PHYSICAL REVIEW A 75, 022319 共2007兲
Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases
Manuel A. Ballester* and Stephanie Wehner†
Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
共Received 27 November 2006; published 21 February 2007兲
We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In
particular, we show that a bound derived from the result by Maassen and Uffink 关Phys. Rev. Lett. 60, 1103
共1988兲兴 for two such measurements can in fact be tight for up to 冑d measurements in mutually unbiased bases.
We then show that using more mutually unbiased bases does not always lead to a better locking effect. We
prove that the optimal bound for the accessible information using up to 冑d specific mutually unbiased bases is
log d / 2, which is the same as can be achieved by using only two bases. Our result indicates that merely using
mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional
properties.
DOI: 10.1103/PhysRevA.75.022319
PACS number共s兲: 03.67.Dd, 03.65.Ta
I. INTRODUCTION
We investigate two related notions that are of importance
in many quantum cryptographic tasks: Entropic uncertainty
relations and locking classical information in quantum states.
Entropic uncertainty relations are an alternative way to
state Heisenberg’s uncertainty principle. They are frequently
a more useful characterization, because the “uncertainty” is
lower bounded by a quantity that does not depend on the
state to be measured 关1,2兴. Recently, entropic uncertainty relations have gained importance in the context of quantum
cryptography in the bounded storage model, where proving
the security of such protocols ultimately reduces to bounding
such relations 关3兴. Proving new entropic uncertainty relations
could thus give rise to new protocols. Such relations are
known for two 关4兴, or d + 1 关5,6兴 mutually unbiased measurements 共see Sec. II for a definition兲. Very little, however, is
known for any other number of measurements 关7兴.
Here we prove tight entropic uncertainty relations for
measurements in a large number of mutually unbiased bases
共MUBs兲 in square dimensions. In particular, we consider any
MUBs derived from mutually orthogonal Latin squares 关8兴,
and any set of MUBs obtained from the set of unitaries of the
form 兵U 丢 U*其, where 兵U其 gives a set of MUBs in dimension
s when applied to the basis elements of the computational
basis. For any s, there are at most s + 1 such MUBs in a
Hilbert space of dimension d = s2. Let B be the set of MUBs
coming from one of these two constructions. We prove that
for any subset T 債 B of size at least 2 of these bases we have
min
兩␾典
兺 H共B,兩␾典兲 =
B苸T
兩T兩
2
log d,
d
兩具␾ 兩 bi典兩2 log 兩具␾ 兩 bi典兩2 is the Shannon
where H共B , 兩␾典兲 = −兺i=1
entropy 关9兴 arising from measuring the state 兩␾典 in the basis
B = 兵兩b1典 , . . . , 兩bd典其.
Our result furthermore shows that one needs to be careful
to think of “maximally incompatible” measurements as being
necessarily mutually unbiased. When we take entropic uncer-
*Email address: [email protected]
†
Email address: [email protected]
1050-2947/2007/75共2兲/022319共8兲
tainty relations as our measure of “incompatibility,” mutually
unbiased measurements are in fact not always the most incompatible when considering more than two observables. In
particular, it has been shown 关10兴 that if we choose approximately 共log d兲4 bases uniformly at random, then
min兩␾典共1 / 兩T兩兲兺B苸TH共B , 兩␾典兲 艌 log d − 3. This means that
there exist 共log d兲4 bases for which this sum of entropies is
very large, i.e., measurements in such bases are very incompatible. However, we showed that when d is large, there exist
冑d, mutually unbiased bases which are much less incompatible according to this measure. When considering entropic
uncertainty relations as a measure of “incompatibility,” we
must therefore look for different properties for the bases to
define incompatible measurements.
Finally, we give an alternative proof that if B is a set of
d + 1 MUBs we have 兺B苸BH共B , 兩␾典兲 艌 共d + 1兲log关共d + 1兲 / 2兴
关5兴. Our proof is based on the fact that such a set forms a
2-design, which may offer new insights.
Locking classical correlations in quantum states is an exciting feature of quantum information 关11兴, intricately related
to entropic uncertainty relations. Consider a two-party protocol with one or more rounds of communication. Intuitively,
one would expect that in each round the amount of correlation between the two parties cannot increase by much more
than the amount of data transmitted. For example, transmitting 2ᐉ classical bits or ᐉ qubits 共and using superdense coding兲 should not increase the amount of correlation by more
than 2ᐉ bits, no matter what the initial state of the two party
system was. This intuition is accurate when we take the classical mutual information Ic as our correlation measure, and
require all communication to be classical. However, when
quantum communication is possible at some point during the
protocol, everything changes: There exist two-party mixed
quantum states, such that transmitting just a single extra bit
of classical communication can result in an arbitrarily large
increase in Ic 关11兴. The magnitude of this increase thereby
only depends on the dimension of the initial mixed state.
Since then, similar locking effects have been observed also
for other correlation measures 关12,13兴. Such effects play a
role in very different scenarios: They have been used to explain physical phenomena related to black holes 关14兴, but
they are also important in cryptographic applications such as
quantum key distribution 关15兴 and quantum bit string com-
022319-1
©2007 The American Physical Society
PHYSICAL REVIEW A 75, 022319 共2007兲
MANUEL A. BALLESTER AND STEPHANIE WEHNER
mitment 关16,17兴. We are thus interested in determining how
exactly we can obtain locking effects, and how dramatic they
can be.
The correlation measure considered here is the classical
mutual information of a bipartite quantum state ␳AB, which is
the maximum classical mutual information that can be obtained by local measurements M A 丢 M B on the state ␳AB 关18兴:
Ic共␳AB兲 = max I共A:B兲.
