Connections Between Mutually Unbiased Bases and Quantum Random Access Codes

Edgar A. Aguilar,1, ∗ Jakub J. Borka la,1 Piotr Mironowicz,2, 3, † and Marcin Paw lowski1 1Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, 80-308, Gdansk, Poland 2Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics, GdańskUniversity of Technology 3National Quantum Information Centre in Gdańsk,81-824 Sopot, Poland (Dated: March 29, 2018) We present a new quantum communication complexity protocol, the promise–Quantum Random Access Code, which allows us to introduce a new measure of unbiasedness for bases of Hilbert spaces. The proposed measure possesses a clear operational meaning and can be used to investigate whether a specific number of mutually unbiased bases exist in a given dimension by employing Semi–Definite Programming techniques.

Introduction.- Mutually unbiased bases (MUBs) play a main idea of this protocol is to use the so-called nd → 1 special role in the formalism of quantum mechanics. In Quantum Random Access Codes (QRACs) with certain particular they serve as complementary quantum tests, constraints. Our main technical result shows that a spe- and find wide applicability in many fields of quantum cific average success probability of the protocol can be information science such as quantum state tomography achieved if and only if n MUBs exist in dimension d. [1, 2], quantum key distribution [3], quantum teleporta- The protocol allows us to create a new measure of un- tion and dense coding [4]. Hence, a general understand- biasedness, which quantifies the amount by which two ing of MUBs is well motivated and of general interest, (or more) bases are mutually unbiased. Other measures see [5] for an extensive review and further references. currently exist and are in use [10], yet the presented one 1 2 Explicitly, two orthonormal bases {|ψi i}i and {|ψj i}j possesses a direct operational interpretation as the suc- of Cd are said to be mutually unbiased if cess probability of a well–defined communication task. Furthermore, the pQRAC game is suitable for numeri- 1 2 2 1 cal optimization techniques like Semi–Definite Program- hψi | ψj i = , ∀i, j ∈ [d], (1) d ming (SDP)[11]. In particular, one may use the see-saw where [d] ≡ {1, 2, . . . , d}. The term unbiased is used be- method [12] to search for n MUBs in dimension d. What 1 is more, pQRACs may be used together with the Navas- cause if we pick any basis vector |ψi i, then performing a 2 cues and Vertesi method [13] to discard the existence of measurement in the {|ψ i}j basis will yield a completely j n MUBs in a particular dimension. This exclusion is a random result (i.e. each outcome |ψ2i will have equal j rigorous statement, in contrast to drawing the conclu- detection probability 1/d). sion out of the failure of trying to find them. As a proof A set of MUBs in dimension d is said to be maximal, of principle, we have applied our method to exclude the if there are d + 1 bases which are all pairwise mutually existence of 5 MUBs in dimension 3, and 6 MUBs in di- unbiased. The construction of maximal sets when d = p, mension 4. We have been unable to rule out the existence a prime number, was described by Ivonovic [1], and later of 4 MUBs in dimension 6, but argue that the problem by Wootters and Fields when d = pk, a prime power is now at arm’s length for future researchers. [2]. The general problem of whether d + 1 bases exist for Methods.- We begin by introducing Random Access arbitrary dimensions remains open for at least the past d 29 years. Codes (RACs)[14]. An n → 1 RAC is a protocol in In particular it is an open question whether a complete which Alice tries to compress an n-dit string into 1 dit, arXiv:1709.04898v2 [quant-ph] 28 Mar 2018 set of MUBs exist even in the simplest case, namely in such that Bob can recover any of the n dits with high dimension 6. Zauner’s conjecture states that no more probability. More precisely, Alice receives a uniformly distributed random input string x = x1x2 ··· xn, xi ∈ [d]. than three MUBs exist in dimension 6 [6]. The task of n proving the conjecture is a research field on its own, see She then uses an encoding function Ec :[d] → [d] (pos- e.g. [7, 8] for partial analytical results supporting the sibly classically probabilistic), and is allowed to send one conjecture. Numerical approaches have also failed to be dit a = Ec(x) to Bob. On the other side, Bob receives conclusive, [9]. an input y ∈ [n] (uniformly distributed), and together with Alice’s message a uses one of n (possibly classically In this paper we introduce a novel protocol named y probabilistic) decoding functions Dc :[d] → [d], to out- promise-Quantum Random Access Code (pQRAC). The y put b = Dc (a) as a guess for xy. If Bob’s guess is correct (i.e. b = xy) then we say that they are successful, other- wise we say that they are unsuccessful or fail. ∗ [email protected] Similarly, we define nd → 1 Quantum Random Ac- † [email protected] cess Codes (QRACs) where Alice encodes her input n- 2

x0 x2 x3 x5 dit string into a d-dimensional quantum system (qudit) x1 x4 x6 xN-1 n d y via Eq :[d] → C , and sends the qudit ρx = Eq(x) to Bob. He then performs one of his decoding functions y Cd Dq : → [d] to output his guess b for xy. The decoding z ρ function is simply a quantum measurement, i.e. he out- Alice Bob y puts his guess b with probability P(b = xy) = tr[ρxMb ], y where the operators Mb are POVMs (i.e. positive and b P y ∀y b Mb = 11). As a ﬁgure of merit, we employ the optimal average success probability for both RACs and FIG. 1: Schematic representation of a (n, m)d → 1 QRACs: promise–Quantum Random Access Code. Here xi ∈ [d], y ∈ [n], and z is a subset of [n] with m ¯ 1 X X Pc,q(n, d) = max P(b = xy) . (2) elements. The bold inputs xk depict k ∈ z. ρ is the {E,D} ndn x y quantum state that Alice sends to Bob.

