Tight entropic uncertainty relations for systems with dimension three to five

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by  Alberto Riccardi

Tight 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 for systems with dimension d=3,4,5: the
k
spinobservablesS ,S andS ,andtheobservablesthathavemutuallyunbiasedbasesaseigenstates.
x y z
Wederivetightentropicuncertaintyrelationsforthesefamilies,intheform(cid:80) H(O )(cid:62)α ,where
k k d
H(O ) is the Shannon entropy of the measurement outcomes of O and α is a constant. We show
k k d
that most of our bounds are stronger than previously known ones. We also give the form of the
states that attain these inequalities.
Entropic uncertainty relations [1–3] express the con- find,fordimensiond=3(whereuptofourMUBsexist):
cept of quantum uncertainty nicely since their lower
7 H(A )+H(A )+H(A )(cid:62)3, (4)
bound is typically state-independent, in contrast to the 1 2 3
1 H(A )+H(A )+H(A )+H(A )(cid:62)4; (5)
0 Heisenberg-Robertson ones [4, 5]. The most used one is 1 2 3 4
2 the Maassen-Uffink relation [3], for dimension d=4 (where up to five MUBs exist):

n H(A )+H(A )+H(A )(cid:62)3, (6)
Ja H(A)+H(B)(cid:62)−2log2c=qMU, (1) H(A1)+H(A12)+H(A23)+H(A34)(cid:62)5, (7)
6 where H(A) and H(B) are the Shannon entropies of the H(A )+H(A )+H(A )+H(A )+H(A )(cid:62)7; (8)
1 2 3 4 5
1 measurement outcomes of two observables A and B, and
and, finally, for dimension d=5:
c = max |(cid:104)a |b (cid:105)| is the maximum overlap between
j,k j k
] theireigenstates. Itisastate-independentbound,mean- H(A )+H(A )+H(A )(cid:62)2log 5 (9)
h 1 2 3 2
p ingful even if the observables share some common eigen- H(A )+H(A )+H(A )+H(A )(cid:62)6.34 (10)
1 2 3 4
- states. The bound (1) is tight if A and B have mu-
t (cid:80)5 H(A )(cid:62)8.33 (11)
n tually unbiased bases (MUBs) as eigenstates. Stronger j=1 j
a bounds for arbitrary observables, which involve the sec- (cid:80)6 H(A )(cid:62)10.25. (12)
u ond largest term in |(cid:104)a |b (cid:105)|, have been found recently j=1 j
q j k In addition to the above bounds, we also provide the
in [6] and [7]. If one considers more than two observ-
[ form of the states that saturate them and we compare
ables, tight bounds were proven only in few cases, most
them to previous results in the literature.
1 ofthemindimensiond=2. ForacompletesetofMUBs
v the strongest bounds were derived by Ivanovic in [8] for Thepaperisorganizedasfollows. InSec.Iweconsider
4 spinobservables. Thecases=1isdevelopedanalytically
odd d, and by Sanchez in [9] for even d. Moreover, some
0 fromarecentparametrizationofthestate[11], whilethe
bounds for an incomplete set of MUBs are in [10].
3 othercasesaresolvednumerically. InSec.IIweconsider
4 In this paper we derive tight entropic uncertainty re-
the observables with MUBs as eigenstates: after a brief
0 lations for more than two observables for systems of di-
review of the previous results, we derive tight entropic
. mensions d = 3,4,5, both for spin observables and for
1 uncertainty relations through numerical methods. In all
arbitrary numbers of MUBs. On one hand, for spin ob-
0 cases, we detail the classes of states that saturate the
7 servables we find
obtained relations. In the appendix, we give the details
1
: H(S )+H(S )+H(S )(cid:62)γ (2) of the numerical procedures we employed.
v x y z s
i
X with γ = 2,3− 3log 3 (cid:39) 3.62,3.12 for spin s = 1,3,2
s 4 2 2 I. ENTROPIC UNCERTAINTY RELATIONS
r respectively. Theseinequalitiesareallstrongerthanpre-
FOR SPIN OBSERVABLES
a viously known results. The case of s = 1 has been de-
rived analytically, while the rest numerically. For half-
We start by considering the entropic uncertainty rela-
integer spins the inequality is saturated by any of the
tions (EUR) relative to the spin observables S , S and
eigenstatesofthethreespinobservables,whileforinteger x y
S for systems of different dimensions.
