A UNIFIED APPROACH TO SIC-POVMs AND MUBs
Olivier Albouy and Maurice R. Kibler
Université de Lyon, Institut de Physique Nucléaire,
Université Lyon 1 and CNRS/IN2P3, 43 bd du 11 novembre 1918,
F–69622 Villeurbanne, France
Electronic mail: o.albouy@ipnl.in2p3.fr, m.kibler@ipnl.in2p3.fr
Abstract
A unified approach to (symmetric informationally complete) positive op-
erator valued measures and mutually unbiased bases is developed in this arti-
cle. The approach is based on the use of Racah unit tensors for the Wigner-
Racah algebra of SU(2) ⊃ U(1). Emphasis is put on similarities and differ-
ences between SIC-POVMs and MUBs.
Keywords: finite–dimensional Hilbert spaces; mutually unbiased bases; positive op-
erator valued measures; SU(2) ⊃ U(1) Wigner–Racah algebra
1 INTRODUCTION
The importance of finite–dimensional spaces for quantum mechanics is well recognized
(see for instance [1]-[3]). In particular, such spaces play a major role in quantum informa-
tion theory, especially for quantum cryptography and quantum state tomography [4]-[27].
Along this vein, a symmetric informationally complete (SIC) positive operator valued
measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also
[3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases
for such a space [15]-[27].
The introduction of POVMs goes back to the seventies [4]-[7]. The most general quan-
tum measurement is represented by a POVM. In the present work, we will be interested
http://arxiv.org/abs/0704.0511v3
in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of
the quantum state. Moreover, those POVMs are endowed with an extra symmetry condi-
tion (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or
explicit in the seminal works of [15]-[18], has been the object of numerous mathematical
and physical investigations during the last two decades in connection with the so-called
complementary observables. Unfortunately, the question to know, for a given Hilbert
space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there
exist has remained an open one.
The aim of this note is to develop a unified approach to SIC-POVMs and MUBs based
on a complex vector space of higher dimension, viz. d2 instead of d. We then give a
specific example of this approach grounded on the Wigner-Racah algebra of the chain
SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin
systems [28] and for an angular momentum study of MUBs [26, 27].
Most of the notations in this work are standard. Let us simply mention that I is the
identity operator, the bar indicates complex conjugation, A† denotes the adjoint of the
operator A, δa,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0
according as a, b and c satisfy or not the triangular inequality.
2 SIC-POVMs
Let Cd be the standard Hilbert space of dimension d endowed with its usual inner product
denoted by 〈 | 〉. As is usual, we will identify a POVM with a nonorthogonal decompo-
sition of the identity. Thus, a discrete SIC-POVM is a set {Px : x = 1, 2, · · · , d2} of d2
nonnegative operators Px acting on C
d, such that:
• they satisfy the trace or symmetry condition
Tr (PxPy) =
, x 6= y; (1)
moreover, we will assume the operators Px are normalized, thus completing this
condition with
= 1; (2)
• they form a decomposition of the identity
Px = I; (3)
• they satisfy a completeness condition: the knowledge of the probabilities px defined
by px = Tr(Pxρ) is sufficient to reconstruct the density matrix ρ.
Now, let us develop each of the operators Px on an orthonormal (with respect to the
Hilbert–Schmidt product) basis {ui : i = 1, 2, · · · , d2} of the space of linear operators on
vi(x)ui, (4)
where the operators ui satisfy Tr(u
iuj) = δi,j . The operators Px are thus considered as
vectors
v(x) = (v1(x), v2(x), · · · , vd2(x)) (5)
in the Hilbert space Cd
of dimension d2 and the determination of the operators Px is
equivalent to the determination of the components vi(x) of v(x). In this language, the
trace property (1) together with the normalization condition (2) give
v(x) · v(y) = 1
(dδx,y + 1) , (6)
where v(x) · v(y) =
i=1 vi(x)vi(y) is the usual Hermitian product in C
In order to compare Eq. (6) with what usually happens in the search for SIC-POVMs,
we suppose from now on that the operators Px are rank-one operators. Therefore, by
putting
Px = |Φx〉〈Φx| (7)
with |φx〉 ∈ Cd, the trace property (1, 2) reads
|〈Φx|Φy〉|2 =
(dδx,y + 1) . (8)
From this point of view, to find d2 operators Px is equivalent to finding d
2 vectors |φx〉
in Cd satisfying Eq. (8). At the price of an increase in the number of components from
d3 (for d2 vectors in Cd) to d4 (for d2 vectors in Cd
), we have got rid of the square
modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove
to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is
independent of any hypothesis on the rank of the operators Px. In fact, there exists a lot
of relations among these d4 coefficients that decrease the effective number of coefficients
to be found and give structural constraints on them. Those relations are highly sensitive
to the choice of the basis {ui : i = 1, 2, · · · , d2} and we are going to exhibit an example
of such a set of relations by choosing the basis to consist of Racah unit tensors.