The maximization is over encoding-decoding strategies {Ec,q, Dc,q} (classical or quantum respectively), and the The proof of this result is by direct calculation (See average is taken over all possible inputs (x, y) of Alice Supplementary Material B for details). This occurs since and Bob. In the quantum case, the optimal average suc- there are inequivalent subsets of MUBs (i.e. not related ¯ cess probability Pq, can be achieved with pure states, by unitary transformations) in higher dimensions. As an ρ = |xihx| [14], where |xi is the eigenvector of P M y x y xy example, let us consider the case n = 3, d = 5. Bob must with largest eigenvalue. In [15], it was shown that for choose 3 different measurement bases, and he can do so 2d → 1 QRACs this maximum is achieved when the op- in 6 = 20 ways. Half of those selections lead to an y 3 erators Mb are (rank 1) projective measurements. There- average success probability of 0.610855, while the other fore, throughout the rest of this letter we will be consid- half give 0.596449. Hence, the choice of the subset of ering only pure-state encoding and von-Neumann mea- MUBs matters. This feature occurs also for other choices surements. of n and d. However, we conjecture that the optimal RACs and QRACs have increasingly become an exper- average success probability for nd → 1 QRACs is indeed imental tool to test the “quantumness” or non-classical achieved with a suitable choice of MUBs. behavior of a system [16, 17]. For fixed n and d, we have Next we define a (n, m)d → 1 promise-QRAC ¯ ¯ d Pc < Pq, and a gap is exploited to show that a system is (pQRAC), m ≤ n, as an n → 1 QRAC with an ex- behaving non-classically . For example a 22 → 1 RAC has tra promise. Let Sn be the set of all possible subsets ¯ m Pc = 0.75, while the corresponding QRAC√ has an optimal of [n] of size m. Then in a pQRAC, Alice receives an ¯ n average success probability of Pq = (2 + 2)/4 ≈ 0.8536 additional input z ∈ Sm, with the promise that y ∈ z. [18]. Thus for a system of dimension 2, observing an That is, Alice knows that Bob will not be questioned over average success probability greater than 0.75 indicates some of Alice’s inputs, see Fig. 1 for an illustration of a non-classical behaviour. pQRAC. The quantum advantage comes from encoding Alice’s Hence, the optimal average success probability (2), is 1 2 d state as a superposition of the bases {|ψi i}i and {|ψj i}j, modified in the case of (n, m) → 1 pQRACs to: 1 2 namely |xi = α|ψx i+β|ψx i, while Bob measures in the y 1 2 {|ψ i}i basis. We have the following: 1 X X X i P˜ (n, m, d) = max tr[ρ M y ], q n x,z xy {ρ,{M}} mdm d m z∈Sn x y∈z Lemma 1. For a 2 → 1 QRAC, the optimal average m z success probability (4) where the summation over xz indicates a summation over x , x , . . . , x such that {i , i , . . . , i } = z , and the ¯ 1 1 i1 i2 im 1 2 m Pq(2, d) = 1 + √ (3) maximization is taken over all quantum encoding and 2 d decoding strategies {ρ, {M}}. Now, we are able to prove is obtained if and only if Bob’s measurement bases our main technical result: 1 2 {|ψ i}i, {|ψ i}j are mutually unbiased. i j Lemma 2. For a (n, 2)d → 1 pQRAC, the following The proof is given in Supplementary Material A. holds: We find it interesting to note here an observation that Lemma 1 cannot be generalized to the case of nd → 1 1 1 P˜q(n, 2, d) ≤ 1 + √ (5) QRACs for n ≥ 3, as stated below: 2 d

Observation 1. The MU condition on Bob’s measure- with equality iﬀ at least n MUBs exist in dimension d. ment bases is not suﬃcient for obtaining the optimal average success probability in nd → 1 QRACs when n ≥ 3. Proof. We begin by writing the optimal average success 3