MA 丢 MB
共1兲
The classical mutual information is defined as I共A : B兲
= H共PA兲 + H共PB兲 − H共PAB兲, where H is the Shannon entropy.
PA, PB, and PAB are the probability distributions corresponding to the individual and joint outcomes of measuring the
state ␳AB with M A 丢 M B. The mutual information between A
and B is a measure of the information that B contains about
A. This measure of correlation is of particular relevance for
quantum bit string commitments 关16,17兴. Furthermore, the
first locking effect was observed for this quantity in the following protocol between two parties: Alice 共A兲 and Bob 共B兲.
Let B = 兵B1 , . . . , Bm其 with Bt = 兵兩bt1典 , . . . , 兩btd典其 a set of m
MUBs in Cd. Alice picks an element k 苸 兵1 , . . . , d其 and a
basis Bt 苸 B uniformly at random. She then sends 兩btk典 to
Bob, while keeping t secret. Such a protocol gives rise to the
joint state
d
m
1
␳AB =
兺 兺 共兩k典具k兩 丢 兩t典具t兩兲A 丢 共兩btk典具btk兩兲B .
md k=1 t=1
II. PRELIMINARIES
Throughout this paper, we use the shorthand notation
关d兴 = 兵1 , . . . , d其. We write
d
H共Bt,兩␾典兲 = − 兺 兩具␾兩btk典兩2 log兩具␾兩btk典兩2 ,
i=1
for the Shannon entropy 关9兴 arising from measuring the pure
state 兩␾典 in basis Bt = 兵兩bt1典 , . . . , 兩btd典其. In general, we will use
兩btk典 with k 苸 关d兴 to denote the kth element of a basis Bt
indexed by t. We also briefly refer to the Rényi entropy of
order 2 共collision entropy兲 of measuring 兩␾典 in basis Bt given
d
兩具␾ 兩 btk典兩4 关20兴.
by H2共Bt , 兩␾典兲 = −log兺i=1
A. Mutually unbiased bases
Clearly, if Alice told her basis choice t to Bob, he could
measure in the right basis and obtain the correct k. Alice and
Bob would then share log d + log m bits of correlation, which
is also their mutual information Ic共␴AB兲, where ␴AB is the
state obtained from ␳AB after the announcement of t. But,
how large is Ic共␳AB兲, when Alice does not announce t to
Bob? It was shown 关11兴 that in dimension d = 2n, using the
two MUBs given by the unitaries I 丢 n and H 丢 n applied to the
computational basis, where H is the Hadamard matrix, we
have Ic共␳AB兲 = 共1 / 2兲log d. This means that the single bit of
basis information Alice transmits to Bob “unlocks”
共1 / 2兲log d bits: Without this bit, the mutual information is
共1 / 2兲log d, but with this bit it is log d + 1. It is also known
that if Alice and Bob randomly choose a large set of unitaries
from the Haar measure to construct B, then Ic can be brought
down to a small constant 关10兴. However, no explicit constructions with more than two bases are known that give
good locking effects. Based on numerical studies for spaces
of prime dimension 3 艋 d 艋 30, one might hope that adding a
third MUB would strengthen the locking effect and give
Ic共␳AB兲 ⬇ 共1 / 3兲log d 关11兴.
Here, however, we show that this intuition fails us. We
prove that for three MUBs given by I 丢 n, H 丢 n, and K 丢 n where
K = 共I + i␴x兲 / 冑2 and dimension d = 2n for some even integer n,
we have
Ic共␳AB兲 = 共1/2兲log d,
eralized Pauli matrices 关19兴, we again obtain Eq. 共2兲, i.e.,
using two or all 冑d of them makes no difference at all.
Finally, we show that for any set of MUBs B based on generalized Pauli matrices in any dimension, Ic共␳AB兲 = log d
− min兩␾典共1 / 兩B兩兲兺B苸BH共B , 兩␾典兲, i.e., it is enough to determine
a bound on the entropic uncertainty relation to determine the
strength of the locking effect.
Although bounds for general MUBs still elude us, our
results show that merely choosing the bases to be mutually
unbiased is not sufficient and we must look elsewhere to find
bases which provide good locking.
共2兲
the same locking effect as with two MUBs. We also
show that for any subset of the MUBs based on Latin
squares and the MUBs in square dimensions based on gen-
We also need the notion of mutually unbiased bases
共MUBs兲, which were initially introduced in the context of
state estimation 关21兴, but appear in many other problems in
quantum information. The following definition closely follows the one given in 关19兴.
Definition 1 MUBs. Let B1 = 兵兩b11典 , . . . , 兩b1d典其 and B2
= 兵兩b21典 , . . . , 兩b2d典其 be two orthonormal bases in Cd. They are
said to be mutually unbiased if 兩具b1k 兩 b2l 典兩 = 1 / 冑d, for every
k , l 苸 关d兴. A set 兵B1 , . . . , Bm其 of orthonormal bases in Cd is
called a set of mutually unbiased bases if each pair of bases
is mutually unbiased.
We use N共d兲 to denote the maximal number of MUBs in
dimension d. In any dimension d, we have that N共d兲 艋 d + 1
关19兴. If d = pk is a prime power, we have that N共d兲 = d + 1 and
explicit constructions are known 关19,21兴. If d = s2 is a square,
N共d兲 艌 MOLS共s兲, where MOLS共s兲 denotes the number of
mutually orthogonal s ⫻ s Latin squares 关8兴. In general, we
have N共nm兲 艌 min兵N共n兲 , N共m兲其 for all n , m 苸 N 关22,23兴. It is
also known that in any dimension, there exists an explicit
construction for three MUBs 关24兴. Unfortunately, not very
much is known for other dimensions. For example, it is still
an open problem whether there exists a set of seven MUBs in
dimension d = 6. We say that a unitary Ut transforms the
computational basis into the tth MUB Bt = 兵兩bt1典 , . . . , 兩btd典其 if
for all k 苸 关d兴 we have 兩btk典 = Ut兩k典. Here we are particularly
concerned with two specific constructions of mutually unbiased bases.
