Tight entropic uncertainty relations for systems with dimension three to five
TTight entropic uncertainty relations for systems with dimension three to five
Alberto Riccardi, Chiara Macchiavello and Lorenzo Maccone
Dip. Fisica and INFN Sez. Pavia, University of Pavia, via Bassi 6, I-27100 Pavia, Italy
We consider two (natural) families of observables O k for systems with dimension d = 3 , ,
5: thespin observables S x , S y and S z , and the observables that have mutually unbiased bases as eigenstates.We derive tight entropic uncertainty relations for these families, in the form (cid:80) k H ( O k ) (cid:62) α d , where H ( O k ) is the Shannon entropy of the measurement outcomes of O k and α d is a constant. We showthat most of our bounds are stronger than previously known ones. We also give the form of thestates that attain these inequalities. Entropic uncertainty relations [1–3] express the con-cept of quantum uncertainty nicely since their lowerbound is typically state-independent, in contrast to theHeisenberg-Robertson ones [4, 5]. The most used one isthe Maassen-Uffink relation [3], H ( A ) + H ( B ) (cid:62) − c = q MU , (1)where H ( A ) and H ( B ) are the Shannon entropies of themeasurement outcomes of two observables A and B , and c = max j,k |(cid:104) a j | b k (cid:105)| is the maximum overlap betweentheir eigenstates. It is a state-independent bound, mean-ingful even if the observables share some common eigen-states. The bound (1) is tight if A and B have mu-tually unbiased bases (MUBs) as eigenstates. Strongerbounds for arbitrary observables, which involve the sec-ond largest term in |(cid:104) a j | b k (cid:105)| , have been found recentlyin [6] and [7]. If one considers more than two observ-ables, tight bounds were proven only in few cases, mostof them in dimension d = 2. For a complete set of MUBsthe strongest bounds were derived by Ivanovic in [8] forodd d , and by Sanchez in [9] for even d . Moreover, somebounds for an incomplete set of MUBs are in [10].In this paper we derive tight entropic uncertainty re-lations for more than two observables for systems of di-mensions d = 3 , ,
5, both for spin observables and forarbitrary numbers of MUBs. On one hand, for spin ob-servables we find H ( S x ) + H ( S y ) + H ( S z ) (cid:62) γ s (2)with γ s = 2 , − log (cid:39) . , .
12 for spin s = 1 , , s = 1 has been de-rived analytically, while the rest numerically. For half-integer spins the inequality is saturated by any of theeigenstates of the three spin observables, while for integerspins the inequality is saturated only by null projectionstates. Moreover, we find H ( S j ) + H ( S k ) (cid:62) ξ s (3)for all j, k = x, y, z ( j (cid:54) = k ) with ξ s = 1 , . , .
56 forspin s = 1 , ,
2. The case s = 1 coincides with (1),but the other cases are stronger than previous results.On the other hand, for observables { A j } with MUBs aseigenstates (the eigenvalues are irrelevant for EURs) we find, for dimension d = 3 (where up to four MUBs exist): H ( A ) + H ( A ) + H ( A ) (cid:62) , (4) H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62)
4; (5)for dimension d = 4 (where up to five MUBs exist): H ( A ) + H ( A ) + H ( A ) (cid:62) , (6) H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62) , (7) H ( A ) + H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62)
7; (8)and, finally, for dimension d = 5: H ( A ) + H ( A ) + H ( A ) (cid:62) H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62) .
34 (10) (cid:80) j =1 H ( A j ) (cid:62) .