The cornerstone of this approach is to identify Cd with a subspace ε(j) of constant
angular momentum j = (d− 1)/2. Such a subspace is spanned by the set {|j,m〉 : m =
−j,−j + 1, · · · , j}, where |j,m〉 is an eigenvector of the square and the z-component
of a generalized angular momentum operator. Let u(k) be the Racah unit tensor [29]
of order k (with k = 0, 1, · · · , 2j) defined by its 2k + 1 components u(k)q (where q =
−k,−k + 1, · · · , k) through
u(k)q =
m′=−j
(−1)j−m
j k j
−m q m′
|j,m〉〈j,m′|, (9)
where (· · ·) denotes a 3–jm Wigner symbol. For fixed j, the (2j + 1)2 operators u(k)q
(with k = 0, 1, · · · , 2j and q = −k,−k + 1, · · · , k) act on ε(j) ∼ Cd and form a basis
of the Hilbert space CN of dimension N = (2j + 1)2, the inner product in CN being the
Hilbert–Schmidt product. The formulas (involving unit tensors, 3–jm and 6–j symbols)
relevant for this work are given in Appendix (see also [29] to [31]). We must remember
that those Racah operators are not normalized to unity (see relation (46)). So this will
generate an extra factor when defining vi(x).
Each operator Px can be developed as a linear combination of the operators u
Hence, we have
ckq(x)u
q , (10)
where the unknown expansion coefficients ckq(x) are a priori complex numbers. The
determination of the operators Px is thus equivalent to the determination of the coefficients
ckq(x), which are formally given by
ckq(x) = (2k + 1)〈Φx|u(k)q |Φx〉, (11)
as can be seen by multiplying each member of Eq. (10) by the adjoint of u
p and then
using Eq. (46) of Appendix.
By defining the vector
v(x) = (v1(x), v2(x), · · · , vN(x)), N = (2j + 1)2 (12)
vi(x) =
2k + 1
ckq(x), i = k
2 + k + q + 1, (13)
the following properties and relations are obtained.
• The first component v1(x) of v(x) does not depend on x since
c00(x) =
2j + 1
for all x ∈ {1, 2, · · · , (2j + 1)2}.
Proof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.
• The components vi(x) of v(x) satisfy the complex conjugation property described
ckq(x) = (−1)qck−q(x) (15)
for all x ∈ {1, 2, · · · , (2j + 1)2}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k + 1, · · · , k}.
Proof: Use the Hermitian property of Px and Eq. (43) of Appendix.
• In terms of ckq, Eq. (6) reads
2k + 1
ckq(x)ckq(y) =
2(j + 1)
[(2j + 1)δx,y + 1] (16)
for all x, y ∈ {1, 2, · · · , (2j + 1)2}, where the sum over q is SO(3) rotationally
invariant.
Proof: The proof is trivial.
• The coefficients ckq(x) are solutions of the nonlinear system given by
2K + 1
cKQ(x) = (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
ckq(x)cℓp(x) (17)
for all x∈ {1, 2, · · · , (2j+1)2}, K ∈ {0, 1, · · · , 2j} and Q∈ {−K,−K+1, · · · , K}.
Proof: Consider P 2x = Px and use the coupling relation (51) of Appendix involving
a 3–jm and a 6–j Wigner symbols.
As a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50)
of Appendix, we get again the normalization relation ‖v(x)‖2 = v(x) · v(x) = 1.
• All coefficients ckq(x) are connected through the sum rule
(2j+1)2
ckq(x)
j k j
−m q m′
= (−1)j−m(2j + 1)δm,m′ , (18)
which turns out to be useful for global checking purposes.