d d i probability of the (n, 2) → 1 pQRAC. Explicitly, given n bases of C , {ψ }i, the maximum attainable average success probability of the pQRAC,p ¯, ˜ 1 X X X y is: Pq(n, 2, d) = max n tr[ρx,zMx ] {ρ,{M}} 2 y 2 2d n   z∈S2 xz y∈z ! i 1 X 1 1 X a b p¯({ψ }i) =  + |hψi |ψj i| , 1 X 1 X X y n 2 2d2 ≤ n max tr[ρx,zMx ] . 2 {a,b}∈Sn i,j∈[d] {ρ,{M}} 2d2 y 2 2 n x y∈z z∈S2 z (8) which comes as a direct conclusion of Lemma 1. We may The inequality follows, since the strategies to maximize normalize (8) such that the minimum value is 0 (obtained the summands might not be compatible with each other iff all bases are the same), and the maximum value is globally. In fact, we recognize the term in parenthesis 1 (obtained iff all bases are pairwise MU) and get the ¯ d as Pq(2, d), the optimal success probability for a 2 → 1 expression: QRAC (2). From Lemma 1, this maximization occurs if and only if the measurement bases corresponding to the i ¯ ¯ i p¯({ψ }i) − Pc(2, d) set z are mutually unbiased. It follows that it is pos- Q({ψ }i) = . (9) P¯q(2, d) − P¯c(2, d) sible to simultaneously satisfy all of these maximization constraints iff there exists n MUBs in dimension d. See Supplementary Material C to see a direct comparison between (9) and (7). The intuition behind Lemma 2, is that Bob must be For illustrative purposes, we have optimized the value ready to measure in all n bases. If there exist n bases of the (4, 2)6 → 1 pQRAC game expression using the which are all pairwise mutually unbiased, then essentially see-saw method [12]. This allows us to show how the op- they are just playing a more complicated version of the timization of (9) may be used to construct MUBs in a d usual 2 → 1 QRAC. If these bases do not exist, then for particular dimension, as well as providing numerical ex- n some z ∈ S2 , the protocol will not be able to achieve amples of how D¯ 2 and Q¯ compare. With this method, the optimal value (3), dropping the entire average. the maximal value of D¯ 2 of four MUBs in dimension 6 Results.- In the context of MUBs, reference [4] has in- we obtained is 0.998284 with Q¯ = 0.998045. On the 1 troduced a distance measure between two bases {|ψi i}i other hand, the bases from [20, 21] have D¯ 2 = 0.998292, 2 and {|ψj i}j which quantifies unbiasedness: and Q¯ = 0.998036. With this result one sees that the two measures, (7) and (9) are not equivalent, and induce 2 2 1 X 1 2 2 1 different partial orderings on the sets of bases. See Sup- D 1 2 = 1 − |hψ |ψ i| − . (6) ψ ψ d − 1 i j d plememntary Material D for more details. i,j∈[d] Our second result is another direct application of Lemma 2, and deals with ruling out if there are n mutu- The measure is symmetric (D2 = D2 ). If the bases ψ1ψ2 ψ2ψ1 ally unbiased bases in dimension d. Explicitly, if it is pos- 2 2 are the same, then Dψ1ψ1 = 0. The maximum Dψ1ψ2 = 1 sible to show that there are no sets of encoded states and is obtained iff the bases are mutually unbiased. For a set measurement bases that would obtain a success probabil- d j j of n bases in C ({ψ } = {|ψ i}i, j ∈ [n]), one can i ity of P¯q(2, d), then one immediately concludes that there analyze the average square distance between all possible does not exist n MUBs in the given dimension. Thus one pairs of bases [10]: may use the SDP hierarchy of relaxations proposed by Navascues and Vertesi (NV) [13]. The method defines a ¯ 2 i 1 X 2 D ({ψ }i) = n Dψaψb . (7) sequence of SDP problems yielding upper bounds to op- 2 n {a,b}∈S2 timization tasks over quantum probability distributions with dimensional constraints. One can show that the Likewise, D¯ 2 = 1 iff all bases are pairwise mutually un- method converges to the accurate quantum values [22]. biased. However, (7) is an abstract distance measure, If at a given level of the hierarchy the upper bound falls lacking an operational interpretation. below the threshold Q¯ = 1, then the conclusion follows. Lemma 2 immediately leads us to our first result. We emphasize that if n MUBs do not exist in a partic- Given a set of n bases in dimension d we define as their ular dimension, then applying the SDP hierarchy to the unbiasedness measure the average success probability in (n, 2)d pQRAC gives an algorithmic way of proving their a (n, 2)d → 1 pQRAC if the bases are used as Bob’s non-existence. On the other hand, if n MUBs do exist, measurement bases. This measure is thus defined op- the proposed method will fail to draw a conclusion. erationally and has the following properties: (1) The Implementing the hierarchy.