1. Latin squares
First of all, we consider MUBs based on mutually orthogonal Latin squares 关8兴. Informally, an s ⫻ s Latin square
022319-2
ENTROPIC UNCERTAINTY RELATIONS AND LOCKING: …
over the symbol set 关s兴 = 兵1 , . . . , s其 is an arrangement of elements of 关s兴 into an s ⫻ s square such that in each row and
each column every element occurs exactly once. Let Lij denote the entry in a Latin square in row i and column j. Two
Latin squares L and L⬘ are called mutually orthogonal if and
only if 兵共Li,j , L⬘i,j兲 兩 i , j 苸 关s兴其 = 兵共u , v兲 兩 u , v 苸 关s兴其. From any s
⫻ s Latin square we can obtain a basis for Cs 丢 Cs. First, we
construct s of the basis vectors from the entries of the Latin
square itself. Let 兩v1,ᐉ典 = 共1 / 冑s兲兺i,j苸关s兴ELi,j共ᐉ兲兩i , j典, where EL is
a predicate such that ELi,j共ᐉ兲 = 1 if and only if Li,j = ᐉ. Note
that for each ᐉ we have exactly s pairs i , j such that Ei,j共ᐉ兲
= 1, because each element of 关s兴 occurs exactly s times in the
Latin square. Secondly, from each such vector we obtain s
− 1 additional vectors by adding successive rows of an s ⫻ s
共complex兲 Hadamard matrix H = 共hij兲 as coefficients to obtain
the remaining 兩vt,j典 for t 苸 关s兴, where hij = ␻ij with i , j
苸 兵0 , . . . , s − 1其 and ␻ = e2␲i/s. Two additional MUBs can then
be obtained in the same way from the two non-Latin squares
where each element occurs for an entire row or column, respectively. From each mutually orthogonal Latin square and
these two extra squares which also satisfy the above orthogonality condition, we obtain one basis. This construction
therefore gives MOLS共s兲 + 2 many MUBs. It is known that if
s = pk is a prime power itself, we obtain pk + 1 ⬇ 冑d MUBs
from this construction. Note, however, that there do exist
many more MUBs in prime power dimensions, namely d
+ 1. If s is not a prime power, it is merely known that
MOLS共s兲 艌 s1/14.8 关8兴.
As an example, consider the following 3 ⫻ 3 Latin square
and the 3 ⫻ 3 Hadamard matrix
PHYSICAL REVIEW A 75, 022319 共2007兲
the resulting bases will be mutually unbiased. Indeed, suppose we are given another such basis, B⬘ = 兵兩ut,ᐉ典 兩 t , ᐉ 苸 关s兴其
belonging to L⬘. We then have for any ᐉ , ᐉ⬘ 苸 关s兴 that
兩具u1,ᐉ⬘ 兩 v1,ᐉ典兩2 = 兩共1 / s兲兺i,j苸关s兴ELi,j⬘共ᐉ⬘兲ELi,j共ᐉ兲兩2 = 1 / s2, as there
exists exactly only one pair ᐉ , ᐉ⬘ 苸 关s兴 such that
ELi,j⬘共ᐉ⬘兲ELi,j共ᐉ兲 = 1. Clearly, the same argument holds for the
additional vectors derived from the Hadamard matrix.
2. Generalized Pauli matrices
The second construction we consider is based on the generalized Pauli matrices Xd and Zd 关19兴, defined by their actions on the computational basis C = 兵兩1典 , . . . , 兩d典其 as follows:
Zd兩k典 = ␻k兩k典,
Xd兩k典 = 兩k + 1典,
∀ 兩k典 苸 C,
2␲i/d
where ␻ = e
. We say that 共Xd兲a1共Zd兲b1 丢 ¯
aN
bN
丢 共Xd兲 共Zd兲
for ak , bk 苸 兵0 , . . . , d − 1其 and k 苸 关N兴 is a
string of Pauli matrices.
If d is a prime, it is known that the d + 1 MUBs constructed first by Wootters and Fields 关21兴 can also be obtained
as
the
eigenvectors
of
the
matrices
k
Zd , Xd , XdZd , XdZ2d , . . . , XdZd−1
d 关19兴. If d = p is a prime power,
consider all d2 − 1 possible strings of Pauli matrices excluding the identity and group them into sets C1 , . . . , Cd+1 such
that 兩Ci兩 = d − 1 and Ci 艛 C j = 兵I其 for i ⫽ j and all elements of
Ci commute. Let Bi be the common eigenbasis of all elements of Ci. Then B1 , . . . , Bd+1 are MUBs 关19兴. A similar
result for d = 2k has also been shown in 关25兴. A special case of
this construction is the three mutually unbiased bases in dimension d = 2k given by the unitaries I 丢 k,H 丢 k, and K 丢 k with
K = 共I + i␴x兲 / 冑2 applied to the computational basis.
B. 2-designs
For the purposes of the present work, spherical t-designs
共see for example Ref. 关26兴兲 can be defined as follows.
