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arXiv:2607.16396

Uncountably many inequivalent maximally entangled measurements for two qutrits

Elna Svegborn, Jef Pauwels, Nicolas Gisin, Alejandro Pozas-Kerstjens

quant-phmath-phSubmitted Jul 17, 2026• Announced Jul 21, 2026• 0 citations
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Abstract

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.

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5+11 pages, 1 figure, RevTeX 4.2

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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.
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.
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.
In this paper, we consider the local unitary classification of the class of qudit bipartite mixed states for which no information can be obtained locally. These states are represented by symmetrical density matrices in which both tracial states are maximally mixed. Interestingly, this symmetry facilitates the local unitary classification of two-qubit states. However, the same formalism fails in the case of systems of higher dimensions. We consider a broader set of states by introducing a family of qudit bipartite mixed states with maximally mixed marginals. For this family of states, we determine several constants which are in variant under local unitary transformations so can be used for entanglement classification. Finally, we consider the two-qutrit case and in particular, a two-parameter family of states for which the local unitary classification is complete. We relate this classification to known entanglement measures such as purity and negativity.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
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.
We introduce a method of investigating qutrit nonclassicality by translating qutrit operators to symmetric two-qubit operators. We show that this procedure sheds light on the discrepancy between maximal qutrit entanglement and maximal nonclassicality of qutrit correlations. Namely, we express Bell operators corresponding to qutrit Bell inequalities in terms of symmetric two-qubit operators and analyze the maximal quantum violation of a given Bell inequality from the qubit perspective. As an example we show that the two-qutrit Collins-Gisin-Linden-Massar-Popescu (CGLMP) Bell inequality can be seen as a combination of Mermin's and Clauser-Horne-Shimony-Holt CHSH) qubit Bell inequalities, and therefore the optimal state violating this combination differs from the one which corresponds to the maximally entangled state of two qutrits. In addition, we discuss the same problem for a three-qutrit inequality. We also demonstrate that the maximal quantum violation of the CGLMP inequality follows from complementarity of correlations.