33 (11) (cid:80) j =1 H ( A j ) (cid:62) . . (12)In addition to the above bounds, we also provide theform of the states that saturate them and we comparethem to previous results in the literature.The paper is organized as follows. In Sec. I we considerspin observables. The case s = 1 is developed analyticallyfrom a recent parametrization of the state [11], while theother cases are solved numerically. In Sec. II we considerthe observables with MUBs as eigenstates: after a briefreview of the previous results, we derive tight entropicuncertainty relations through numerical methods. In allcases, we detail the classes of states that saturate theobtained relations. In the appendix, we give the detailsof the numerical procedures we employed. I. ENTROPIC UNCERTAINTY RELATIONSFOR SPIN OBSERVABLES
We start by considering the entropic uncertainty rela-tions (EUR) relative to the spin observables S x , S y and S z for systems of different dimensions. A. Spin The state of a three-dimensional system can be writtenin terms of S x , S y and S z as [11] ρ = (cid:88) j = x,y,z (cid:18) ω j (cid:0) I − S j (cid:1) + a j S j + q j Q j (cid:19) , (13) a r X i v : . [ qu a n t - ph ] J a n where Q j is the anti-commutator of S k and S l , with j (cid:54) = k, l , i.e. Q j = { S k , S l } , and ω j = 1 − (cid:104) S j (cid:105) , a j = (cid:104) S j (cid:105) , q j = (cid:104) Q j (cid:105) , (14)with 0 ≤ ω j ≤ | a j | ≤ . In matrix form (13) is ρ = ω x − ia z − q z ia y − q y ia z − q z ω y − ia x − q x − ia y − q y ia x − q x ω z . (15)The condition Tr[ ρ ] = 1 implies ω x + ω y + ω z = 1 . (16)Since ρ is positive-semidefinite, all principal minors of theright-hand-side of (15) are non-negative, which impliesthe three inequalities 4 ω k ω l (cid:62) a j , for k, l = x, y, z and j (cid:54) = k , j (cid:54) = l . These inequalities can be expressed also as − √ ω k ω j ≤ a j ≤ √ ω k ω l . (17)In the representation where S j are diagonal, the spincomponents are S x = − i i , S y = i − i , (18) S z = − i i . (19)The eigenstates of S x are then given by: | S x = 0 (cid:105) = , | S x = ± (cid:105) = 1 √ ∓ i , (20)and similar relations for the other observables. The prob-abilities of S j are then given by p m =0 = ω j , p m = ± = 12 (1 − ω j ∓ a j ) , (21)whence one can calculate the Shannon entropies of S j as H ( S j ) = −
12 (1 − ω j + a j ) log (cid:20)
12 (1 − ω j + a j ) (cid:21) (22) −
12 (1 − ω j − a j ) log (cid:20)
12 (1 − ω j − a j ) (cid:21) − ω j log ω j . For two observables we find the optimal EUR (1), H ( S i ) + H ( S j ) (cid:62) , (23)indeed c = √ and moreover the above inequality is tightwhen calculated on the null projection state of any ofthe two observables. For three observables we obtain anEUR by finding an upper bound to − (cid:80) j H ( S j ). To Figure 1. Plot of the function Γ ( ω x , ω y ). this aim, we can use the conditions (17), employing themonotonicity of the logarithm as12 (1 − ω j ± a j ) log (cid:20)
12 (1 − ω j ± a j ) (cid:21) (24) ≤
12 (1 − ω j + 2 √ ω k ω l ) log (cid:20)
12 (1 − ω j + 2 √ ω k ω l ) (cid:21) . Then we have − (cid:88) j H ( S j ) ≤ (cid:88) j ω j log ω j + (1 − ω j + 2 √ ω k ω l ) log (cid:20) − ω j + 2 √ ω k ω l (cid:21) . (25)The right-hand side is a function − Γ ( ω x , ω y ) which de-pends only ω x and ω y since the ω j s are constrained by(16). Inverting the inequality, we find the EUR (cid:80) j H ( S j ) (cid:62) Γ ( ω x , ω y ) (cid:62) . (26)The lower bound Γ is plotted in Fig. 1. Its minimumvalue Γ = 2 is found for ω j = 1 and ω k = ω l = 0 . Theseconditions imply that a j = 0 for all j through (17). Thus, (cid:80) j H ( S j ) = 2, is attained on null projection states.This result shows a different behavior of the EUR forspin observables in the case of integer spin with respectto the half-integer case. A simple example of the latteris the qubit case: it was shown in [9] that for qubits wehave (cid:80) H ( S j ) (cid:62)
2, but the minimum is achieved byany of the eigenstates of one of the S j in contrast to thequtrit case obtained here. This difference in behaviorbetween integer and half-integer spins is true also forlarger spin numbers (see below).A straightforward generalization of (23) is obtained byrepeating that inequality for pairs of observables, obtain-ing (cid:80) H ( S j ) (cid:62) . It is weaker than our bound (26). B. Spin For a four-dimensional system, we are unaware of arepresentation of the density matrix in terms of the spinobservables and we cannot reproduce the derivation givenfor s = 1. We thus develop a simple computationalmethod that gives tight EUR for small system dimen-sions d .An arbitrary pure state | ψ (cid:105) of a d -dimensional systemdepends on 2 d − | ψ (cid:105) of the measurement outcomes is p ( a k ) = |(cid:104) a k | ψ (cid:105)| for an arbitrary observable A = (cid:80) k a k | a k (cid:105) (cid:104) a k | ,whence the entropy is H ( A ) = (cid:80) k − p ( a k ) log [ p ( a k )].Considering n observables A , A , ..A n we can calculatethe quantity (cid:80) nj =1 H ( A j ), which can be seen as a func-tion of the 2 d − H ( S j ) + H ( S k ) (cid:62) . , (27)with j, k = x, y, z and j (cid:54) = k .To compare this result with the previous results of [6]and [7], we can express these as [12] H ( A ) + H ( B ) (cid:62) max( q CP , q RP Z ) , with (28) q CP = 2 (cid:20) − log c + 12 (cid:0) − √ c (cid:1) log cc (cid:21) , (29) q RP Z = 2 (cid:104) − log c − log (cid:16) b + c c (cid:0) − b (cid:1)(cid:17)(cid:105) , (30)where b = √ c , c = max j,k |(cid:104) a j | b k (cid:105)| is the maximumoverlap among eigenstates of A and B , and c is the sec-ond maximum overlap. Both q CP and q RP Z are greaterthan q MU of (1). Our result (27) is an even strongerbound than both q CP and q RP Z . Indeed for s = 1we have c = (cid:113) and c = √ , so q CP = 1 .