Proof: Take the jm–jm′ matrix element of the resolution of the identity in terms of
the operators Px/(2j + 1).
3 MUBs
A complete set of MUBs in the Hilbert space Cd is a set of d(d + 1) vectors |aα〉 ∈ Cd
such that
|〈aα|bβ〉|2 = δα,βδa,b +
(1− δa,b), (19)
where a = 0, 1, · · · , d and α = 0, 1, · · · , d − 1. The indices of type a refer to the bases
and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.
We know that such a complete set exists if d is a prime or the power of a prime (e.g., see
[16]-[24]).
The approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let
us suppose that it is possible to find d+ 1 sets Sa (with a = 0, 1, · · · , d) of vectors in Cd,
each set Sa = {|aα〉 : α = 0, 1, · · · , d − 1} containing d vectors |aα〉 such that Eq. (19)
be satisfied. This amounts to finding d(d+ 1) projection operators
Πaα = |aα〉〈aα| (20)
satisfying the trace condition
Tr (ΠaαΠbβ) = δα,βδa,b +
(1− δa,b), (21)
where the trace is taken on Cd. Therefore, they also form a nonorthogonal decomposition
of the identity
Πaα = I. (22)
As in Sec. 2, we develop each operator Πaα on an orthonormal basis with expansion
coefficients wi(aα). Thus we get vectors w(aα) in C
w(aα) = (w1(aα), w2(aα), · · · , wd2(aα)) (23)
such that
w(aα) · w(bβ) = δα,βδa,b +
(1− δa,b) (24)
for all a, b ∈ {0, 1, · · · , d} and α, β ∈ {0, 1, · · · , d− 1}.
Now we draw the same relations as for POVMs by choosing the Racah operators to be
our basis in Cd
. We assume once again that the Hilbert space Cd is realized by ε(j) with
j = (d − 1)/2. Then, each operator Πaα can be developed on the basis of the (2j + 1)2
operators u
Πaα =
dkq(aα)u
q , (25)
to be compared with Eq. (10). The expansion coefficients are
dkq(aα) = (2k + 1)〈aα|u(k)q |aα〉 (26)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k +
1, · · · , k}. For a and α fixed, the complex coefficients dkq(aα) define a vector
w(aα) = (w1(aα), w2(aα), · · · , wN(aα)) , N = (2j + 1)2 (27)
in the Hilbert space CN , the components of which are given by
wi(aα) =
2k + 1
dkq(aα), i = k
2 + k + q + 1. (28)
We are thus led to the following properties and relations. The proofs are similar to those
in Sec. 2.
• First component w1(aα) of w(aα):
d00(aα) =
2j + 1
for all a ∈ {0, 1, · · · , 2j + 1} and α ∈ {0, 1, · · · , 2j}.
• Complex conjugation property:
dkq(aα) = (−1)qdk−q(aα) (30)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈
{−k,−k + 1, · · · , k}.
• Rotational invariance:
2k + 1
dkq(aα)dkq(bβ) = δα,βδa,b +
2j + 1
(1− δa,b) (31)
for all a, b ∈ {0, 1, · · · , 2j + 1} and α, β ∈ {0, 1, · · · , 2j}.
• Tensor product formula:
2K + 1
dKQ(aα) = (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
dkq(aα)dℓp(aα) (32)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, K ∈ {0, 1, · · · , 2j} and Q ∈
{−K,−K + 1, · · · , K}.
• Sum rule:
dkq(aα)
j k j
−m q m′
= (−1)j−m2(2j + 1)δm,m′ (33)
which involves all coefficients dkq(aα).
4 CONCLUSIONS
Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other
hand is very similar, there are deep differences between the two sets of results. The
similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to
finite affine planes [12, 13, 22, 23, 25] and to complex projective 2–designs [8, 10, 19, 24].
On the other side, there are two arguments in favor of the differences between relations (6)
and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent
to the existence of an affine plane of order d [12, 13]. Second, there is a consensus around
the conjecture according to which there exists a complete set of MUBs in dimension d if
and only if there exists an affine plane of order d [22].