spins the inequality is saturated only by null projection z
states. Moreover, we find
H(S )+H(S )(cid:62)ξ (3) A. Spin 1
j k s
for all j,k = x,y,z (j (cid:54)= k) with ξ = 1,1.71,1.56 for Thestateofathree-dimensionalsystemcanbewritten
s
spin s = 1,3,2. The case s = 1 coincides with (1), in terms of S , S and S as [11]
2 x y z
but the other cases are stronger than previous results. (cid:18) (cid:19)
On the other hand, for observables {Aj} with MUBs as ρ= (cid:88) ωj(cid:0)I−Sj2(cid:1)+ ajSj +2 qjQj , (13)
eigenstates (the eigenvalues are irrelevant for EURs) we
j=x,y,z
2
where Q is the anti-commutator of S and S , with j (cid:54)=
j k l
k,l, i.e. Q ={S ,S }, and
j k l
ω =1−(cid:104)S2(cid:105), a =(cid:104)S (cid:105),q =(cid:104)Q (cid:105), (14)
j j j j j j
with 0≤ω ≤1 and |a |≤1. In matrix form (13) is
j j
 ωx −iaz2−qz iay2−qy  Figure 1. Plot of the function Γ(ωx,ωy).
ρ= iaz2−qz ωy −iax2−qx . (15)
−iay−qy iax−qx ω
2 2 z
this aim, we can use the conditions (17), employing the
The condition Tr[ρ]=1 implies monotonicity of the logarithm as
(cid:20) (cid:21)
ω +ω +ω =1. (16) 1 1
x y z (1−ω ±a )log (1−ω ±a ) (24)
2 j j 2 2 j j
Sinceρispositive-semidefinite,allprincipalminorsofthe (cid:20) (cid:21)
1 √ 1 √
right-hand-side of (15) are non-negative, which implies ≤ 2(1−ωj +2 ωkωl)log2 2(1−ωj +2 ωkωl) .
the three inequalities 4ω ω (cid:62) a2, for k,l = x,y,z and
k l j
j (cid:54)=k, j (cid:54)=l. These inequalities can be expressed also as
Then we have
√ √
−2 ω ω ≤a ≤2 ω ω . (17) (cid:88) (cid:88)
k j j k l − H(S )≤ ω log ω
j j 2 j
In the representation where S2 are diagonal, the spin j j √
j √ (cid:20)1−ω +2 ω ω (cid:21)
components are +(1−ω +2 ω ω )log j k l . (25)
j k l 2 2
   
0 0 0 0 0 i
Sx =0 0 −i, Sy = 0 0 0, (18) The right-hand side is a function −Γ(ωx,ωy) which de-
0 i 0 −i 0 0 pends only ω and ω since the ω s are constrained by
x y j
  (16). Inverting the inequality, we find the EUR
0 −i 0
Sz = i 0 0. (19) (cid:80) H(S )(cid:62)Γ(ω ,ω )(cid:62)2. (26)
0 0 0 j j x y
The lower bound Γ is plotted in Fig. 1. Its minimum
The eigenstates of S are then given by:
x value Γ = 2 is found for ω = 1 and ω = ω = 0. These
j k l
1 1  0  c(cid:80)ondHiti(oSns)i=mp2l,yisthaatttaaijne=d0onfornualllljprtohjreocutgiohn(1st7a)t.eTs.hus,
|Sx =0(cid:105)=0, |Sx =±1(cid:105)= √2∓i, (20) Tj his rejsult shows a different behavior of the EUR for
0 1
spin observables in the case of integer spin with respect
to the half-integer case. A simple example of the latter
andsimilarrelationsfortheotherobservables. Theprob-
is the qubit case: it was shown in [9] that for qubits we
abilities of S are then given by
j have (cid:80)H(S ) (cid:62) 2, but the minimum is achieved by
j
1 any of the eigenstates of one of the Sj in contrast to the
pm=0 =ωj, pm=±1 = 2(1−ωj ∓aj), (21) qutrit case obtained here. This difference in behavior
between integer and half-integer spins is true also for
whence one can calculate the Shannon entropies of S as larger spin numbers (see below).
j
(cid:20) (cid:21)
1 1 Astraightforwardgeneralizationof(23)isobtainedby
H(S )=− (1−ω +a )log (1−ω +a ) (22)
j 2 j j 2 2 j j repeatingthatinequalityforpairsofobservables,obtain-
(cid:20) (cid:21) ing (cid:80)H(S )(cid:62) 3. It is weaker than our bound (26).
