number of parties.