Constructions on Real Approximate Mutually Unbiased Bases

Yang, Ming; Zhang, Aixian; Wen, Jin‐Kun; Feng, Keqin

doi:10.48550/arxiv.2110.06665

Cited by 2 publications

(3 citation statements)

References 11 publications

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“…Nevertheless, not all overlaps are the same, hence the eigenvectors corresponding to q ̸ = q ′ are not MUB. However, the modulus square of their overlap is bounded by twice the inverse of the system's dimension, therefore in large Hlibert spaces the corresponding eigenbases are AMUB [6][7][8][9][10][11]. It is natural to ask what happens for non-prime d. We observed that for some choices of q ̸ = q ′ the overlaps between the corresponding bases are bounded by 2/D as well, however in general this overlap exceeds 2/D.…”

Section: Resultsmentioning

confidence: 90%

“…In realistic situations perfect MUB are hard to implement. This lead to the concept of approximate MUB (AMUB) [6,7] in complex [8,9] and in real spaces [10,11]. In case of AMUB one relaxes the condition (1) and instead looks for bases whose vectors obey…”

Section: Introductionmentioning

confidence: 99%

“…where c is a real positive constant independent of D [9]. Despite significant impact on our understanding of nature, majority of research on MUB and AMUB is purely mathematical.…”

Section: Introductionmentioning

confidence: 99%

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Complementarity in quantum walks

Grudka

Kurzyński

Polak

et al. 2023

J. Phys. A: Math. Theor.

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The eigenbases of two quantum observables, $\{|a_i\rangle\}_{i=1}^D$ and $\{|b_j\rangle\}_{j=1}^D$, form Mutually Unbiassed Bases (MUB) if $|\langle a_i |b_j \rangle| = 1/\sqrt{D}$ for all $i$ and $j$. In realistic situations MUB are hard to obtain and one looks for Approximate MUB (AMUB), in which case the corresponding eigenbases obey $|\langle a_i |b_j \rangle| \leq c/\sqrt{D}$, where $c$ is some positive constant independent of $D$. In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks on $d$-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter $q$. We solve the model analytically and observe that for prime $d$ the eigenvectors of two quantum walk evolution operators form AMUB. Namely, if $d$ is prime the corresponding eigenvectors of the evolution operators, that act in the $D$-dimensional Hilbert space ($D=2d$), obey $|\langle v_q|v'_{q'} \rangle| \leq \sqrt{2}/\sqrt{D}$ for $q\neq q'$ and for all $|v_q\rangle$ and $|v'_{q'}\rangle$. Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.

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