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arXiv:1708.05181 · Physical Review A

Entanglement witnesses from mutually unbiased bases

Dariusz Chruściński, Gniewomir Sarbicki, Filip Wudarski

Published Mar 16, 2018• 32 citations
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Abstract

We provide a class of entanglement witnesses constructed in terms of mutually unbiased bases (MUBs). This construction reproduces many well-known examples such as the celebrated reduction map and the Choi map together with its generalizations. We illustrate our construction by a detailed analysis of the three-dimensional case: In this case, one obtains a family of entanglement witnesses parametrized by an $L$-dimensional torus ($L=2,3,4$ being a number of MUBs used in the construction).

DOI
10.1103/physreva.97.032318

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arXiv:1911.08761•Nov 20, 2019•1 citation
We provide a class of entanglement witnesses constructed in terms of mutually unbiased bases (MUBs). This construction reproduces many well-known examples such as the celebrated reduction map and the Choi map together with its generalizations. We illustrate our construction by a detailed analysis of the three-dimensional case: In this case, one obtains a family of entanglement witnesses parametrized by an LLL-dimensional torus (L=2,3,4L=2,3,4L=2,3,4 being a number of MUBs used in the construction).
In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives no information about the outcomes of measurements in another basis. Here we relax the no-information condition by allowing the ddd outcomes to be predicted according to a predefined probability distribution $q=({q}_{0},...,{q}_{d\ensuremath{-}1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of mutually equibiased bases (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For d=2,3d=2,3d=2,3, we derive a set of d+1d+1d+1 MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution qqq, leading, for d=3d=3d=3, to the restriction $1/3\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1/2$, where $\ensuremath{\mu}={\ensuremath{\sum}}_{k=0}^{2}{q}_{k}^{2}$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $\ensuremath{\mu}$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\ifmmode\times\else\texttimes\fi{}3$ system. Our results reveal that not all bases in a set of LLL MEBs can contribute to the entanglement detection. A constraint, dependent on the probability distribution qqq, limits the number of MEBs that can be employed.
Certifying entanglement is an important step in the development of many quantum technologies, especially for higher-dimensional systems, where entanglement promises increased capabilities for quantum communication and computation. A key feature distinguishing entanglement from classical correlations is the occurrence of correlations for complementary measurement bases. In particular, mutually unbiased bases (MUBs) are a paradigmatic example that is well-understood and routinely employed for entanglement certification. However, implementing unbiased measurements exactly is challenging and not generically possible for all physical platforms. Here, we extend the entanglement-certification toolbox from correlations in MUBs to arbitrary bases. This practically significant simplification paves the way for efficient characterizations of high-dimensional entanglement in a wide range of physical systems. Furthermore, we introduce a simple three-MUBs construction for all dimensions without using the Wootters-Fields construction, potentially simplifying experimental requirements when measurements in more than two MUBs are needed, especially in high-dimensional settings.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.
We construct two mutually unbiased bases by maximally entangled states (MUMEBsss) in C2⊗C3\mathbb{C}^{2}\otimes \mathbb{C}^{3}C2⊗C3. This is the first example of MUMEBsss in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ when d∤d′d\nmid d'd∤d′, namely d′d'd′ is not divisible by ddd. We show that they cannot be extended to four MUBs in C6\mathbb{C}^6C6. We propose a recursive construction of mutually unbiased bases formed by special entangled states with a fixed Schmidt number kkk (MUSEBkkks). It shows that min⁡{t1,t2}\min \{t_{1},t_{2}\}min{t1​,t2​} MUSEBk1k2k_{1}k_{2}k1​k2​s in Cpd⊗Cqd′\mathbb{C}^{pd}\otimes \mathbb{C}^{qd'}Cpd⊗Cqd′ can be constructed from t1t_{1}t1​ MUSEBk1k_{1}k1​s in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ and t2t_{2}t2​ MUSEBk2k_{2}k2​s in Cp⊗Cq\mathbb{C}^{p}\otimes \mathbb{C}^{q}Cp⊗Cq for any d,d′,p,qd,d',p,qd,d′,p,q. Further, we show that three MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for any d,d′d,d'd,d′ with d∣d′d\mid d'd∣d′, and two MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for infinitely many d,d′d,d'd,d′ with d∤d′d\nmid d'd∤d′.