1 1 j 2
− (1−ω −a )log (1−ω −a ) −ω log ω .
2 j j 2 2 j j j 2 j
For two observables we find the optimal EUR (1), B. Spin 3
2
H(Si)+H(Sj)(cid:62)1, (23) For a four-dimensional system, we are unaware of a
representation of the density matrix in terms of the spin
indeedc= √1 andmoreovertheaboveinequalityistight observablesandwecannotreproducethederivationgiven
2
when calculated on the null projection state of any of for s = 1. We thus develop a simple computational
the two observables. For three observables we obtain an method that gives tight EUR for small system dimen-
(cid:80)
EUR by finding an upper bound to − H(S ). To sions d.
j j
3
An arbitrary pure state |ψ(cid:105) of a d-dimensional system 3·1.68=2.52: alsointhiscaseourresult(32)isstronger
2
dependson2d−2realparameters. Itissufficienttocon- than previous ones.
siderpurestatesbecauseoftheconcavityoftheShannon The lower bound (32) is achieved by the eigenstates of
entropy: mixed states have greater entropy. The prob- any of three observables S . This generalizes the result
j
ability on |ψ(cid:105) of the measurement outcomes is p(a ) = found by Sanchez in [9]: indeed in this case the MUBs
k
|(cid:104)a |ψ(cid:105)|2 foranarbitraryobservableA=(cid:80) a |a (cid:105)(cid:104)a |, representalsothespincomponents. Asmentionedabove,
whkence the entropy is H(A) = (cid:80) −p(a )klokg [pk(a k)]. the EUR for half-integer and integer spin values are at-
k k 2 k
Considering n observables A ,A ,..A we can calculate tained for different classes of states.
1 2 n
the quantity (cid:80)n H(A ), which can be seen as a func-
j=1 j
tionofthe2d−2parametersrepresentingthestate. This
functioncanthenbenumericallyminimizedoverthispa- C. Spin 2
rameter space. In addition to finding the minimum, we
thenalsofindthestatesthatsaturatethebounds, which A spin 2 system has dimension d=5. Using the same
are then tight. In the Appendix we give more details on algorithm detailed in the previous section, we find
the computational procedure, here we present only the
results. H(Sj)+H(Sk)(cid:62)1.56 (34)
For the case of two spin observables we find H(S )+H(S )+H(S )(cid:62)3.12. (35)
x y y
H(Sj)+H(Sk)(cid:62)1.71, (27) Both the above inequalities are saturated by the eigen-
states corresponding to the eigenvalue 0 of any of the
with j,k =x,y,z and j (cid:54)=k.
three observables S , the null projection state (as in the
j
To compare this result with the previous results of [6]
case s=1). For example, the above inequalities are sat-
and [7], we can express these as [12]
urated by the state
H(A)+H(B)(cid:62)max(qCP,qRPZ), with (28) 1(cid:114)3 1 1(cid:114)3
q =2(cid:20)−log c+ 1(cid:0)1−√c(cid:1)log c (cid:21), (29) |Sx =0(cid:105)= 2 2|0(cid:105)− 2|2(cid:105)+ 2 2|4(cid:105). (36)
CP 2 2 2 c
2
q =2(cid:104)−log c−log (cid:16)b2+ c2 (cid:0)1−b2(cid:1)(cid:17)(cid:105), (30) The comparison of (34) with the previously known
RPZ 2 2 c bounds qMU, qCP and qRCZ shows that, again, our re-
(cid:113)
√ sult is stronger. In fact, in this case we have c = 1 3
where b = 1+ c, c = max |(cid:104)a |b (cid:105)| is the maximum 2 2
overlap among2eigenstates ojf,kA anjdkB, and c2 is the sec- and c2 = 21. Therefore, qMU = 1.41, qCP = 1.48 and
ond maximum overlap. Both qCP and qRPZ are greater qRPZ = 1.53, which are weaker than (34). Instead, the
than q of (1). Our result (27) is an even stronger numerical bound found in [7] agrees with ours, but we
MU
bound than both q and q . Indeed for s = 1 also provide the states that saturate it. If we consider
CP RPZ
(cid:113) theapplicationof(35)tothreespinobservableswewould
we have c = 21 32 and c2 = 2√12, so qCP = 1.59 and obtain H(S ) + H(S ) + H(S ) (cid:62) 3 · 1.56 = 2.34,
q =1.68. x y y 2
RPZ which is weaker than (35): the three-observable bound
The bound (27) is not saturated by one of the eigen-
is again stronger than the ones obtained by joining two-
states of S , indeed for any eigenstate we have H(S )+
j j observable bounds.