- Let us try to directly ap- ¯ 1 −1/2 d maximum of Pq(2, d) = 2 (1 + d ) is attainable iff ply the NV hierarchy to the (n, 2) → 1 pQRAC. To all bases are pairwise unbiased, (2) It is symmetric un- implement the k-th level of the hierarchy, Qk, the set k der permutation of bases, and (3) The minimal value of Sd of all feasible moment matrices of order 2k arising ¯ 1 −1 Pc(2, d) = 2 (1+d ) is achieved iff all bases are the same. from quantum systems of dimension d must be calcu- d j The optimal classical success probability of n → 1 RACs lated. For this, moment matrices Γk are randomly gen- j k is shown in [19]. erated from this set until span({Γk}j) = Sd. In practice, 4 the algorithm keeps creating new moment matrices j = case (n, 2)d Q¯ for Q1 Q¯ for Q1+succ vk+1 j vk 2 {1, 2, . . . , vk} and stops when Γk ∈ span({Γk}j=1). (3, 2) 0.99999999 0.99999999 The method requires an assumption on the rank of the (4, 2)2 0.99999999 0.99999992 projectors {M y}, but in our scenario Bob’s optimal strat- b (4, 2)3 1.13165247 0.99999995 egy is to implement d-dimensional von Neumann mea- 3 surements, therefore all operators are rank 1. (5, 2) * 1.13165249 0.99989895 j n 2 (5, 2)4 * 1.24264066 0.99999996 In order to generate Γk, we randomly choose A = 2 d states for Alice to encode and B = nd measurement (6, 2)4 * 1.24264066 0.99991655 Cd j 6 operators for Bob (n bases of ). Then, Γk con- (3, 2) 1.42882541 0.99999998 tains the traces of all strings of size less than or equal (4, 2)6 * 1.42882538 0.99999997 to 2k constructed from Alice’s states and Bob’s opera- j 1 tors. For example, typical matrix elements of Γ1 include TABLE I: Results of implementing hierarchy levels Q 0 j j j y,j y,j y ,j and Q1+succ to (n, 2)d → 1 pQRACs. The * indicates an tr[ρx,zρx0,z0 ], tr[ρx,zMb ], and tr[Mb Mb0 ]. While in 0 00 000 k j j y,j y ,j y ,j j y ,j optimization over G(Sd). Other cases were executed for Γ , we can find tr[ρ M M M ρ 0 0 M ], etc. 3 x,z b b0 b00 x ,z b000 both k and G( k) to test our code implementation. We write the k-th order relaxation to our problem as Sd Sd the following semidefinite program [13]: ˆj k Clearly Γ ∈ G(Sd), and this is repeated until P˜q(n, 2, d) = max tr[Bˆ Γk] k (10) ˆj k k span({Γk}j) = G(Sd). To illustrate the power of the pro- s.t. Γk ∈ Sd, (Γk)1,1 = 1, Γk ≥ 0, posed method, we report that for a (4, 2)5 → 1 pQRAC, 1 dim(S5) = 13672 and the SDP running time was 22.5h where we call Bˆ the pQRAC game matrix, and construct on a desktop computer, whereas dim(G( 1)) = 7 and had y S5 it to “pick out” the values tr[ρx,zMb ] from Γk such that a total run-time of 50s. b = xy and y ∈ z. Using this, we have implemented Q1 and a subset of 1 4k 1+succ Roughly 2 (A + B) real-valued numbers need to be the “almost quantum” level [24] (Q ) for some rel- stored in a computer’s RAM in order to describe the set evant pQRAC cases, see Table I. The level Q1+succ in- k of all feasible moment matrices Sd. Below, we describe a cludes traces of strings of length ≤ 2 from the set of op- y,j j potentially quadratic reduction in the problem’s memory j zi,j erators {{ρx,z}, {Mb }, {ρ Mx }}. That is, xz1 ,xz2 ,{z1,z2} zi requirements. See Supplementary Material E for details. we also included pairs of states and measurements which ˆ ˆT Note that B = B , and is a sparse matrix with a lot lead to successful trials. The details of the implementa- of symmetries. In this case, we employ the symmetries tion are found in Supplementary Material F. As a proof corresponding to relabeling measurement device outputs, of principle, we note that we have been able to rigorously and the ones corresponding to permuting the labels of rule out the existence of d+2 MUBs in dimensions d = 3 the measurement devices themselves. This approach has and d = 4. However, with our numerical precision and been followed on the NPA hierarchy in the Bell-test sce- at this hierarchy level we have been unable to rigorously nario [23]. exclude the existence of 4 MUBs in dimension 6. ˆ Let B be invariant under the group of transformations We notice that the hierarchy level Q1+succ was also G. In other words, for every representation G of an el- unable to rule out the existence of 4 MUBs in d = 2. If ˆ T ˆ ement g ∈ G, GBG = B. Then, if we apply a group these four bases existed, together with the d = 3 MUBs action on the game matrix inside the objective function one could create four MUBs in dimension 6. We conjec- (10), this would be equivalent to applying a group action ture that in order for a level of the hierarchy to be able to ˆ ˆ T on Γk. Namely, tr[B Γk] = tr[B G ΓkG]. Therefore, it is rule out the existence of 4 bases in dimension 6, it must unnecessary to consider the full space of feasible moment first rule out the existence of 4 MUBs in d = 2. In future k matrices Sd and can simplify (10) into: work, we wish to find more efficient ways of calculating (11), and higher levels of the hierarchy. Conclusions.- In this paper we give a new class of quan- ˜ ˆ ˆ Pq(n, 2, d) = max tr[B Γk] tum games, pQRACs, which serve as an operational way (11) ˆ k ˆ ˆ of testing unbiasedness. It also enables one to reformu- s.t. Γk ∈ G(Sd), Γk = 1, Γk ≥ 0, 1,1 late the problem of searching for a given number of MUBs in a particular dimension as a problem of optimizing the k where we denote G(Sd) as the set of feasible moment strategy of the pQRAC game. In particular if one is able matrices which are G-invariant. In order to implement to get a proper upper bound on the value of the game, ˆj this, we generate random invariant moment matrices Γk then our formulation allows to exclude the existence of j by first creating a moment matrix Γk and averaging it a given number of MUBs in the considered dimension. out over all of the group elements: We have exploited the symmetries of the pQRAC game matrix in the Navascues and Vertesi hierarchy. We hope 1 X Γˆj = G Γj GT . (12) this will lead to rigorously proving Zauner’s conjecture k |G| k G by considering higher levels. 5