59 and q RP Z = 1 . S j , indeed for any eigenstate we have H ( S j ) + H ( S k ) (cid:62) . . Instead, it is saturated by the state | ψ (cid:105) = sin (15 ◦ ) | (cid:105) + cos (15 ◦ ) | (cid:105) , (31)and by similar superpositions weighted by the angle α =15 ◦ . The bound (27) is in agreement with the numericalbound found in [7], but here we find also the state thatachieves the minimum.For the case of three spin observables we find H ( S x ) + H ( S y ) + H ( S z ) (cid:62) −
34 log . . (32)If we employ (27) to obtain a bound for three observables,(by applying it to each pair of observables) we find H ( S x ) + H ( S y ) + H ( S z ) (cid:62) · (1 .
71) = 2 . , (33)which is weaker than (32). The same argument appliedto q RP Z of (30) leads to H ( S x ) + H ( S y ) + H ( S z ) (cid:62) · .
68 = 2 .
52: also in this case our result (32) is strongerthan previous ones.The lower bound (32) is achieved by the eigenstates ofany of three observables S j . This generalizes the resultfound by Sanchez in [9]: indeed in this case the MUBsrepresent also the spin components. As mentioned above,the EUR for half-integer and integer spin values are at-tained for different classes of states. C. Spin A spin 2 system has dimension d = 5. Using the samealgorithm detailed in the previous section, we find H ( S j ) + H ( S k ) (cid:62) .
56 (34) H ( S x ) + H ( S y ) + H ( S y ) (cid:62) . . (35)Both the above inequalities are saturated by the eigen-states corresponding to the eigenvalue 0 of any of thethree observables S j , the null projection state (as in thecase s = 1). For example, the above inequalities are sat-urated by the state | S x = 0 (cid:105) = 12 (cid:114) | (cid:105) − | (cid:105) + 12 (cid:114) | (cid:105) . (36)The comparison of (34) with the previously knownbounds q MU , q CP and q RCZ shows that, again, our re-sult is stronger. In fact, in this case we have c = (cid:113) and c = . Therefore, q MU = 1 . q CP = 1 .
48 and q RP Z = 1 .
53, which are weaker than (34). Instead, thenumerical bound found in [7] agrees with ours, but wealso provide the states that saturate it. If we considerthe application of (35) to three spin observables we wouldobtain H ( S x ) + H ( S y ) + H ( S y ) (cid:62) · .