H(S )(cid:62)1.81. Instead, it is saturated by the state
k
|ψ(cid:105)=sin(15◦)|0(cid:105)+cos(15◦)|2(cid:105), (31)
II. ENTROPIC UNCERTAINTY RELATIONS
FOR ARBITRARY NUMBERS OF MUBS
and by similar superpositions weighted by the angle α=
15◦. The bound (27) is in agreement with the numerical
bound found in [7], but here we find also the state that WenowconsidertheEURsrelativetoobservablesthat
achieves the minimum. havemutuallyunbiasedbases(MUBs)aseigenstates(the
For the case of three spin observables we find eigenvalues are irrelevant for the EURs). To obtain the
EURs we use the same procedure detailed in Sec. IB.
3 However, we must also calculate the MUBs for each di-
H(S )+H(S )+H(S )(cid:62)3− log 3=3.62. (32)
x y z 4 2 mension. In a d-dimensional Hilbert space there exist
d+1 MUBs if d is a power of a prime, otherwise only
Ifweemploy(27)toobtainaboundforthreeobservables,
three bases are known to exist [13]. The proprieties of
(by applying it to each pair of observables) we find
MUBsstronglydependonthedimension,e.g.foraqubit,
MUBs are also the eigenbases of the spin observables,
3
H(Sx)+H(Sy)+H(Sz)(cid:62) 2 ·(1.71)=2.56, (33) but this is not true for d > 2. The problem of finding
MUBs can be translated into finding Hadamard matri-
which is weaker than (32). The same argument applied ces: the columns of such matrices are the states of the
to q of (30) leads to H(S ) + H(S ) + H(S ) (cid:62) MUBs. This problem was solved in [14] for dimensions
RPZ x y z
4
d=2,3,4,5. HereweusethatresulttostudyEURs: for H(A ) = 0, then we have H(A )+H(A )+H(A ) =
1 1 2 3
eachdimensiond=3,4,5weconsideruptod+1observ- 2log 3. In contrast to the qubit case d = 2, this is not
2
ables A ,A ,...,A that have MUBs as eigenstates. thestatethatgivesthestrongestEURford=3. Indeed,
1 2 d+1
We now briefly review previous results for EURs with the state √1 (|1(cid:105)−|2(cid:105)) has entropies for all MUBs equal
2
MUBs observables. For any number L of these observ- to 1: H(A )=1. Therefore,
i
ables, we can construct an EUR with a trivial general-
ization of Maassen and Uffink’s relation (1) by applying H(A )+H(A )+H(A )(cid:62)3 (43)
1 2 3
(1) to pairs of bases, obtaining H(A )+H(A )+H(A )+H(A )(cid:62)4. (44)
1 2 3 4
L
(cid:88) L We have numerically shown that the above inequalities
H(A )(cid:62) log d. (37)
i 2 2 aretheoptimalones. Inadditiontotheabovestate,they
i=1
are saturated also by the following states
However, this inequality is almost never tight. A better
bound was given in [8] for L=d+1: eiϕ|0(cid:105)+|1(cid:105) eiϕ|0(cid:105)+|2(cid:105) eiϕ|1(cid:105)+|2(cid:105)
√ , √ , √ , (45)
2 2 2
(cid:80)L H(A )(cid:62)(d+1)(log (d+1)−1)=q , (38)
i=1 i 2 I
where ϕ = π,π,5π. Our bound (43) is stronger than
3 3
which is also not always tight. For even dimension d, a (37), whichinthiscasegivesH(A )+H(A )+H(A )=
1 2 3
stronger bound was given in [9]: 3log 3 = 2.38. For L = 3 the bound (41) gives q =
2 2 A
2.54, that is also weaker than (43). For a complete set
L (cid:18) (cid:19)
(cid:88)H(A )(cid:62) dlog d + d+1log d+1 =q , of MUBs L = 4, the bound (38) gives qI = 4 and is
i 2 2 2 2 2 2 S then equal to our relation (44). However, here we have
i=1 proventhat(44)isatightrelationford=3,andwehave
(39)
provided the states achieve the minimum.