Acknowledgments.- The paper was supported by EU EA acknowledges support from CONACyT. The SDP grant RAQUEL, ERC AdG QOLAPS, National Science optimizations have been performed using OCTAVE [25] Centre (NCN) Grant No. 2014/14/E/ST2/00020 and DS with SDPT3 solver [26, 27], SeDuMi [28], and YALMIP Programs of the Faculty of Electronics, Telecommunica- toolbox [29]. We thank D. Saha and M. Farkas for discus- tions and Informatics, Gda´nskUniversity of Technology. sions, and I. Bengtsson for guidance with the literature.

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Appendix A: Optimal average success probability of where p↓ (q↓) is a vector with the same components as 2d → 1 QRAC p (q), but written in descending order. A function F : RN → R is said to be Schur concave if p q implies Let {|ψii}i and {|φji}j be two orthonormal bases used F (q) ≥ F (p). In particular, if f : R → R is a concave P on Bob’s side to perform the von Neumann measure- function , then F = f(xi) is Schur concave. 0 i ment, and we write their projectors as |ψiihψi| = M and P 2 i Finally, note that αx = d, is a constant. Let us |φ ihφ | = M 1. The optimization of the average success x j j j deﬁne the probability distribution a = (a1, a2, . . . , ad2 ), probability is simply a maximization over the measure- with a = 1 α2, ∀i. Then (A2) may be rewritten as: 0 1 i d i ment bases {Mi }, {Mj } of the expression: 1 X p d−1 P¯ = 1 + da . (A7) 1 q 2d2 x X 0 1 2 p¯q = tr[ρx ,x (M + M )]. (A1) x∈[d ] 2d2 0 1 x0 x1 x0,x1=0 By the remark above this function is Schur concave, and

The maximum is achieved when ρx0,x1 (a pure state) is therefore maximized by the uniform distribution u = 1 is an eigenvector, corresponding to the largest eigen- d2 (1, 1,..., 1) [30]. This is because u is majorized by value of (M 0 + M 1 ). This can be seen by writ- 2 x0 x1 all probability distributions. Explicitly, we obtain αx = −1 ing ρx ,x = |ξihξ| , and expressing the pure state as d , ∀x, which is precisely the condition of unbiasedness 0 P1 |ξi = ck|ki, where {|ki}k is the eigenbasis of the (1). sum of the operators (M 0 + M 1 = P λ |kihk|). Then x0 x1 k tr[ρ (M 0 +M 1 )] = P |c |2λ , which is clearly max- x0,x1 x0 x1 k k imal when ρx0,x1 = |λmaxihλmax|. Appendix B: Insufficiency of MUBs for optimality in Thus we are trying to maximize the sum 3d → 1 QRACs P λ (M 0 + M 1 ). Direct calculations consid- X max x0 x1 ering only two dimensional subspaces at a time yields: In this section we would like to find the maximum d d−1 success probability for 3 → 1 QRACs, and prove that 1 X the maximum is achieved when using MUBs. To do this P¯ = (1 + α ) , (A2) q 2d2 x0,x1 we need to find the optimal encoding/decoding strategy x ,x =0 0 1 for Alice and Bob. where α = ptr[M 0 M 1 ] = |hψ |φ i| is simply the x0,x1 x0 x1 x0 x1 modulus of the inner product between two elements of For a given encoding/decoding strategy, the average the different bases. success probability is given by (2), which we write out here as: To see this, we first fix x0 and x1. Then, any 2 linearly independent vectors | ψx0 i, | φx1 i span a 2 dimensional d−1 d−1 d−1 d 1 X X X subspace of C . Let | 0i, | 1i be the basis vectors of such p¯ = tr[ρ (Ψ + Φ + Θ )], (B1) 3d3 ijk i j k a subspace and for simplicity we choose them such that: i=0 j=0 k=0 | ψ i = | 0i (A3) x0 where Ψi = |ψiihψi|,Φj = |φjihφj|,Θk = |θkihθk| are