In dimension d, to find d2 operators Px of a SIC-POVM acting on the Hilbert space
d amounts to find d2 vectors v(x) in the Hilbert space CN with N = d2 satisfying
‖vx‖ = 1, v(x) · v(y) =
for x 6= y (34)
(the norm ‖v(x)‖ of each vector v(x) is 1 and the angle ωxy of any pair of vectors v(x)
and v(y) is ωxy = cos
−1[1/(d+ 1)] for x 6= y).
In a similar way, to find d + 1 MUBs of Cd is equivalent to find d + 1 sets Sa (with
a = 0, 1, · · · , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in CN with N = d2
satisfying
w(aα) · w(aβ) = δα,β, w(aα) · w(bβ) =
for a 6= b (35)
(each set Sa consists of d orthonormalized vectors and the angle ωaαbβ of any vector
w(aα) of a set Sa with any vector w(bβ) of a set Sb is ωaαbβ = cos
−1(1/d) for a 6= b).
According to a well accepted conjecture [8, 10], SIC-POVMs should exist in any
dimension. The present study shows that in order to prove this conjecture it is sufficient
to prove that Eq. (34) admits solutions for any value of d.
The situation is different for MUBs. In dimension d, it is known that there exist d+ 1
sets of d vectors of type |aα〉 in Cd satisfying Eq. (19) when d is a prime or the power of
a prime. This shows that Eq. (35) can be solved for d prime or power of a prime. For d
prime, it is possible to find an explicit solution of Eq. (19). In fact, we have [26, 27]
|aα〉 =
2j + 1
ω(j+m)(j−m+1)a/2+(j+m)α|j,m〉, (36)
ω = exp
2j + 1
, j =
(d− 1) (37)
for a, α ∈ {0, 1, · · · , 2j} while
|aα〉 = |j,m〉 (38)
for a = 2j + 1 and α = j +m = 0, 1, · · · , 2j. Then, Eq. (26) yields
dkq(aα) =
2k + 1
2j + 1
m′=−j
ωθ(m,m
′)(−1)j−m
j k j
−m q m′
, (39)
θ(m,m′) = (m−m′)
(1−m−m′)a+ α
for a, α ∈ {0, 1, · · · , 2j} while
dkq(aα) = δq,0(2k + 1)(−1)j−m
j k j
−m 0 m
for a = 2j + 1 and α = j + m = 0, 1, · · · , 2j. It can be shown that Eqs. (40) and (41)
are in agreement with the results of Sec. 3. We thus have a solution of the equations for
the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find
an explicit solution for the coefficients dkq(aα) when d = 2j + 1 is any positive power
of a prime. Finally, note that to prove (or disprove) the conjecture according to which a
complete set of MUBs in dimension d exists only if d is a prime or the power of a prime
is equivalent to prove (or disprove) that Eq. (35) has a solution only if d is a prime or the
power of a prime.
APPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃
We limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain
SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations
in this appendix have to be extended to the allowed values for the involved magnetic and
angular momentum quantum numbers.
The definition (9) of the components u
q of the Racah unit tensor u(k) yields
〈j,m|u(k)q |j,m′〉 = (−1)j−m
j k j
−m q m′
, (42)
from which we easily obtain the Hermitian conjugation property
u(k)q
= (−1)qu(k)−q . (43)
The 3–jm Wigner symbol in Eq. (42) satisfies the orthogonality relations
j j′ k
m m′ q
j j′ ℓ
m m′ p
2k + 1
δk,ℓδq,p∆(j, j
′, k) (44)
(2k + 1)
j j′ k
m m′ q
j j′ k
M M ′ q
= δm,Mδm′,M ′. (45)
The trace relation on the space ε(j)
u(k)q
u(ℓ)p
2k + 1
δk,ℓδq,p∆(j, j, k) (46)
easily follows by combining Eqs. (42) and (44). Furthermore, by introducing
j j′ 0
m −m′ 0
= δj,j′δm,m′(−1)j−m
2j + 1
in Eq. (44), we obtain the sum rule
(−1)j−m
j k j
−m q m
2j + 1δk,0δq,0∆(j, k, j), (48)
known in spectroscopy as the barycenter theorem.