56 = 2 . , which is weaker than (35): the three-observable boundis again stronger than the ones obtained by joining two-observable bounds. II. ENTROPIC UNCERTAINTY RELATIONSFOR ARBITRARY NUMBERS OF MUBS
We now consider the EURs relative to observables thathave mutually unbiased bases (MUBs) as eigenstates (theeigenvalues are irrelevant for the EURs). To obtain theEURs we use the same procedure detailed in Sec. I B.However, we must also calculate the MUBs for each di-mension. In a d -dimensional Hilbert space there exist d + 1 MUBs if d is a power of a prime, otherwise onlythree bases are known to exist [13]. The proprieties ofMUBs strongly depend on the dimension, e.g. for a qubit,MUBs are also the eigenbases of the spin observables,but this is not true for d >
2. The problem of findingMUBs can be translated into finding Hadamard matri-ces: the columns of such matrices are the states of theMUBs. This problem was solved in [14] for dimensions d = 2 , , ,
5. Here we use that result to study EURs: foreach dimension d = 3 , , d + 1 observ-ables A , A , ..., A d +1 that have MUBs as eigenstates.We now briefly review previous results for EURs withMUBs observables. For any number L of these observ-ables, we can construct an EUR with a trivial general-ization of Maassen and Uffink’s relation (1) by applying(1) to pairs of bases, obtaining L (cid:88) i =1 H ( A i ) (cid:62) L d. (37)However, this inequality is almost never tight. A betterbound was given in [8] for L = d + 1: (cid:80) Li =1 H ( A i ) (cid:62) ( d + 1) (log ( d + 1) −
1) = q I , (38)which is also not always tight. For even dimension d , astronger bound was given in [9]: L (cid:88) i =1 H ( A i ) (cid:62) (cid:18) d d d + 12 log d + 12 (cid:19) = q S , (39)which is tight only in dimension two. For L < d +1 in [15]it has been shown that if the Hilbert space dimension isa square, that is d = r , then for L < r + 1 the inequality(37) is tight, namely L (cid:88) i =1 H ( A i ) (cid:62) L d = q BW . (40)A further bound for L < d + 1 was given in [10]: L (cid:88) i =1 H ( A i ) (cid:62) − L log (cid:18) d + L − d · L (cid:19) = q A . (41)For more details on the above bounds, we refer to [12].We now present our results which are tight for all dimen-sions and all numbers L of MUBs. A. Dimension Three
In dimension d = 3 four MUBs exist A , A , A and A , whose states are respectively given by the columnsof the Hadamard matrices M = , M = √ ω ω ω ω , (42) M = √ ω ω ω ω , M = √ ω ω ω ω , with ω = exp (cid:0) πi (cid:1) . If the system is prepared in an eigen-state of any of the MUBs, the entropy of that observ-able is null while the other entropies are maximal: e.g. if H ( A ) = 0, then we have H ( A ) + H ( A ) + H ( A ) =2 log
3. In contrast to the qubit case d = 2, this is not the state that gives the strongest EUR for d = 3. Indeed,the state √ ( | (cid:105) − | (cid:105) ) has entropies for all MUBs equalto 1: H ( A i ) = 1. Therefore, H ( A ) + H ( A ) + H ( A ) (cid:62) H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62) . (44)We have numerically shown that the above inequalitiesare the optimal ones. In addition to the above state, theyare saturated also by the following states e iϕ | (cid:105) + | (cid:105)√ , e iϕ | (cid:105) + | (cid:105)√ , e iϕ | (cid:105) + | (cid:105)√ , (45)where ϕ = π , π, π . Our bound (43) is stronger than(37), which in this case gives H ( A ) + H ( A ) + H ( A ) = log .
38. For L = 3 the bound (41) gives q A =2 .
54, that is also weaker than (43). For a complete setof MUBs L = 4, the bound (38) gives q I = 4 and isthen equal to our relation (44). However, here we haveproven that (44) is a tight relation for d = 3, and we haveprovided the states achieve the minimum. B. Dimension Four