| φx1 i = α| 0i + β| 1i, (A4) three projectors which are used by Bob to measure the message from Alice (encoded as ρijk). To maximize the 2 2 where |α| + |β| = 1, and α = αx0,x1 = |hψx0 |φx1 i|. above expression, we will let Alice’s encoding ρijk be a Then, we solve the characteristic equation for the mea- pure state corresponding to the eigenvector of the largest surement operator, 0 = det(| ψx0 ihψx0 | + | φx1 ihφx1 | − eigenvalue of the measurement operators λmax(Ψi +Φj + λI), to get the following two solutions: Θk). For the moment we analyze the eigenvalues of such operators, and drop the i, j, k subscripts. In short, we p 2 λ± = 1 ± 1 − |β| = 1 ± α, (A5) want to solve the following characteristic equation: where λ+ is the maximum eigenvalue used in (A2). det(| ψihψ | + | φihφ | + | θihθ | − λI) = 0, (B2) 7 which leads to the following third degree polynomial for d the eigenvalues: 5 7 8 9 11 13 16 3 n 4 − λ3 + 3λ2 + mλ + n = 0, (B3a) 5 m = |hψ |φi|2 + |hψ |θi|2 + |hφ |θi|2 − 3, (B3b) n = det(| ψihψ | + | φihφ | + | θihθ |). (B3c) TABLE II: Table showing which pairs of n, d values produce diﬀerent average success probabilities depending on the choice of subsets of MUBs, i.e. an The 3 vectors | ψi, | φi, | θi span at most a 3 dimen- anomaly. The symbol means that we observe the sional subspace of Cd. Let | 0i, | 1i, | 2i be the basis vec- anomaly for this particular number of inputs and tors of such a subspace and write: dimension and means that there is no anomaly.

| ψi = | 0i, (B4a) | φi = α| 0i + β| 1i, (B4b) not uniform). We believe that this effect is somehow re- | θi = a| 0i + b| 1i + c| 2i. (B4c) lated to the complexity of higher dimension Cd spaces. d In general given n vectors {|vii}i in C , it is not possible Then n = |cβ|2. We observe that the parameter m ∈ to find another normalized vector |ψi with equal overlap 2 [−3, 0] and n ∈ [0, 1] but since | ψihψ | + | φihφ | + | θihθ | |hvi|ψi| to all n vectors. is a Hermitian operator, we need to focus only on the There is no obvious pattern which could suggest in subdomain of m and n where all eigenvalues are real. which case those anomalies are present. It is surprising Solving equation (B3) gives three eigenvalues λ1, λ2, λ3 as for example that in 8 dimensions we did not find this functions of m, n. Let λ1 be the largest of the eigenvalues. effect. One of the research directions in this topic is The maximal average success probability of the 3d → 1 answering the question: are there other protocols which QRAC is given by the expression will be affected by this sensitivity for choosing different subsets of MUBs. Another question is: is there any X max λ1(mijk, nijk). (B5) pattern which could help us predict which cases of n, d i,j,k have a so-called MUB anomaly.

From this formula one can easily see that this depends not only on unbiasedness of the basis, {mijk}, but also on the additional parameters {nijk}. Now, we show that in order to maximize the success Appendix C: The Unbiasedness Measure Q¯ probability of a 3d → 1 QRAC in general it is not sufficient to take arbitrary MUBs as measurements. In a d To obtain (9), we simply substitute the expressions 3 → 1 QRAC protocol, let us use 3 out of the d+1 mutu- ¯ d d+1 Pc,q(2, d) of the optimal classical and quantum 2 → 1 ally unbiased bases. There are such subsets. From i 3 Random Access Codes. Dropping the argument {ψ }i, numerical results we observe that the choice of bases sub- this yields: set of MUBs affects the final average success probability (2).   As an example there are 20 distinct subsets of 3 MUBs ¯ 1 X 1 X a b 1 Q =  √ |hψi |ψj i| −  . in d = 5. By direct calculation, using 10 of those subsets n d( d − 1) d 2 {a,b}∈Sn i,j∈[d] we obtain an average success probability of 0.610855, and 2 with the other 10 subsets 0.596449. Hence the MUB con- (C1) dition alone does not guarantee the highest average suc- We rewrite (7), to have a side-by-side comparison. cess probability. We call this rather surprising behaviour,  2 an anomaly. We observe this effect also in higher dimen- 1 X 1 X 1 sions. We present our numerical observations for higher D¯ 2 = 1 − |hψ1|ψ2i|2 − . n  d − 1 i j d  2 n dimensions in Tab. II. {a,b}∈S2 i,j∈[d] From our numerical analysis we have observed some- (C2) thing else which might be of interest. Consider the subset Notice that D¯ 2 is a function of the ”probabilities” a b 2 of MUBs which give the highest observed average success |hψi |ψj i| , and tries to measure how much these values probability. Here, for every pure state that Alice encodes, differ from the uniform distribution. This has a very aes- the success probability, p(b = xy), does not depend on thetic mathematical interpretation. However, it lacks a the basis y Bob uses to measure. In contrast, for the clear operational meaning which our proposed measure cases where a lower success probability was observed, Q¯ posseses. Finally, we point out that D¯ 2 is not a func- the successful collapse probability of the encoded state tion of Q¯ or vice-versa, except for the trivial cases (0 and depended on the measurement basis (and was therefore 1). 8

case (n, m)d D¯ 2 Q¯ (4, 2)3 0.99999959 0.99999972 (5, 2)4 0.99999922 0.99999939 0.8 1 (6, 2)5 0.99999904 0.99999910 (3, 2)6 0.99999869 0.98639054 0.8 (4, 2)6 0.99828388 0.99804689 0.75 (4, 2)7 0.99237197 0.97792916 0.6 2