Definition 2 共t-design兲. Let 兵兩␶1典 , . . . , 兩␶m典其 be a set of state
vectors in Cd. They are said to form a t-design if
where ␻ = e2␲i/3. First, we obtain vectors
m
⌸+共t,d兲
1
关兩␶i典具␶i兩兴 丢 t =
,
兺
m i=1
Tr ⌸+共t,d兲
兩v1,1典 = 共兩1,1典 + 兩2,3典 + 兩3,2典兲/冑3,
where ⌸+共t , d兲 is a projector onto the completely symmetric
subspace of Cd 丢 t and
兩v1,2典 = 共兩1,2典 + 兩2,1典 + 兩3,3典兲/冑3,
兩v1,3典 = 共兩1,3典 + 兩2,2典 + 兩3,1典兲/冑3.
With the help of H we obtain three additional vectors from
the ones above. From the vector 兩v1,1典, for example, we obtain
兩v1,1典 = 共兩1,1典 + 兩2,3典 + 兩3,2典兲/冑3,
兩v2,1典 = 共兩1,1典 + ␻兩2,3典 + ␻2兩3,2典兲/冑3,
Tr ⌸+共t,d兲 =
冉
d+t−1
d−1
冊
=
共d + t − 1兲!
,
共d − 1兲!t!
is its dimension.
Any set B of d + 1 MUBs forms a spherical 2-design
关26,27兴, i.e., we have for B = 兵B1 , . . . , Bd+1其 with Bt
= 兵兩bt1典 , . . . , 兩btd典其 that
d+1 d
⌸共2,d兲
1
.
关兩btk典具btk兩兴 丢 2 = 2 +
兺
兺
d共d + 1兲 t=1 k=1
d共d + 1兲
兩v3,1典 = 共兩1,1典 + ␻2兩2,3典 + ␻兩3,2典兲/冑3.
This gives us basis B = 兵兩vt,ᐉ典 兩 t , ᐉ 苸 关s兴其 for s = 3. The construction of another basis follows in exactly the same way
from a mutually orthogonal Latin square. The fact that two
such squares L and L⬘ are mutually orthogonal ensures that
III. UNCERTAINTY RELATIONS
We now prove tight entropic uncertainty for measurements in MUBs in square dimensions. The main result of 关4兴,
022319-3
PHYSICAL REVIEW A 75, 022319 共2007兲
MANUEL A. BALLESTER AND STEPHANIE WEHNER
which will be very useful for us, is stated next.
Theorem 1 共Maassen and Uffink兲. Let B1 and B2 be two
orthonormal bases in a Hilbert space of dimension d. Then
for all pure states 兩␺典
s
=
1
关H共B1,兩␺典兲 + H共B2,兩␺典兲兴 艌 − log c共B1,B2兲,
2
共3兲
where c共B1 , B2兲 = max兵兩具b1 兩 b2典兩 : 兩b1典 苸 B1 , 兩b2典 苸 B2其.
The case when B1 and B2 are MUBs is of special interest
for us. More generally, when one has a set of MUBs a trivial
application of Eq. 共3兲 leads to the following corollary also
noted in 关7兴.
Corollary 1. Let B = 兵B1 , . . . , Bm其, be a set of MUBs in a
Hilbert space of dimension d. Then
s
1
1
具␺兩U 丢 U 兩␺典 = 兺 具k兩U兩l典具k兩U*兩l典 = 兺 具k兩U兩l典具l兩U†兩k典
s k,l=1
s k,l=1
*
1
TrUU† = 1.
s
Therefore, for any t 苸 关m兴 we have that
H共Vt,兩␺典兲 = − 兺 兩具kl兩Ut 丢 U*t 兩␺典兩2 log兩具kl兩Ut 丢 U*t 兩␺典兩2 =
kl
− 兺 兩具kl兩␺典兩2 log兩具kl兩␺典兩2 = log s =
kl
log d
.
2
Taking the average of the previous equation we get the desired result.
䊏
m
log d
1
兺 H共Bt,兩␺典兲 艌 2 .
m t=1
Proof. Using Eq. 共3兲, one gets that for any pair of MUBs
Bt and Bt⬘ with t ⫽ t⬘
log d
1
关H共Bt, ␺兲 + H共Bt⬘, ␺兲兴 艌
.
2
2
B. MUBs based on Latin squares
共4兲
共5兲
Adding up the resulting equation for all pairs t ⫽ t⬘ we get
the desired result 共4兲.
䊏
Here, we now show that this bound can in fact be tight for
a large set of MUBs.
A. MUBs in square dimensions
Corollary 1 gives a lower bound on the average of the
entropies of a set of MUBs. The obvious question is whether
that bound is tight. We show that the bound is indeed tight
when we consider product MUBs in a Hilbert space of
square dimension.
Theorem 2. Let B = 兵B1 , . . . , Bm其 with m 艌 2 be a set of
MUBs in a Hilbert space H of dimension s. Let Ut be the
unitary operator that transforms the computational basis to
Bt. Then V = 兵V1 , . . . , Vm其, where
We now consider mutually unbiased bases based on Latin
squares 关8兴 as described in Sec. II. Our proof again follows
by providing a state that achieves the bound in corollary 1,
which turns out to have a very simple form.
Lemma 1. Let B = 兵B1 , . . . , Bm其 with m 艌 2 be any set of
MUBs in a Hilbert space of dimension d = s2 constructed on
the basis of Latin squares. Then
min
兩␺典
Proof. Consider the state 兩␺典 = 兩1 , 1典 and fix a basis Bt
= 兵兩vti,j典 兩 i , j 苸 关s兴其 苸 B coming from a Latin square. It is easy
to see that there exists exactly one j 苸 关s兴 such that
具vt1,j 兩 1 , 1典 = 1 / 冑s. Namely this will be the j 苸 关s兴 at position
共1,1兲 in the Latin square. Fix this j. For any other ᐉ 苸 关s兴,
t
兩 1 , 1典 = 0. But this means that there exist
ᐉ ⫽ j, we have 具v1,ᐉ
exactly s vectors in B such that 兩具vti,j 兩 1 , 1典兩2 = 1 / s, namely
exactly the s vectors derived from 兩vt1,j典 via the Hadamard
matrix. The same argument holds for any such basis B 苸 T.