whichistightonlyindimensiontwo. ForL<d+1in[15]
it has been shown that if the Hilbert space dimension is
asquare,thatisd=r2,thenforL<r+1theinequality
B. Dimension Four
(37) is tight, namely
In dimension d = 4 five MUBs exist, whose states are
L
(cid:88)H(A )(cid:62) Llog d=q . (40) given by the columns of the Hadamard matrices
i 2 2 BW
 
i=1 1 0 0 0
0 1 0 0
A further bound for L<d+1 was given in [10]: M1=0 0 1 0, (46)
0 0 0 1
(cid:88)L (cid:18)d+L−1(cid:19) 1 1 1 1   1 1 1 1 
H(A )(cid:62)−Llog =q . (41)
i=1 i 2 d·L A M2= 1211 −11 −−11 −11, M3= 21−1i 1i −i1 −−1i ,
1 −1 1 −1 i −i i −i
For more details on the above bounds, we refer to [12].
   
1 1 1 1 1 1 1 1
Wenowpresentourresultswhicharetightforalldimen-
sions and all numbers L of MUBs. M4= 21−i1 −−1i 1i −1i, M5= 21 ii −ii −ii −ii.
i −i −i i −1 −1 1 1
Since d = 4 = r2 is a square for r = 2, then for L <
A. Dimension Three
r+1 = 3 the inequality (37) is tight [15]. This is then
the best bound up to L=2. However, for d=4 we can
In dimension d = 3 four MUBs exist A , A , A and
1 2 3 consider up to L = 5. For example, for L = 3 we find
A , whose states are respectively given by the columns
4 that the optimal bound is
of the Hadamard matrices
    H(A )+H(A )+H(A )(cid:62)3. (47)
1 0 0 1 1 1 1 2 3
M1 =0 1 0, M2 = √131 ω ω2 , (42) It is achieved by the four states
0 0 1 1 ω2 ω
√ √
   
1 1 1 1 1 1 (|0(cid:105)±|1(cid:105)) 2, (|2(cid:105)±|3(cid:105))/ 2. (48)
M3 = √1 ω2 ω 1 , M4 = √1 ω ω2 1 ,
3 1 ω ω2 3 1 ω2 ω By symmetry, similar relations holds by permuting the
MUBs observables, but involving the superposition of
withω =exp(cid:0)2πi(cid:1). Ifthesystemispreparedinaneigen- different eigenstates. For example H(A1) + H(A2)√+
state of any of3the MUBs, the entropy of that observ- H(A4) (cid:62) 3 has√lower bound achieved by (|0(cid:105)±|2(cid:105))/ 2
able is null while the other entropies are maximal: e.g. if and (|1(cid:105)±|3(cid:105))/ 2.
