Choice of mutually unbiased bases and outcome labelling affects measurement outcome secrecy Mirdit Doda,1, 2 Matej Pivoluska,1, 3 and Martin Plesch1, 3 1Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia 2Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 3Institute of Computer Science, Masaryk University, 602 00 Brno, Czech Republic (Dated: March 11, 2021) Mutually unbiased bases (MUBs) are a crucial ingredient for many protocols in quantum infor- mation processing. Measurements performed in these bases are unbiased to the maximally possible extent, which is used to prove randomness or secrecy of measurement results. In this work we show that certain properties of sets of MUBs crucially depend on their specific choice, including, somewhat surprisingly, measurement outcome labelling. If measurements are chosen in a coherent way, the secrecy of the result can be completely lost for specific sets of MUB measurements, while partially retained for others. This can potentially impact a broad spectrum of applications, where MUBs are utilized. I. INTRODUCTION The natural question of the number of unbiased bases in a given dimension d turned out to be unexpectedly One of the defining features of quantum mechanics is complicated. While the answer is rather simple for qubits the impossibility to simultaneously measure a certain set { there are three pairwise mutually unbiased bases, de- of physical quantities. This fact led to the definition of fined as eigenvectors of Pauli σx; σy; σz operators up the famous Heisenberg uncertainty principle [1] or under- to unitary equivalencies, in general, the construction of standing of the quantum model of hydrogen atom [2]. If a MUBs is a very difficult task. It is known that the num- simultaneous measurement of two quantities is not possi- ber of MUBs has to be smaller than d + 1 for any di- ble, or, in other words, if a measurement of one quantity mension and the constructions of d + 1 MUBs are known r influences the expectation of the other measurement, we for d = p , where p is a prime. However, for non-prime- call these two measurements incompatible. In this con- power d only the trivial tensor product construction is text a very natural question arises { how much incom- known. patible a pair of measurements can be? The answer to Fortunately, in many applications one needs to use only this question is simple { for any quantum system, one k ≤ d+1 MUB measurements. Clearly, there are different can find a pair of measurements where irrespective of the ways to pick the subset of k out of all MUBs. In fact, it starting state of the system, after performing one of the is known that different sets of MUBs are not necessarily measurements the result of the other one is completely equivalent under different mathematical operations, such random. as global unitary operations, changing individual vector A straightforward generalization is at hand { can one phases, relabelling of outcomes, relabelling of moments or form a larger set of measurements that are pairwise fully introducing complex conjugation [27]. This mathemati- incompatible? Here again one can answer affirmatively { cal inequivalence is however irrelevant in many practical for each system one can find at least three such measure- applications where just satisfying the defining property ments and the size of this set depends on the dimension (1) is required for the task. of the system. More interestingly, it was recently shown that differ- In order to tackle with these questions more formally, ent subsets of MUBs of can be inequivalent operationally the notion of mutually unbiased bases (MUBs) [3{6] was as well. For example, MUBs turn out to be an optimal arXiv:2006.08226v2 [quant-ph] 10 Mar 2021 introduced. Two d-dimensional bases fj iigi=0;:::;d−1 strategy in a communication task called quantum ran- and fj'jigj=0;:::;d−1 corresponding to two full projective dom access coding (QRAC) [28]. measurements are mutually unbiased, when In [29] it was shown that in a certain variant of QRAC, different subsets of k out of d + 1 MUBs lead to differ- 1 8i; j : jh ij'jij = p : (1) ent strategies with different average success rates. More d recently, it was shown that different subsets of k out of Due to their properties, mutually unbiased bases have d + 1 MUBs behave differently under a measure called become an important cornerstone of contemporary quan- incompatibility robustness [30]. Last but not least, very tum information processing [7]. They are being used specific MUBs are required to obtain Bell inequalities for quantum tomography [4,6], uncertainty relations [22], which are maximally violated by maximally entan- [5,8,9], quantum key distribution [10{13], quantum er- gled states and MUBs. ror correction [14], as well as for witnessing entanglement The full definition of a measurement consist of specify- [15{21], design of Bell inequalities [22, 23] and more gen- ing the basis as a set of states and labelling these states. eral forms of quantum correlations [24{26]. Two measurements consisting of the same set of states 2 are in principle different, even if they measure the same y property and their results can be classically transformed ρB a at any later stage. From the experimental and opera- Ui tional point of view it makes sense to distinguish be- tween different measurements that only differ in labelling (we call this a classical difference) and two measurements that differ in the states per se (quantum difference). One can then naturally ask, to what extent the properties of ρC Mb b MUBs do change if one only makes a classical change in them. In other words, do the properties of the sub- sets change by simple re-labelling of their vectors? In FIG. 1. Guessing game description. Alice measures the probe this work, we affirmatively answer this question by intro- state ρB with one out of d possible measurements. Alice's ducing a quantum information task called guessing game. measurements choice is implemented coherently, via a con- trolled unitary Pd−1 U y ⊗ jiihij, where U y maps the basis There a subset of d out of d+1 MUBs is used to hide and i=0 i i guess information between two parties. We show that vectors of the i-th basis onto the computational basis. Alice then measures in the computational basis and her outcome is this simple choice of removing a single MUB from the denoted a. Bob's goal is to guess Alice's outcome by prepar- full set critically affects achievable results in the game. ing a probe state ρB and an optimal measurement described Even more interestingly, for a suitable chosen subset of d−1 by POVM elements fMbgb=0 , through which he obtains his d out of d + 1 MUBs, we observe the full spectrum of guess b. Bob wins when b = a. In the classical coin case, results { perfect guessing and maximal hiding { just by the control state ρC is fully mixed , and in the quantum coin relabelling the measurement outcomes. case, ρC is a superposition of computational basis vectors. II. RESULTS shown that for qubits (d = 2), in the quantum coin sce- nario Bob can guess Alice's outcome with certainty. In The incompatibility of measurements can be demon- contrast, this was not the case for higher dimensions. strated and examined with the help of a very simple They have concluded that in case of two measurements quantum game, studied in [31, 32]. Here Alice realizes the control state is always a two dimensional state and it one of m possible measurements on a d-dimensional sys- is impossible to use it to determine a higher dimensional tem and records the result a of this measurement. The outcome. task of Bob is to guess this result using the following In [32] we have further analyzed the guessing game strategy: first, he prepares the state for Alice to be mea- with the quantum coin and we have shown that for sured and second, he receives information about which qubits, with any number of measurements (independent measurement was performed (see the next section for the on their level of compatibility) it is always possible for full definition of the guessing game). Bob to obtain the result of Alice with probability 1. In If the game is described by classical physics, a pure contrast, for higher dimensions this is not the case, so state has a determined outcome for all possible mea- even if Bob receives a large enough control state, he will surements. Therefore, trivially, Bob can prepare a state not be able to guess the result perfectly for a specific set which leads to a deterministic outcome irrespective on of MUBs chosen by Alice. measurement performed by Alice. Here we analyze the problem further. We fix the num- One can make the scenario partially quantum, by mak- ber of measurements to m = d, which will make the size ing Bob's probe state as well as the measurements quan- of the measurement outcomes alphabet equal to the di- tum, but keep the information about the measurement mension of the control state available to Bob. First we chosen by Alice classical { we call this a classical coin study quantum coin scenario with this choice for different scenario. This is the traditional way to demonstrate in- sets of d MUBs and for each prime d we construct a set of compatibility of quantum measurements { for compati- d MUBs, which allow Bob to guess Alice's measurement ble measurements Bob still can guess with certainly, but outcomes with certainty.