There are several relations involving 3–jm and 6–j symbols. In particular, we have
(−1)j−M
j k j
−m q M
j ℓ j
−M p m′
j K j
−m Q m′
= (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
, (49)
where {· · ·} denotes a 6–j Wigner symbol (or W Racah coefficient). Note that the intro-
duction of
k ℓ 0
j j J
= δk,ℓ(−1)j+k+J
(2k + 1)(2j + 1)
in Eq. (49) gives back Eq. (44). Equation (49) is central in the derivation of the coupling
relation
u(k)q u
(−1)2j−Q(2K + 1)
k ℓ K
−q −p Q
k ℓ K
j j j
Q . (51)
Equation (51) makes it possible to calculate the commutator [u
q , u
p ] which shows that
the set {u(k)q : k = 0, 1, · · · , 2j; q = −k,−k + 1, · · · , k} can be used to span the Lie
algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions
(17) and (32).
Note added in version 3
After the submission of the present paper for publication in Journal of Russian Laser
Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32].
The main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this
result, Eq. (34) admits solutions in any dimension.
Acknowledgements
This work was presented at the International Conference on Squeezed States and Un-
certainty Relations, University of Bradford, England (ICSSUR’07). The authors wish to
thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man’ko and M.
Planat for interesting comments.
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http://arxiv.org/abs/quant-ph/0205155
http://arxiv.org/abs/quant-ph/0502031
	INTRODUCTION
	SIC-POVMs
	MUBs
	CONCLUSIONS

  A unified approach to (symmetric informationally complete) positive operator
valued measures and mutually unbiased bases is developed in this article. The
approach is based on the use of operator equivalents expanded in the enveloping
algebra of SU(2). Emphasis is put on similarities and differences between
SIC-POVMs and MUBs.

A UNIFIED APPROACH TO SIC-POVMs AND MUBs
Olivier Albouy and Maurice R. Kibler
Université de Lyon, Institut de Physique Nucléaire,
Université Lyon 1 and CNRS/IN2P3, 43 bd du 11 novembre 1918,
F–69622 Villeurbanne, France
Electronic mail: o.albouy@ipnl.in2p3.fr, m.kibler@ipnl.in2p3.fr
Abstract
A unified approach to (symmetric informationally complete) positive op-
erator valued measures and mutually unbiased bases is developed in this arti-
cle. The approach is based on the use of Racah unit tensors for the Wigner-
Racah algebra of SU(2) ⊃ U(1). Emphasis is put on similarities and differ-
ences between SIC-POVMs and MUBs.
Keywords: finite–dimensional Hilbert spaces; mutually unbiased bases; positive op-
erator valued measures; SU(2) ⊃ U(1) Wigner–Racah algebra
1 INTRODUCTION
The importance of finite–dimensional spaces for quantum mechanics is well recognized
(see for instance [1]-[3]). In particular, such spaces play a major role in quantum informa-
tion theory, especially for quantum cryptography and quantum state tomography [4]-[27].
Along this vein, a symmetric informationally complete (SIC) positive operator valued
measure (POVM) is a set of operators acting on a finite Hilbert space [4]-[14] (see also
[3] for an infinite Hilbert space) and mutually unbiased bases (MUBs) are specific bases
for such a space [15]-[27].
The introduction of POVMs goes back to the seventies [4]-[7]. The most general quan-
tum measurement is represented by a POVM. In the present work, we will be interested
http://arxiv.org/abs/0704.0511v3
in SIC-POVMs, for which the statistics of the measurement allows the reconstruction of
the quantum state. Moreover, those POVMs are endowed with an extra symmetry condi-
tion (see definition in Sec. 2). The notion of MUBs (see definition in Sec. 3), implicit or
explicit in the seminal works of [15]-[18], has been the object of numerous mathematical
and physical investigations during the last two decades in connection with the so-called
complementary observables. Unfortunately, the question to know, for a given Hilbert
space of finite dimension d, whether there exist SIC-POVMs and how many MUBs there
exist has remained an open one.
The aim of this note is to develop a unified approach to SIC-POVMs and MUBs based
on a complex vector space of higher dimension, viz. d2 instead of d. We then give a
specific example of this approach grounded on the Wigner-Racah algebra of the chain
SU(2) ⊃ U(1) recently used for a study of entanglement of rotationally invariant spin
systems [28] and for an angular momentum study of MUBs [26, 27].