In dimension d = 4 five MUBs exist, whose states aregiven by the columns of the Hadamard matrices M = , (46) M = − − − − − − , M = − − − i i i − ii − i i − i ,M = i − i i − i − − i − i − i i , M = i − i i − ii i − i i − − . Since d = 4 = r is a square for r = 2, then for L 77. Therefore, our inequalityis stronger than both.In the case of L = 5 = d + 1 observables (the completeset of MUBs), we find H ( A ) + H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62) , (51)which is saturated by states of the following form: | ψ jk (cid:105) = 1 √ (cid:16) | j (cid:105) ± ( i ) t | k (cid:105) (cid:17) , (52)with t = 0 , j and k are the eigenstates of A . For d = 4 the inequality (39) gives q S = 2 + log . C. Dimension Five In dimension d = 5 six MUBs exist, whose states aregiven by the columns of the Hadamard matrices M = , (53) M = 1 √ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω , (54) M = 1 √ ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ω , (55) M = 1 √ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω , (56) M = 1 √ ω ω ω ωω ω ω ωω ω ω ω ω ω ω ω , (57) M = 1 √ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω , (58) with ω = exp (cid:0) πi (cid:1) . For three MUBs observables wefind that the optimal bound is H ( A ) + H ( A ) + H ( A ) (cid:62) , (59)which is saturated by any eigenstate of any of the threeMUBs, as in the qubit d = 2 case (also there the EURfor three complementary observables is saturated by theeigenstates of the observables). The bound (59) is theonly known entropic uncertainty relation, apart from thequbit case, with more than two observables that hasthis property. In this respect, it is somewhat similar toMaassen and Uffink’s (1): they are both achieved by theeigenstates of one of the observables (so that the entropiesof the others are maximum). For L = 3 in (41) we have q A = 3 . 30 while 2 log . . Our bound is strongerthan these also in this case.For four MUBs we find that the optimal bound is H ( A ) + H ( A ) + H ( A ) + H ( A ) (cid:62) . , (60)and the minimum is achieved by states that are superpo-sition of four basis states, e.g. | ψ (cid:105) = 0 . e i π | (cid:105) +0 . | (cid:105) +0 . e i π | (cid:105) +0 . | (cid:105) . (61)In this case we have q A = 5 . 28, that is again weaker thanour bound (60). For five MUBs we find H ( A )+ H ( A )+ H ( A )+ H ( A )+ H ( A ) (cid:62) . . (62)and, finally, for the complete set of six MUBs we find (cid:88) i =1 H ( A i ) (cid:62) . . (63)The two above inequalities are again minimized by statesthat can be expressed by the superposition of four basisstates, having the same form of (61). For L = 5 we cancompare (62) to (41) which gives a weaker bound q A =7 . 34, while for the complete set of MUBs we can compare(63) to (38), which gives a weaker bound q I = 9 . III. CONCLUSIONS In this paper we have found several tight entropic un-certainty relations for two classes of observables: the spinobservables S x , S y , S z and the observables { A j } withMUBs eigenstates.For the case of spin observables, for s = 1 we founda tight relation (26) for the complete set of spin observ-ables, its minimum value is achieved by null projectionstates of any of three observables. The same types ofstates saturate also the inequality (35) for the case of s = 2. Instead, in the case s = the inequality (32)is minimized by eigenstates of any of three spin observ-ables. For both s = and s = 2 we have also foundtight inequalities for two spin observables, which are inagreement with the optimal bound found in [7], and wehave given the states that minimize them. In the case of s = 2 they are the null projections states.For the case of MUBs observables, we have derivedseveral tight inequalities for dimensions d = 3 , , 5. For d = 3 the results (44) equals the previous bound (38) buthere we also found the class of states that saturates it. In contrast, for d = 4 , 5, the bounds (51) and (63) representstronger EUR than known ones. New inequalities havebeen also found for incomplete sets of MUBs in everydimension: in each case the new bounds are tight and wehave derived the states that achieve the minimum. Wenote the peculiar behavior of (59), which is achieved byany eigenstate of one of the three MUBs, resembling thebehavior of qubit systems. [1] D. Deutsch, Uncertainty in Quantum Measurements ,Phys. Rev. Lett. 50, 631 (1983).[2] K. Kraus, Complementary observables and uncertaintyrelations , Phys. Rev. D 35, 3070 (1987).[3] H. Maassen, J.B.M. Uffink, Generalized entropic uncer-tainty relations , Phys. Rev. Lett. 60, 1103 (1988).[4] W. Heisenberg, Uber den anschaulichen Inhalt der quan-tentheoretischen Kinematik und Mechanik , Zeitschrift furPhysik 43 (3-4), 1721927).[5] H. P. Robertson, The Uncertainty Principle, Phys. Rev.34 (1), 163 (1929).[6] P. J. Coles and M. Piani, Improved entropic uncertaintyrelations and information exclusion relations , Phys. Rev.A 89, 022112 (2014).[7] (cid:32)L. Rudnicki, Z. Pucha(cid:32)la, and K. ˙Zyczkowski, Strong ma-jorization entropic uncertainty relations, Phys. Rev. A89, 052115 (2014).