TABLE III: The results of execution of the see-saw D 0.7 optimization for diﬀerent pQRACs in search of n MUBs 0.4 in dimension d. The table shows the value of D2 0.65 parameter for measurements obtained in optimizations 0.2 and the success probability of the pQRAC with these measurement basis. We conclude that for d = 3, 4, 5 we 0.6 0 have found the maximal amount of MUBs using the 0.66 0.67 0.68 method. p‾

Appendix D: See-Saw for (n, 2)6 → 1 pQRACs FIG. 2: The values represents relative densities of the number of different basis giving similar values of the Here we use the see-saw method and our unbiasedness pair (¯p, D2). measure (Q¯ orp ¯) to add to the pre-existing evidence that 4 mutually unbiased bases do not exist in dimension 6. In the see-saw method one interlaces two steps of SDP then the SVD decomposition is performed giving matri- optimizations: For every other iteration one optimizes ces U, S and V satisfying USV † = A, with U and V Alice’s preparation states for a given set of Bob’s mea- being unitary. We form the basis for the measurement surements, and for the remaining steps one optimizes by taking subsequent columns of the U matrix. Bob’s measurements for a given set of Alice’s preparation In our Monte Carlo experiment we randomized 10000 states [12]. Even though this method does not guarantee instances of random sets of measurements, calculated op- the convergence to the global maximum, it has proved to timal states and the value of the (4, 2)6 pQRAC game, be efficient in Bell-type scenarios, see e.g. [31]. p¯, (8), and the value of D¯ 2, (7). The efficiency of the see-saw method can be improved b The meaning of this result is the following. If one in the following way. Let {My } be the set of measure- considers a set of basis constructed in a natural way, then ments of Bob. Note that when Alice gets as input the in- he can expect that the higher the unbiasedness, D2, the formation that Bob has either y1 or y2 setting, her strat- higher the success probabilityp ¯ in the (4, 2)6 pQRAC egy should be to send a state ρ maximizing the value x x game, and vice versa. of Tr ρ M y1 + M y2 . This is obtained if the state y1 y2 The Spearman [32, 33] rank correlation coefficient is is in the subspace of vectors with maximal eigenvalue of x x a way of measuring the rank correlation between two the operator M y1 + M y2 . Thus the see-saw step op- y1 y2 (random) variables. It is a generalization of the Pearson timizing the states can be replaced with a construction coefficient, where the latter measures only linear depen- of the states basing on eigenvectors decomposition of the dence, and the former is able to express any monotonic measurement, cf. remark below (2). relationship, with 0 meaning no correlation and ±1 mean- We have used the see-saw optimization for several cases d ing perfect monotonicity. We have calculated the value of (n, k) → 1 pQRACs. The results are given in Tab. III. of Spearman rank correlation coefficient for the random- 6 Above we observed that for the (4, 2) pQRAC game ized data points and obtained that it’s value is about 0.91 to obtain the maximal guessing probability (9), the mea- meaning very strong dependence, but also displaying that surements of Bob has to be unbiased, and the value of these measures are subtly different. (7) is 1. We will now use the Monte Carlo method to investigate the relation between the quantities (7) and the games’ average success probability (8). Our results show Appendix E: Estimating the Memory Requirements that the majority of highly unbiased measurements in the of the Navascues Vertesi hierarchy meaning of (7) gives large values of the (4, 2)6 pQRAC game, (8) and vice versa. The random moment matrices Γj are created by gen- We have randomized instances of measurements of Bob k erating A = nd2 random pure states {ρj }, and in the following way. For each measurement we random- 2 x,z y,j ize a 6 × 6 complex matrix A with each entry given by B = nd projective measurement operators {Mb }. Let j y,j uniform distribution on the set {x + iy : x, y ∈ [0, 1]}; wk = wk({{ρx,z}, {Mb }}) be the vector containing all 9 strings of operators of length smaller than or equal to Appendix F: Symmetries k. For convenience we relabel the indicess of the states j A j B {ρα}α=1, and of the measurements {Mβ}β=1. Then, We find it easier to describe the symmetry group G by 1 j j j j e.g. w1 = ( , ρ1, . . . , ρA,M1 ,...,MB), and w2 would analyzing that there are two different types of possible re- further contain the elements (ρj ρj , ρj M j , M j ρj , labelings. Let Bob have n different work stations, and on α1 α2 α1 β1 β1 α1 j j each station he has a quantum measurement device. The M M ), ∀α1,2 ∈ [A], β1,2 ∈ [B]. The moment matrix is β1 β2 success probability should not depend on which physical constructed as follows: device is situated at which work station, so