Most of the notations in this work are standard. Let us simply mention that I is the
identity operator, the bar indicates complex conjugation, A† denotes the adjoint of the
operator A, δa,b stands for the Kronecker symbol for a and b, and ∆(a, b, c) is 1 or 0
according as a, b and c satisfy or not the triangular inequality.
2 SIC-POVMs
Let Cd be the standard Hilbert space of dimension d endowed with its usual inner product
denoted by 〈 | 〉. As is usual, we will identify a POVM with a nonorthogonal decompo-
sition of the identity. Thus, a discrete SIC-POVM is a set {Px : x = 1, 2, · · · , d2} of d2
nonnegative operators Px acting on C
d, such that:
• they satisfy the trace or symmetry condition
Tr (PxPy) =
, x 6= y; (1)
moreover, we will assume the operators Px are normalized, thus completing this
condition with
= 1; (2)
• they form a decomposition of the identity
Px = I; (3)
• they satisfy a completeness condition: the knowledge of the probabilities px defined
by px = Tr(Pxρ) is sufficient to reconstruct the density matrix ρ.
Now, let us develop each of the operators Px on an orthonormal (with respect to the
Hilbert–Schmidt product) basis {ui : i = 1, 2, · · · , d2} of the space of linear operators on
vi(x)ui, (4)
where the operators ui satisfy Tr(u
iuj) = δi,j . The operators Px are thus considered as
vectors
v(x) = (v1(x), v2(x), · · · , vd2(x)) (5)
in the Hilbert space Cd
of dimension d2 and the determination of the operators Px is
equivalent to the determination of the components vi(x) of v(x). In this language, the
trace property (1) together with the normalization condition (2) give
v(x) · v(y) = 1
(dδx,y + 1) , (6)
where v(x) · v(y) =
i=1 vi(x)vi(y) is the usual Hermitian product in C
In order to compare Eq. (6) with what usually happens in the search for SIC-POVMs,
we suppose from now on that the operators Px are rank-one operators. Therefore, by
putting
Px = |Φx〉〈Φx| (7)
with |φx〉 ∈ Cd, the trace property (1, 2) reads
|〈Φx|Φy〉|2 =
(dδx,y + 1) . (8)
From this point of view, to find d2 operators Px is equivalent to finding d
2 vectors |φx〉
in Cd satisfying Eq. (8). At the price of an increase in the number of components from
d3 (for d2 vectors in Cd) to d4 (for d2 vectors in Cd
), we have got rid of the square
modulus to result in a single scalar product (compare Eqs. (6) and (8)), what may prove
to be suitable for another way to search for SIC-POVMs. Moreover, our relation (6) is
independent of any hypothesis on the rank of the operators Px. In fact, there exists a lot
of relations among these d4 coefficients that decrease the effective number of coefficients
to be found and give structural constraints on them. Those relations are highly sensitive
to the choice of the basis {ui : i = 1, 2, · · · , d2} and we are going to exhibit an example
of such a set of relations by choosing the basis to consist of Racah unit tensors.
The cornerstone of this approach is to identify Cd with a subspace ε(j) of constant
angular momentum j = (d− 1)/2. Such a subspace is spanned by the set {|j,m〉 : m =
−j,−j + 1, · · · , j}, where |j,m〉 is an eigenvector of the square and the z-component
of a generalized angular momentum operator. Let u(k) be the Racah unit tensor [29]
of order k (with k = 0, 1, · · · , 2j) defined by its 2k + 1 components u(k)q (where q =
−k,−k + 1, · · · , k) through
u(k)q =
m′=−j
(−1)j−m
j k j
−m q m′
|j,m〉〈j,m′|, (9)
where (· · ·) denotes a 3–jm Wigner symbol. For fixed j, the (2j + 1)2 operators u(k)q
(with k = 0, 1, · · · , 2j and q = −k,−k + 1, · · · , k) act on ε(j) ∼ Cd and form a basis
of the Hilbert space CN of dimension N = (2j + 1)2, the inner product in CN being the
Hilbert–Schmidt product. The formulas (involving unit tensors, 3–jm and 6–j symbols)
relevant for this work are given in Appendix (see also [29] to [31]). We must remember
that those Racah operators are not normalized to unity (see relation (46)). So this will
generate an extra factor when defining vi(x).