5
In the case of L=4 observables we find with ω = exp(cid:0)2πi(cid:1). For three MUBs observables we
5
find that the optimal bound is
H(A )+H(A )+H(A )+H(A )(cid:62)5 (49)
1 2 3 4
H(A )+H(A )+H(A )(cid:62)2log 5, (59)
1 2 3 2
as the optimal bound, which is saturated by the states
which is saturated by any eigenstate of any of the three
√ √ √
(|0(cid:105)±|1(cid:105))/ 2, (|0(cid:105)±|2(cid:105))/ 2, (|0(cid:105)±i|3(cid:105))/ 2, (50) MUBs, as in the qubit d = 2 case (also there the EUR
√ √ √ for three complementary observables is saturated by the
(|1(cid:105)±i|2(cid:105))/ 2, (|1(cid:105)±|3(cid:105))/ 2, (|2(cid:105)±|3(cid:105))/ 2.
eigenstates of the observables). The bound (59) is the
onlyknownentropicuncertaintyrelation,apartfromthe
Compare our bound (49) to (40) and (41): for L=4 we
qubit case, with more than two observables that has
find q = 4 and q = 4.77. Therefore, our inequality
BW A this property. In this respect, it is somewhat similar to
is stronger than both.
Maassen and Uffink’s (1): they are both achieved by the
In the case of L=5=d+1 observables (the complete
eigenstatesofoneoftheobservables(sothattheentropies
set of MUBs), we find
of the others are maximum). For L = 3 in (41) we have
q = 3.30 while 2log 5 = 4.64. Our bound is stronger
H(A )+H(A )+H(A )+H(A )+H(A )(cid:62)7, (51) A 2
1 2 3 4 5 than these also in this case.
For four MUBs we find that the optimal bound is
which is saturated by states of the following form:
H(A )+H(A )+H(A )+H(A )(cid:62)6.34, (60)
1 (cid:16) (cid:17) 1 2 3 4
|ψ (cid:105)= √ |j(cid:105)±(i)t|k(cid:105) , (52)
jk
2 andtheminimumisachievedbystatesthataresuperpo-
sition of four basis states, e.g.
with t = 0,1 and j and k are the eigenstates of A . For
1
d = 4 the inequality (39) gives qS = 2+ 52log25 = 5.30, |ψ(cid:105)=0.19ei35π|0(cid:105)+0.19|1(cid:105)+0.68ei95π|3(cid:105)+0.68|4(cid:105). (61)
so that (39) is weaker than our bound (51) in this case.
Inthiscasewehaveq =5.28,thatisagainweakerthan
A
our bound (60). For five MUBs we find
C. Dimension Five
H(A )+H(A )+H(A )+H(A )+H(A )(cid:62)8.33. (62)
1 2 3 4 5
In dimension d = 5 six MUBs exist, whose states are
given by the columns of the Hadamard matrices and, finally, for the complete set of six MUBs we find
1 0 0 0 0 6
0 1 0 0 0 (cid:88)H(A )(cid:62)10.25. (63)
M1=0 0 1 0 0, (53) i
0 0 0 1 0 i=1
0 0 0 0 1
Thetwoaboveinequalitiesareagainminimizedbystates
1 1 1 1 1 
that can be expressed by the superposition of four basis
1 ω ω2 ω3 ω4
1   states, having the same form of (61). For L = 5 we can
M2= √511 ωω32 ωω4 ωω4 ωω23 , (54) compare (62) to (41) which gives a weaker bound qA =
1 ω4 ω3 ω2 ω 7.34,whileforthecompletesetofMUBswecancompare
 1 1 1 1 1  (63) to (38), which gives a weaker bound qI =9.51.
ω ω2 ω3 ω4 1
1  
M3= √5ωω44 ωω2 ω13 ω13 ωω2 , (55) III. CONCLUSIONS
ω 1 ω4 ω3 ω2
 1 1 1 1 1 
In this paper we have found several tight entropic un-
ω3 ω4 1 ω ω2
M4= √15ωω22 ω14 ωω3 ωω3 ω14 , (56) coebrstearivnatbylereslaStio,nSsf,orStwoancdlastsheesoofbosbersvearvbalebsle{s:Ath}ewspitinh
x y z j
ω3 ω2 ω3 1 ω4 MUBs eigenstates.