We get
兺 H共B,兩1,1典兲 = B苸B
兺 兺 兩具vti,j兩1,1典兩2 log兩具vti,j兩1,1典兩2
i,j苸关s兴
B苸B
Vt = 兵Ut兩k典 丢 U*t 兩l典:k,l 苸 关s兴其,
1
1
log d
.
= 兩T兩s log = 兩T兩
s
s
2
is a set of MUBs in H 丢 H, and it holds that
The result then follows directly from corollary 1.
m
log d
1
min 兺 H共Vt,兩␺典兲 =
,
m
2
兩␺典
t=1
共6兲
where d = dim共H 丢 H兲 = s2.
Proof. It is easy to check that V is indeed a set of MUBs.
Our proof works by constructing a state 兩␺典 that achieves the
bound in corollary 1. It is easy to see that the maximally
entangled state
兩␺典 =
1
s
兩kk典,
冑s 兺
k=1
log d
1
兺 H共B,兩␺典兲 = 2 .
m B苸B
䊏
C. Using a full set of MUBs
We now provide an alternative proof of an entropic uncertainty relation for a full set of mutually unbiased bases.
This has previously been proved in 关5兴. Nevertheless, because our proof is so simple using existing results about
2-designs we include it here for completeness, in the hope
that if may offer additional insight.
Lemma 2. Let B be a set of d + 1 MUBs in a Hilbert space
of dimension d. Then
冉 冊
1
d+1
H2共B,兩␺典兲 艌 log
.
兺
2
d + 1 B苸B
satisfies U 丢 U 兩␺典 = 兩␺典 for any U 苸 U共d兲. Indeed,
*
022319-4
ENTROPIC UNCERTAINTY RELATIONS AND LOCKING: …
Proof. Let Bt = 兵兩bt1典 , . . . , 兩btd典其 and B = 兵B1 , . . . , Bd+1其. We
can then write
冉
d+1 d
1
兺 兺 兩具bt 兩␺典兩4
d + 1 t=1 k=1 k
冉 冊
= log
A. An example
冊
d+1
,
2
where the first inequality follows from the concavity of the
log, and the final inequality follows directly from the fact
that a full set of MUBs forms a 2-design and 共关27兴, theorem
1兲.
䊏
We then obtain the original result by Sanchez-Ruiz 关5兴 by
noting that H共·兲 艌 H2共·兲.
Corollary 2. Let B be a set of d + 1 MUBs in a Hilbert
space of dimension d. Then
冉 冊
d+1
1
H共B,兩␺典兲 艌 log
.
兺
2
d + 1 B苸B
We first consider a very simple example with only three
MUBs that provides the intuition behind the remainder of
our paper. The three MUBs we consider now are generated
by the unitaries I, H, and K = 共I + i␴x兲 / 冑2 when applied to the
computational basis. For this small example, we also investigate the role of the prior over the bases and the encoded
basis elements. It turns out that this does not affect the
strength of the locking effect positively. Actually, it is possible to show the same for encodings in many other bases.
However, we do not consider this case in full generality as to
not obscure our main line of argument.
Lemma 3. Let U0 = I 丢 n, U1 = H 丢 n, and U2 = K 丢 n, where k
苸 兵0 , 1其n and n is an even integer. Let 兵pt其 with t 苸 关2兴 be a
probability distribution over the set S = 兵U1 , U2 , U3其. Suppose
that p1 , p2 , p3 艋 1 / 2 and let pt,k = pt共1 / d兲. Consider the ensemble E = 兵pt共1 / d兲 , Ut兩k典具k兩U†t 其; then
n
Iacc共E兲 = .
2
IV. LOCKING
We now turn our attention to locking. We first explain the
connection between locking and entropic uncertainty relations. In particular, we show that for MUBs based on generalized Pauli matrices, we only need to look at such uncertainty relations to determine the exact strength of the locking
effect. We then consider how good MUBs based on Latin
squares are for locking.
In order to determine how large the locking effect is for
some set of mutually unbiased bases B, and the state
兩B兩
Ic共␳AB兲 = Iacc共E兲. We are now ready to prove our locking results.
d
d+1
1
1
兩具btk兩␺典兩4
H2共B,兩␺典兲 = −
兺
兺 log 兺
d + 1 B苸B
d + 1 t=1
k=1
艌 log
PHYSICAL REVIEW A 75, 022319 共2007兲
If, on the other hand, there exists a t 苸 关2兴 such that pt
⬎ 1 / 2, then Iacc共E兲 ⬎ n / 2.