Each operator Px can be developed as a linear combination of the operators u
Hence, we have
ckq(x)u
q , (10)
where the unknown expansion coefficients ckq(x) are a priori complex numbers. The
determination of the operators Px is thus equivalent to the determination of the coefficients
ckq(x), which are formally given by
ckq(x) = (2k + 1)〈Φx|u(k)q |Φx〉, (11)
as can be seen by multiplying each member of Eq. (10) by the adjoint of u
p and then
using Eq. (46) of Appendix.
By defining the vector
v(x) = (v1(x), v2(x), · · · , vN(x)), N = (2j + 1)2 (12)
vi(x) =
2k + 1
ckq(x), i = k
2 + k + q + 1, (13)
the following properties and relations are obtained.
• The first component v1(x) of v(x) does not depend on x since
c00(x) =
2j + 1
for all x ∈ {1, 2, · · · , (2j + 1)2}.
Proof: Take the trace of Eq. (10) and use Eq. (48) of Appendix.
• The components vi(x) of v(x) satisfy the complex conjugation property described
ckq(x) = (−1)qck−q(x) (15)
for all x ∈ {1, 2, · · · , (2j + 1)2}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k + 1, · · · , k}.
Proof: Use the Hermitian property of Px and Eq. (43) of Appendix.
• In terms of ckq, Eq. (6) reads
2k + 1
ckq(x)ckq(y) =
2(j + 1)
[(2j + 1)δx,y + 1] (16)
for all x, y ∈ {1, 2, · · · , (2j + 1)2}, where the sum over q is SO(3) rotationally
invariant.
Proof: The proof is trivial.
• The coefficients ckq(x) are solutions of the nonlinear system given by
2K + 1
cKQ(x) = (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
ckq(x)cℓp(x) (17)
for all x∈ {1, 2, · · · , (2j+1)2}, K ∈ {0, 1, · · · , 2j} and Q∈ {−K,−K+1, · · · , K}.
Proof: Consider P 2x = Px and use the coupling relation (51) of Appendix involving
a 3–jm and a 6–j Wigner symbols.
As a corollary of the latter property, by taking K = 0 and using Eqs. (47) and (50)
of Appendix, we get again the normalization relation ‖v(x)‖2 = v(x) · v(x) = 1.
• All coefficients ckq(x) are connected through the sum rule
(2j+1)2
ckq(x)
j k j
−m q m′
= (−1)j−m(2j + 1)δm,m′ , (18)
which turns out to be useful for global checking purposes.
Proof: Take the jm–jm′ matrix element of the resolution of the identity in terms of
the operators Px/(2j + 1).
3 MUBs
A complete set of MUBs in the Hilbert space Cd is a set of d(d + 1) vectors |aα〉 ∈ Cd
such that
|〈aα|bβ〉|2 = δα,βδa,b +
(1− δa,b), (19)
where a = 0, 1, · · · , d and α = 0, 1, · · · , d − 1. The indices of type a refer to the bases
and, for fixed a, the index α refers to one of the d vectors of the basis corresponding to a.
We know that such a complete set exists if d is a prime or the power of a prime (e.g., see
[16]-[24]).
The approach developed in Sec. 2 for SIC-POVMs can be applied to MUBs too. Let
us suppose that it is possible to find d+ 1 sets Sa (with a = 0, 1, · · · , d) of vectors in Cd,
each set Sa = {|aα〉 : α = 0, 1, · · · , d − 1} containing d vectors |aα〉 such that Eq. (19)
be satisfied. This amounts to finding d(d+ 1) projection operators
Πaα = |aα〉〈aα| (20)
satisfying the trace condition
Tr (ΠaαΠbβ) = δα,βδa,b +
(1− δa,b), (21)
where the trace is taken on Cd. Therefore, they also form a nonorthogonal decomposition
of the identity
Πaα = I. (22)
As in Sec. 2, we develop each operator Πaα on an orthonormal basis with expansion
coefficients wi(aα). Thus we get vectors w(aα) in C
w(aα) = (w1(aα), w2(aα), · · · , wd2(aα)) (23)
such that
w(aα) · w(bβ) = δα,βδa,b +
(1− δa,b) (24)
for all a, b ∈ {0, 1, · · · , d} and α, β ∈ {0, 1, · · · , d− 1}.
