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arXiv:2011.15078 · New Journal of Physics

Detecting entanglement can be more effective with inequivalent mutually unbiased bases

Beatrix C. Hiesmayr, D McNulty, S Baek, S Singha Roy, J Bae, D Chruściński

Published Aug 25, 2021• 22 citations
arXivHTMLPDF

Abstract

Abstract Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entanglement. Furthermore, we provide an efficient and systematic method to search for inequivalent MUBs and show that such sets occur regularly within the Heisenberg–Weyl MUBs, as the dimension increases. Our findings are particularly useful for experimentalists since they demonstrate that a clever selection of MUBs allows for entanglement detection with fewer measurements.

DOI
10.1088/1367-2630/ac20ea

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Abstract Mutually unbiased bases (MUBs) provide a standard tool in the verification of quantum states, especially when harnessing a complete set for optimal quantum state tomography. In this work, we investigate the detection of entanglement via inequivalent sets of MUBs, with a particular focus on unextendible MUBs. These are bases for which an additional unbiased basis cannot be constructed and, consequently, are unsuitable for quantum state verification. Here, we show that unextendible MUBs, as well as other inequivalent sets in higher dimensions, can be more effective in the verification of entanglement. Furthermore, we provide an efficient and systematic method to search for inequivalent MUBs and show that such sets occur regularly within the Heisenberg–Weyl MUBs, as the dimension increases. Our findings are particularly useful for experimentalists since they demonstrate that a clever selection of MUBs allows for entanglement detection with fewer measurements.
Abstract From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite ( d × d ) -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than d / 2 + 1 . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.
The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.
Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions 333, 444, and 555 demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension 666 where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.
We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion for bipartite systems with equal dimensions is based on measurements from subsets of a quantum 2-design, e.g. mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for quantum state tomography. Secondly, our method is twice as effective as standard witnesses in the sense that both upper and lower bounds can be derived. Thirdly, the scheme can be readily applied to measurement-device-independent scenarios.
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions (d=p2d=p^{2}d=p2) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of p+1p+1p+1 MUBs that saturate both a Shannon (H1H_{1}H1​) and a collision (H2H_{2}H2​) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.