Proof. We first give an explicit measurement strategy and
then prove a matching upper bound on Iacc. Consider the
Bell basis vectors 兩⌫00典 = 共兩00典 + 兩11典兲 / 冑2, 兩⌫01典 = 共兩00典
兩⌫10典 = 共兩01典 + 兩10典兲 / 冑2,
and
兩⌫11典 = 共兩01典
− 兩11典兲 / 冑2,
冑
− 兩10典兲 / 2. Note that we can write for the computational basis
共7兲
t=1 k=1
we must find an optimal bound for Ic共␳AB兲. Here, 兵pt,k其 is a
probability distribution over B ⫻ 关d兴. That is, we must find a
positive operator valued measure 共POVM兲 M A 丢 M B that
maximizes Eq. 共1兲. It has been shown in 关11兴 that we can
restrict ourselves to taking M A to be the local measurement
determined by the projectors 兵兩k典具k兩 丢 兩t典具t兩其. It is also known
that we can limit ourselves to take the measurement M B consisting of rank one elements 兵␣i兩⌽i典具⌽i兩其 only 关28兴, where
␣i 艌 0 and 兩⌽i典 is normalized. Maximizing over M B then corresponds to maximizing Bob’s accessible information 关关29兴,
Eq. 共9.75兲兴 for the ensemble E = 兵pk,t , 兩btk典具btk兩其
冉
Iacc共E兲 = max − 兺 pk,t log pk,t
M
k,t
+ 兺 兺 pk,t␣i具⌽i兩␳k,t兩⌽i典log
i
冑2 共兩⌫00典 + 兩⌫01典兲,
兩01典 =
冑2 共兩⌫10典 + 兩⌫11典兲,
兩10典 =
冑2 共兩⌫10典 − 兩⌫11典兲,
兩11典 =
冑2 共兩⌫00典 − 兩⌫01典兲.
d
␳AB = 兺 兺 pt,k共兩k典具k兩 丢 兩t典具t兩兲A 丢 共兩btk典具btk兩兲B ,
k,t
冊
pk,t具⌽i兩␳k,t兩⌽i典
,
具⌽i兩␮兩⌽i典
共8兲
where ␮ = 兺k,t pk,t␳k,t and
␳k,t = 兩btk典具btk兩.
Therefore, we have
1
兩00典 =
1
1
1
The crucial fact to note is that if we fix some k1k2, then there
exist exactly two Bell basis vectors 兩⌫i1i2典 such that
兩具⌫i1i2 兩 k1k2典兩2 = 1 / 2. For the remaining two basis vectors the
inner product with 兩k1k2典 will be zero. A simple calculation
shows that we can express the two qubit basis states of the
other two mutually unbiased bases analogously: For each
two qubit basis state there are exactly two Bell basis vectors
such that the inner product is zero and for the other two the
inner product squared is 1 / 2.
We now take the measurement given by 兵兩⌫i典具⌫i兩其 with
兩⌫i典 = 兩⌫i1i2典 丢 . . . 丢 兩⌫in−1in典 for the binary expansion of i
= i1 , i2 , . . . , in. Fix a k = k1 , k2 , . . . , kn. By the above argument,
022319-5
PHYSICAL REVIEW A 75, 022319 共2007兲
MANUEL A. BALLESTER AND STEPHANIE WEHNER
there exist exactly 2n/2 strings i 苸 兵0 , 1其n such that 兩具⌫i 兩 k典兩2
= 1 / 共2n/2兲. Putting everything together, Eq. 共8兲 now gives us
for any prior distribution 兵pt,k其 that
− 兺 具⌫i兩␮兩⌫i典log具⌫i兩␮兩⌫i典 −
i
n
艋 Iacc共E兲.
2
共9兲
⌬共pk,t兲 = H共pk,t兲 − Iacc共E兲 − H共pt兲,
substracting the basis information itself. Consider the postmeasurement state ␯ = 兺i具⌫i兩␮兩⌫i典兩⌫i典具⌫i兩. Using Eq. 共9兲 we
obtain
⌬共pk,t兲 艋 H共pk,t兲 − S共␯兲 + n/2 − H共pt兲,
where S is the von Neuman entropy. Considering the state
For our particular distribution we have ␮ = I / d and thus
d
k=1 t=1
We now prove a matching upper bound that shows that
our measurement is optimal. For our distribution, we can
rewrite Eq. 共8兲 for the POVM given by 兵␣i兩⌽i典具⌽i兩其 to
冉
i
␣i
兺 pt兩具⌽i兩Ut兩k典兩2 log兩具⌽i兩Ut兩k典兩2
d k,t
冉
= max log d − 兺
M
we have that
S共␳12兲 = H共pk,t兲 艋 S共␳1兲 + S共␳2兲 = H共pt兲 + S共␮兲 艋 H共pt兲
+ S共␯兲.
Using Eq. 共10兲 and the previous equation we get
Iacc共E兲 = max log d
+兺
3
␳12 = 兺 兺 pk,t共兩t典具t兩兲1 丢 共Ut兩k典具k兩U†t 兲2 ,
n
艋 Iacc共E兲.
2
M
共10兲
i
冊
⌬共pk,t兲 艋 n/2,
冊
for any prior distribution. This bound is saturated by the
uniform prior and therefore we conclude that the uniform
prior results in the largest gap possible.
␣i
兺 ptH共Bt,兩⌽i典兲 .
d t
B. MUBs from generalized Pauli matrices
It follows from corollary 1 that ∀i 苸 兵0 , 1其n and
p1 , p2 , p3 艋 1 / 2,
共1/2 − p1兲关H共B2,兩⌽i典兲 + H共B3,兩⌽i典兲兴 + 共1/2 − p2兲关H共B1,兩⌽i典兲
+ H共B3,兩⌽i典兲兴 + 共1/2 − p3兲关H共B1,兩⌽i典兲 + H共B2,兩⌽i典兲兴
艌 n/2.
3
ptH共Bt , 兩⌽i典兲 艌 n / 2.
Reordering the terms we now get 兺t=1
Putting things together and using the fact that 兺i␣i = d, we
obtain
n
Iacc共E兲 艋 ,
2
We first consider MUBs based on the generalized Pauli
matrices Xd and Zd as described in Sec. II. We consider a
uniform prior over the elements of each basis and the set of
bases. Choosing a nonuniform prior does not lead to a better
locking effect.