[8] I. D. Ivanovic, An inequality for the sum of entropies ofunbiased quantum measurements , J. Phys. A: Math. Gen.25 (7), 363 (1992).[9] J. Sanchez, Improved bounds in the entropic uncertaintyand certainty relations for complementary observables ,Phys. Lett. A 201, 125 (1995).[10] A. Azarchs, Entropic uncertainty relations for incom-plete sets of mutually unbiased observables, arXiv:quant-ph/0412083 (2004).[11] P. Kurzy´nski, A. Ko(cid:32)lodziejski, W. Laskowski, and M.Markiewicz, Three-dimensional visualization of a qutrit. Phys. Rev. A 93, 062126 (2016).[12] P.J. Coles, M. Berta, M. Tomamichel and S. Wehner, Entropic uncertainty relations and their applications ,arXiv:1511.04857. (2015).[13] T. Durt, B. Englert,I. Bengtsson and K. ˙Zyczkowski , Onmutually unbiased bases , Int. J. Quantum Inform. 08,535 (2010).[14] S. Brierley, S. Weigert and I. Bengtsson, All mutuallyunbiased base in dimension two to five. Quantum Info.and Comp. Vol 10, 0803-0820 (2010).[15] M.A. Ballester and S. Wehner, Entropic uncertainty re-lations for more than two observables , Phys. Rev. A. 75,022319 (2007). Appendix A: Numerical methods Here we detail the numerical methods used to derivemost of our entropic uncertainty relations. We have usedthe software package Mathematica. For the sake of illus-tration, we consider the case of s = . The most general pure state of a quantum system for d = 4 is | ψ (cid:105) = e iχ sin a sin a cos a | (cid:105) + e iχ sin a sin a sin a | (cid:105) ++ e iχ sin a cos a | (cid:105) + cos a | (cid:105) (A1)where a i ∈ (cid:2) , π (cid:3) and χ ∈ [0 , π ] . To compute the prob-ability distributions of S x , S y and S z over the state | ψ (cid:105) we work in the representation of eigenstates of S z . Inthis representation, the spin matrices are S x = 12 √ √ √ 30 0 √ , (A2) S y = 12 i √ −√ − √ 30 0 −√ , (A3) S z = 12 − − . (A4)The probability distribution of S z is p ( S z = +2) = sin a sin a cos a ; (A5) p ( S z = +1) = sin a sin a sin a ; (A6) p ( S z = − 1) = sin a cos a ; (A7) p ( S z = − a . (A8)Then the entropy is H ( S z ) = − (cid:80) l p l ( S z = l ) log p l ( S z = l ) , which depends only onthe three parameters a j .To calculate the entropy for S x , consider its eigenstates | S x = ± (cid:105) = 12 √ (cid:16) | (cid:105) ± √ | (cid:105) − √ | (cid:105) ± | (cid:105) (cid:17) ; (A9) | S x = ± (cid:105) = 12 √ (cid:16) √ | (cid:105) ± | (cid:105) − | (cid:105) ∓ √ | (cid:105) (cid:17) . We can compute the probability distribution of S x over | ψ (cid:105) with p ( S x = ± l ) = |(cid:104) ψ | S x = ± l (cid:105)| , (A10)This expression depends on all six parameters of | ψ (cid:105) ,and we can use it to calculate the entropy H ( S x ) = − (cid:80) l p l ( S x = l ) log p l ( S x = l ) . An analogous procedure can be used for S y , whose eigen-states are | S y = ± (cid:105) = 12 √ (cid:16) | (cid:105) ± i √ | (cid:105) − √ | (cid:105) ∓ i | (cid:105) (cid:17) ;(A11) | S y = ± (cid:105) = 12 √ (cid:16) √ | (cid:105) ± i | (cid:105) + | (cid:105) ± i √ | (cid:105) (cid:17) , (A12)whence we can calculate the probabilities and the en-tropy.To obtain the optimal EUR, we need to minimize thesum of two or three of above entropies. Due to theirnon linear dependence on the parameters, it is highlynontrivial to find the minimum analytically. We havetherefore resorted to numerical methods: Mathematicapermits the minimization of a function f ( x , ..., x n ) thatdepends on n parameters with the routine N M inimize [ { f ( x , ..., x n ) , γ ( x , .., x n ) } , { x , .., x n } ] , (A13)where γ represents possible constraints. This routine re-turns both the minimum value of the function and alsothe parameter values that attain it, which in our caseidentify the states that minimize the EUR. For example, if we define f ( a , a , a , χ , χ , χ ) = H ( S x ) + H ( S z ) , (A14)the instruction N M inimize [ f ( a i , χ i ) , { a , a , a , χ , χ , χ } ] , (A15)returns (cid:110) . , (cid:110) a → π , a → π , a → π , χ → π (cid:111)(cid:111) , (A16)which implies (34). The other relations we derived canbe similarly obtained.For example, for the case of spin 2 we can repeat theabove procedure. Again, we can choose the representa-tion of eigenbasis of S z which gives S x = 12 √ √ √ √ , (A17) S y = 12 i − √ −√ √ −√ − , (A18) S z = − ..