 1 1 1 1 1  For the case of spin observables, for s = 1 we found
1 ω2 ω3 ω4 1 ω  a tight relation (26) for the complete set of spin observ-
M5= √5ωω33 ω1 ωω22 ωω24 ω1 , (57) ables, its minimum value is achieved by null projection
states of any of three observables. The same types of
ω2 ω 1 ω4 ω3
states saturate also the inequality (35) for the case of
 1 1 1 1 1 
s = 2. Instead, in the case s = 3 the inequality (32)
M6= √15ωωω4 ωω143 ωω12 ωω122 ωωω343 , (58) iasbmlesin.imFoizredbobtyhesig=ens3tataensdosf a=ny22owf tehhreaevespailnsoobfoseurnvd-
2
ω4 ω3 ω2 ω 1 tight inequalities for two spin observables, which are in
6
agreement with the optimal bound found in [7], and we contrast,ford=4,5,thebounds(51)and(63)represent
have given the states that minimize them. In the case of stronger EUR than known ones. New inequalities have
s=2 they are the null projections states. been also found for incomplete sets of MUBs in every
dimension: ineachcasethenewboundsaretightandwe
For the case of MUBs observables, we have derived have derived the states that achieve the minimum. We
several tight inequalities for dimensions d = 3,4,5. For note the peculiar behavior of (59), which is achieved by
d=3theresults(44)equalsthepreviousbound(38)but any eigenstate of one of the three MUBs, resembling the
herewealsofoundtheclassofstatesthatsaturatesit. In behavior of qubit systems.
[1] D. Deutsch, Uncertainty in Quantum Measurements, pure state of a quantum system for d=4 is
Phys. Rev. Lett. 50, 631 (1983).
[2] K. Kraus, Complementary observables and uncertainty |ψ(cid:105)=eiχ0sina sina cosa |0(cid:105)+eiχ1sina sina sina |1(cid:105)+
0 1 2 0 1 2
relations, Phys. Rev. D 35, 3070 (1987).
+eiχ2sina cosa |2(cid:105)+cosa |1(cid:105) (A1)
[3] H. Maassen, J.B.M. Uffink, Generalized entropic uncer- 0 1 0
tainty relations, Phys. Rev. Lett. 60, 1103 (1988).
[4] W.Heisenberg,Uber den anschaulichen Inhalt der quan- whereai ∈(cid:2)0,π2(cid:3)andχ0 ∈[0,2π].Tocomputetheprob-
tentheoretischenKinematikundMechanik,Zeitschriftfur ability distributions of S ,S and S over the state |ψ(cid:105)
x y z
Physik 43 (3-4), 1721927). we work in the representation of eigenstates of S . In
z
[5] H. P. Robertson, The Uncertainty Principle, Phys. Rev. this representation, the spin matrices are
34 (1), 163 (1929).
√
[6] P. J. Coles and M. Piani, Improved entropic uncertainty  
0 3 0 0
relationsandinformationexclusionrelations,Phys.Rev. √
[7] AL(cid:32).8R9u,d0n2i2c1k1i,2Z(.2P01u4c)h.al(cid:32)a,andK.Z˙yczkowski,Strong ma- Sx = 12 03 20 √20 √03, (A2)
jorization entropic uncertainty relations, Phys. Rev. A 0 0 3 0
√
89, 052115 (2014).  
0 3 0 0
[8] I. D. Ivanovic, An inequality for the sum of entropies of √
u25nb(i7a)s,ed36q3ua(1n9tu9m2).measurements,J.Phys.A:Math.Gen. Sy = 21i−0 3 −02 √02 √03, (A3)
[9] J. Sanchez, Improved bounds in the entropic uncertainty 0 0 − 3 0
and certainty relations for complementary observables, 3 0 0 0 
Phys. Lett. A 201, 125 (1995).
1 0 1 0 0
[10] A. Azarchs, Entropic uncertainty relations for incom- S =  . (A4)
plete sets of mutually unbiased observables, arXiv:quant- z 20 0 −1 0 
0 0 0 −3
ph/0412083 (2004).
[11] P. Kurzyn´ski, A. Ko(cid:32)lodziejski, W. Laskowski, and M.
Markiewicz, Three-dimensional visualization of a qutrit. The probability distribution of Sz is
Phys. Rev. A 93, 062126 (2016).