long as Alice is aware of which measurement basis is being used for every input y of Bob. Likewise, Bob is free to relabel the j h † i Γk = tr (wk)i (wk)j , (E1) outputs of every work station at will without dropping i,j the success rate if Alice applies the same permutation to her inputs. In order to properly describe the group, we will use the with (·)i,j corresponding to the components of the ma- picture where Bob first moves the measurement appara- trix, and similarly for vectors. We note that by construc- tuses around with a permutation ω ∈ Sn, and afterwards j j j tion Γk = Γk , and Γk ≥ 0. he relabels the device in work station i with the permuta- i,j j,i tion πi ∈ Sd. We stress that the permutations π1, π2,... k address a specific work station - as opposed to a specific We wish to estimate the size of Sd. The moment matrices are |wk|-by-|wk|, and by construction we see measurement device. Then, any relabeling of device out- Pk i puts, and devices themselves can be achieved with these that the length of the vectors |wk| = i=0(A + B) = O (A + B)k. In general, it is unknown how many ma- operations. We symbolically denote this abstract group j j element g ∈ G as: trices Γk need to be generated until the set {Γk}j spans k the full space Sd. We can upper bound this number by g = π1, π2, . . . , πn; ω. (F1) 1 |w |2 trivially (since the Γj are real-symmetric). We 2 k k If,g ˜ =π ˜ ,..., π˜ ;ω ˜ ∈ G, thengg ˜ ∈ G: note that e.g. for a (4, 2)5 → 1 pQRAC the actual number 1 n of linearly independent matrices was just 6.5% lower than gg˜ =π ˜1πω˜ −1(1),..., πñπω˜ −1(n);ωω. ˜ (F2) this trivial bound. Putting everything together, we see k 4k that the space Sd is described by roughly O (A + B) The inverse of g is: real parameters. −1 −1 −1 −1 −1 g = πω−1(1), πω−1(2), . . . , πω−1(n); ω . (F3) For the case of 4 MUBs in dimension 6, (A + B) = 240 ≈ 28. The k-th level of the hierarchy thus requires More importantly though, is to see the action of g on roughly 232k bits of memory just for indices and a fur- Bob’s measurements and Alice’s encoded states: ther 26 bits if each number is stored in double precision 0 g(M y) 7→ M y (F4a) floating-point format. This exponential growth on mem- b b0 g(ρx ,x ,{z ,z }) 7→ ρx0 ,x0 ,{z0 ,z0 }), (F4b) ory requirements means that a typical 32-bit operating z1 z2 1 2 z0 z0 1 2 system (or mathematical software) cannot compute the 1 2 first level of the hierarchy since it will run out of available such that memory indices. In fact, we encountered this problem 0 b = π 0 (b) when executing the program without symmetries in 32- y 0 bit OCTAVE. More drastically, a 64-bit system would be y = ω(y) 0 unable to calculate the second level, and all of the world’s xz0 = πz0 (xz0 ) ∀i = 1, 2 storage space is insufficient for the third level (approxi- i i i z0 = ω(z ) ∀i = 1, 2. mately 274 bits of information were generated in 2016 i i [34]). It is therefore clear, that if we apply an element g to Requirements considering symmetries.- Even if we re- the vector of operator strings wk, it just acts as a per- strict our optimization to the symmetric subspace G( k), mutation. The operators are the same, just re-indexed. Sd Hence, we can represent the elements g as the permu- the moment matrices are still of size |wk|-by-|wk|. Just as we do not know a priori the dimension of the space tation matrices G. What is more, by construction we k know that there is an order in which these elements are of feasible moment matrices, dim(Sd), we cannot predict k applied, and G can be seen as a product of permutation the dimension of the symmetric subspace, dim(G(Sd)). Therefore, our proposal can at most be a quadratic im- matrices: provement, i.e. of O (A + B)2k. However, as we have G = Π1Π2 ··· ΠnΩ, (F5) portrayed, this reduction is sufficient to calculate the first 1+AB level of the hierarchy as well as a subset of the Q where Πi is the matrix representation of πi, and sim- level. ilarly Ω is the matrix representation of ω. When this 10 representation acts on a moment matrix, it will first shift Crucially for implementations, the most important thing j T the indices of the measurement devices with ΩΓkΩ , and about (F6) is that the sum may be rewritten as: later it will reshuffle the output labels of each work sta- ! ! ! tion. Notice that the representations of the output labels X X X j T T T Π1 ··· Πn ΩΓ Ω Π ··· Π . commute amongst each other ([Πi, Πj] = 0), but not with k n 1 Ω. We are now in a position to calculate the average of Π1 Πn Ω the moment matrix over the group representation (12). (F7) This reduces the amount of operations needed to perform the average from |S ||S |n to |S | + n|S |. These meth- 1 n d n d ˆj X X X j T T T ods apply to a wide variety of SDP relaxation problems, Γk = n ··· Π1 ··· ΠnΩΓkΩ Πn ··· Π1 |Sn||Sd| Π1 Πn Ω and will be further discussed in [35] in a more rigorous (F6) fashion.