Lemma 4. Let B = 兵B1 , . . . , Bm其 be any set of MUBs constructed on the basis of generalized Pauli matrices in a Hilbert space of prime power dimension d = pN. Consider the
ensemble E = 兵1 / 共dm兲 , 兩btk典具btk兩其. Then
Iacc共E兲 = log d −
1
min 兺 H共Bt,兩␺典兲.
m 兩␺典 Bt苸B
Proof. We can rewrite Eq. 共8兲 for the POVM given by
兵␣i兩⌽i典具⌽i兩其 to
from which the result follows.
If, on the other hand, there exists a t 苸 关2兴 such that pt
⬎ 1 / 2, then by measuring in the basis Bt we obtain Iacc共E兲
䊏
艌 ptn ⬎ n / 2.
Above, we have only considered a nonuniform prior over
the set of bases. In 关30兴 it is observed that when we want to
guess the XOR of a string of length 2 encoded in one 共unknown to us兲 of these three bases, the uniform prior on the
strings is not the one that gives the smallest probability of
success. This might lead one to think that a similar phenomenon could be observed in the present setting, i.e., that one
might obtain better locking with three basis for a nonuniform
prior on the strings. In what follows, however, we show that
this is not the case.
Let pt = 兺k pk,t be the marginal distribution on the basis,
then the difference in Bob’s knowledge between receiving
only the quantum state and receiving the quantum state and
the basis information is given by
冉
冉
Iacc共E兲 = max log d + 兺
M
i
= max log d − 兺
M
i
␣i
兺 兩具⌽i兩btk典兩2 log兩具⌽i兩btk典兩2
dm k,t
冊
冊
␣i
兺 ptH共Bt,兩⌽i典兲 .
d t
For convenience, we split up the index i into i = ab with a
= a1 , . . . , aN and b = b1 , . . . , bN, where aᐉ , bᐉ 苸 兵0 , . . . , p − 1其 in
the following.
We first show that applying generalized Pauli matrices to
the basis vectors of a MUB merely permutes those vectors.
Claim 1. Let Bt = 兵兩bt1典 , . . . , 兩btd典其 be a basis based on generalized Pauli matrices 共Sec. II兲 with d = pN. Then ∀ a, b
苸 兵0 , . . . , p − 1其N, ∀ k 苸 关d兴 we have that ∃ k⬘ 苸 关d兴, such that
兩bkt ⬘典 = Xad1Zbd1 丢 . . . 丢 XadNZbdN兩btk典.
Proof. Let ⌺ip for i 苸 兵0 , 1 , 2 , 3其 denote the generalized
Pauli matrices ⌺0p = I p, ⌺1p = X p, ⌺3p = Z p, and ⌺2p = X pZ p. Note
that XupZvp = ␻uvZvpXup, where ␻ = e2␲i/p. Furthermore, define
022319-6
ENTROPIC UNCERTAINTY RELATIONS AND LOCKING: …
i
i
丢 共x−1兲
丢 ⌺ p 丢 IN−x to be the Pauli operator ⌺ p applied
⌺i,共x兲
p =I
to the xth qupit. Recall from Sec. II that the basis Bt is the
unique simultaneous eigenbasis of the set of operators in Ct,
i.e., for all k 苸 关d兴 and f, g 苸 关N兴, 兩btk典 苸 Bt and ctf,g 苸 Ct, we
have ctf,g兩btk典 = ␭tk,f,g兩btk典 for some value ␭tk,f,g. Note that any
vector 兩v典 that satisfies this equation is proportional to a vector in Bt. To prove that any application of one of the generalized Pauli matrices merely permutes the vectors in Bt is
t
therefore equivalent to proving that ⌺i,共x兲
p 兩bk典 are eigenvectors
t
of c f,g for any f, g 苸 关k兴 and i 苸 兵1 , 3其. This can be seen as
3,共n兲 gN
N
fN
共⌺1,共n兲
for f
follows: Note that ctf,g = 丢 n=1
p 兲 共⌺ p 兲
= 共f 1 , . . . , f N兲 and g = 共g1 , . . . , gN兲 with f N , gN 苸 兵0 , . . . , p − 1其
关19兴. A calculation then shows that
i,共x兲 t
t
t
ctf,g⌺i,共x兲
p 兩bk典 = ␶ f x,gx,i␭k,f,g⌺ p 兩bk典,
where ␶ f x,gx,i = ␻gx for i = 1 and ␶ f x,gx,i = ␻−f x for i = 3. Thus
t
t
⌺i,共x兲
p 兩bk典 is an eigenvector of c f,g for all t, f, g, and i, which
proves our claim.
䊏
Suppose we are given 兩␺典 that minimizes 兺Bt苸TH共Bt , 兩␺典兲.
We can then construct a full POVM with d2 elements by
taking
兵共1 / d兲兩⌽ab典具⌽ab兩其
with
兩⌽ab典 = 共Xad1Zbd1 丢 . . .
aN bN †
丢 Xd Zd 兲 兩␺典. However, it follows from our claim above
that ∀ a, b, k, ∃ k⬘ such that 兩具⌽ab 兩 btk典兩2 = 兩具␺ 兩 bkt ⬘典兩2, and thus
H共Bt , 兩␺典兲 = H共B , 兩⌽ab典兲 from which the result follows.