[12] P.J. Coles, M. Berta, M. Tomamichel and S. Wehner, p(S =+2)=sin2a sin2a cos2a ; (A5)
z 0 1 2
Entropic uncertainty relations and their applications,
p(S =+1)=sin2a sin2a sin2a ; (A6)
arXiv:1511.04857. (2015). z 0 1 2
[13] T.Durt,B.Englert,I.BengtssonandK.Z˙yczkowski, On p(S =−1)=sin2a cos2a ; (A7)
z 0 1
mutually unbiased bases , Int. J. Quantum Inform. 08,
p(S =−2))=cos2a . (A8)
535 (2010). z 0
[14] S. Brierley, S. Weigert and I. Bengtsson, All mutually
Then the entropy is H(S ) =
unbiased base in dimension two to five. Quantum Info. z
(cid:80)
− p (S =l)log p (S =l), which depends only on
and Comp. Vol 10, 0803-0820 (2010). l l z 2 l z
[15] M.A. Ballester and S. Wehner, Entropic uncertainty re- the three parameters aj.
lations for more than two observables, Phys. Rev. A. 75, To calculate the entropy for Sx, consider its eigenstates
022319 (2007).
1 (cid:16) √ √ (cid:17)
|S =±2(cid:105)= √ |0(cid:105)± 3|1(cid:105)− 3|2(cid:105)±|3(cid:105) ; (A9)
x
2 2
1 (cid:16)√ √ (cid:17)
Appendix A: Numerical methods |Sx =±1(cid:105)= √ 3|0(cid:105)±|1(cid:105)−|2(cid:105)∓ 3|3(cid:105) .
2 2
Here we detail the numerical methods used to derive We can compute the probability distribution of S over
x
mostofourentropicuncertaintyrelations. Wehaveused |ψ(cid:105) with
the software package Mathematica. For the sake of illus-
tration, we consider the case of s= 3. The most general p(S =±l)=|(cid:104)ψ|S =±l(cid:105)|2, (A10)
2 x x
7
This expression depends on all six parameters of |ψ(cid:105), if we define
and we can use it to calculate the entropy H(S ) =
x
−(cid:80)lpl(Sx =l)log2pl(Sx =l). f(a0,a1,a2,χ0,χ1,χ2)=H(Sx)+H(Sz), (A14)
AnanalogousprocedurecanbeusedforS ,whoseeigen-
y
states are the instruction
NMinimize[f(a ,χ ),{a ,a ,a ,χ ,χ ,χ }], (A15)
i i 0 1 2 0 1 2
1 (cid:16) √ √ (cid:17)
|S =±2(cid:105)= √ |0(cid:105)±i 3|1(cid:105)− 3|2(cid:105)∓i|3(cid:105) ; returns
y
2 2
(cid:110) (cid:110) π π π (cid:111)(cid:111)
(A11) 1.71, a → ,a → ,a → ,χ →π , (A16)
1 (cid:16)√ √ (cid:17) 0 12 1 4 2 4 1
|S =±1(cid:105)= √ 3|0(cid:105)±i|1(cid:105)+|2(cid:105)±i 3|3(cid:105) ,
y
2 2 which implies (34). The other relations we derived can
(A12) be similarly obtained.
For example, for the case of spin 2 we can repeat the
above procedure. Again, we can choose the representa-
tion of eigenbasis of S which gives
whence we can calculate the probabilities and the en- z
tropy.
 
0 2 0 0 0
√
To obtain the optimal EUR, we need to minimize the 12 √0 6 √0 0
sum of two or three of above entropies. Due to their Sx = 20 6 √0 6 0, (A17)
non linear dependence on the parameters, it is highly 0 0 6 0 2
nontrivial to find the minimum analytically. We have 0 0 0 2 0
therefore resorted to numerical methods: Mathematica  
0 2 0 0 0
permits the minimization of a function f(x ,...,x ) that √
depends on n parameters with the routine1 n 1 −2 √0 6 √0 0
Sy = 2i 0 − 6 √0 6 0, (A18)
 0 0 − 6 0 2
0 0 0 −2 0
NMinimize[{f(x ,...,x ),γ(x ,..,x )},{x ,..,x }],
1 n 1 n 1 n 2 0 0 0 0
(A13)
0 1 0 0 0
where γ represents possible constraints. This routine re-  
S =0 0 0 0 0. (A19)
turns both the minimum value of the function and also z  
0 0 0 −1 0
the parameter values that attain it, which in our case
0 0 0 0 1
identify the states that minimize the EUR. For example,