䊏
Determining the strength of the locking effects for such
MUBs is thus equivalent to proving bounds on entropic uncertainty relations. We thus obtain as a corollary of theorem
2 and lemma 4 that, for dimensions which are the square of
a prime power d = p2N, using any product MUBs based on
generalized Pauli matrices does not give us any better locking than just using 2 MUBs.
Corollary 3. Let S = 兵S1 , . . . , Sm其 with m 艌 2 be any set of
MUBs constructed on the basis of generalized Pauli matrices
in a Hilbert space of prime 共power兲 dimension s = pN. Define
Ut as the unitary that transforms the computational basis into
the tth MUB, i.e., St = 兵Ut兩1典 , . . . , Ut兩s典其. Let B
= 兵B1 , . . . , Bm其 be the set of product MUBs with Bt = 兵Ut
*
*
丢 Ut 兩1典 , . . . , Ut 丢 Ut 兩d典其 in dimension d = s2. Consider the ensemble E = 兵1 / 共dm兲 , 兩btk典具btk兩其. Then
Iacc共E兲 =
log d
.
2
Proof. The claim follows from theorem 2 and the proof of
lemma 4 by constructing a similar measurement formed from
vectors 兩⌽̂âbˆ典 = Ka1b1 丢 Ka*2b2兩␺典 with â = a1a2 and b̂ = b1b2,
where a1, a2 and b1, b2 are defined like a and b in the proof
of lemma 4 and Kab = 共Xad1Zbd1 丢 . . . 丢 XadNZbdN兲† from above.䊏
The simple example we considered above is in fact a special case of corollary 3. It shows that if the vector that minimizes the sum of entropies has certain symmetries, such as
for example the Bell states, the resulting POVM can even be
much simpler.
C. MUBs from Latin squares
At first glance, one might think that maybe the product
MUBs based on generalized Paulis are not well suited for
PHYSICAL REVIEW A 75, 022319 共2007兲
locking just because of their product form. Perhaps MUBs
with entangled basis vectors do not exhibit this problem. To
this end, we examine how well MUBs based on Latin
squares can lock classical information in a quantum state. All
such MUBs are highly entangled, with the exception of the
two extra MUBs based on non-Latin squares. Surprisingly, it
turns out, however, that any set of at least two MUBs based
on Latin squares does equally well at locking as using just
two such MUBs. Thus such MUBs perform equally “badly,”
i.e., we cannot improve the strength of the locking effect by
using more MUBs of this type.
Lemma 5. Let B = 兵B1 , . . . , Bm其 with m 艌 2 be any set of
MUBs in a Hilbert space of dimension d = s2 constructed on
the basis of Latin squares. Consider the ensemble E
= 兵1 / 共dm兲 , 兩btk典具btk兩其. Then
Iacc共E兲 =
log d
.
2
Proof. Note that we can again rewrite Iacc共E兲 as in the
proof of lemma 4. Consider the simple measurement in the
computational basis 兵兩i , j典具i , j 储 i , j 苸 关s兴其. The result then follows by the same argument as in lemma 1.
䊏
V. CONCLUSION AND OPEN QUESTIONS
We have shown tight bounds on entropic uncertainty relations and locking for specific sets of mutually unbiased
bases. Surprisingly, it turns out that using a more mutually
unbiased basis does not always lead to a better locking effect. It is interesting to consider what may make these bases
so special. The example of three MUBs considered in lemma
3 may provide a clue. These three bases are given by the
common eigenbases of 兵␴x 丢 ␴x , ␴x 丢 I , I 丢 ␴x其, 兵␴z 丢 ␴z , ␴z
丢 I , I 丢 ␴z其, and 兵␴ y 丢 ␴ y , ␴ y 丢 I , I 丢 ␴ y 其, respectively 关19兴.
However, ␴x 丢 ␴x, ␴z 丢 ␴z, and ␴y 丢 ␴y commute and thus
also share a common eigenbasis, namely the Bell basis. This
is exactly the basis we will use as our measurement. For all
MUBs based on generalized Pauli matrices, the MUBs in
prime power dimensions are given as the common eigenbasis
of similar sets consisting of strings of Pauli matrices. It
would be interesting to determine the strength of the locking
effect on the basis of the commutation relations of elements
of different sets. Perhaps it is possible to obtain good locking
from a subset of such MUBs where none of the elements
from different sets commute.
It is also worth noting that the numerics of 关11兴 indicate
that at least in dimension p using more than three bases does
indeed lead to a stronger locking effect. It would be interesting to know whether the strength of the locking effect depends not only on the number of bases, but also on the dimension of the system in question.
Whereas general bounds still elude us, we have shown
that merely choosing mutually unbiased bases is not sufficient to obtain good locking effects or high lower bounds for
entropic uncertainty relations. We thus have to look for different properties.
022319-7
PHYSICAL REVIEW A 75, 022319 共2007兲
MANUEL A. BALLESTER AND STEPHANIE WEHNER
ACKNOWLEDGMENTS
We would like to thank Harry Buhrman, Hartwig Bosse,
Matthias Christandl, Richard Cleve, Debbie Leung, Serge
Massar, David Poulin, and Ben Toner for discussions. We
would especially like to thank Andris Ambainis and Andreas
Winter for many helpful comments and interesting discussions. We would also like to thank Debbie Leung, John Smo-
lin, and Barbara Terhal for providing us with explicit details
on the numerical studies conducted in 关11兴. Thanks also to
Matthias Christandl and Serge Massar for discussions on errors in string commitment protocols, to which end claim 1
was proved in the first place. Thanks also to Matthias Christandl and Ronald de Wolf for helpful comments on an earlier version of this paper. We are supported by an NWO vici
grant 2004-2009 and by the EU project QAP 共IST 015848兲.
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