Now we draw the same relations as for POVMs by choosing the Racah operators to be
our basis in Cd
. We assume once again that the Hilbert space Cd is realized by ε(j) with
j = (d − 1)/2. Then, each operator Πaα can be developed on the basis of the (2j + 1)2
operators u
Πaα =
dkq(aα)u
q , (25)
to be compared with Eq. (10). The expansion coefficients are
dkq(aα) = (2k + 1)〈aα|u(k)q |aα〉 (26)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈ {−k,−k +
1, · · · , k}. For a and α fixed, the complex coefficients dkq(aα) define a vector
w(aα) = (w1(aα), w2(aα), · · · , wN(aα)) , N = (2j + 1)2 (27)
in the Hilbert space CN , the components of which are given by
wi(aα) =
2k + 1
dkq(aα), i = k
2 + k + q + 1. (28)
We are thus led to the following properties and relations. The proofs are similar to those
in Sec. 2.
• First component w1(aα) of w(aα):
d00(aα) =
2j + 1
for all a ∈ {0, 1, · · · , 2j + 1} and α ∈ {0, 1, · · · , 2j}.
• Complex conjugation property:
dkq(aα) = (−1)qdk−q(aα) (30)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, k ∈ {0, 1, · · · , 2j} and q ∈
{−k,−k + 1, · · · , k}.
• Rotational invariance:
2k + 1
dkq(aα)dkq(bβ) = δα,βδa,b +
2j + 1
(1− δa,b) (31)
for all a, b ∈ {0, 1, · · · , 2j + 1} and α, β ∈ {0, 1, · · · , 2j}.
• Tensor product formula:
2K + 1
dKQ(aα) = (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
dkq(aα)dℓp(aα) (32)
for all a ∈ {0, 1, · · · , 2j + 1}, α ∈ {0, 1, · · · , 2j}, K ∈ {0, 1, · · · , 2j} and Q ∈
{−K,−K + 1, · · · , K}.
• Sum rule:
dkq(aα)
j k j
−m q m′
= (−1)j−m2(2j + 1)δm,m′ (33)
which involves all coefficients dkq(aα).
4 CONCLUSIONS
Although the structure of the relations in Sec. 1 on the one hand and Sec. 2 on the other
hand is very similar, there are deep differences between the two sets of results. The
similarities are reminiscent of the fact that both MUBs and SIC-POVMs can be linked to
finite affine planes [12, 13, 22, 23, 25] and to complex projective 2–designs [8, 10, 19, 24].
On the other side, there are two arguments in favor of the differences between relations (6)
and (24). First, the problem of constructing SIC-POVMs in dimension d is not equivalent
to the existence of an affine plane of order d [12, 13]. Second, there is a consensus around
the conjecture according to which there exists a complete set of MUBs in dimension d if
and only if there exists an affine plane of order d [22].
In dimension d, to find d2 operators Px of a SIC-POVM acting on the Hilbert space
d amounts to find d2 vectors v(x) in the Hilbert space CN with N = d2 satisfying
‖vx‖ = 1, v(x) · v(y) =
for x 6= y (34)
(the norm ‖v(x)‖ of each vector v(x) is 1 and the angle ωxy of any pair of vectors v(x)
and v(y) is ωxy = cos
−1[1/(d+ 1)] for x 6= y).
In a similar way, to find d + 1 MUBs of Cd is equivalent to find d + 1 sets Sa (with
a = 0, 1, · · · , d) of d vectors, i.e., d(d + 1) vectors in all, w(aα) in CN with N = d2
satisfying
w(aα) · w(aβ) = δα,β, w(aα) · w(bβ) =
for a 6= b (35)
(each set Sa consists of d orthonormalized vectors and the angle ωaαbβ of any vector
w(aα) of a set Sa with any vector w(bβ) of a set Sb is ωaαbβ = cos
−1(1/d) for a 6= b).
According to a well accepted conjecture [8, 10], SIC-POVMs should exist in any
dimension. The present study shows that in order to prove this conjecture it is sufficient
to prove that Eq. (34) admits solutions for any value of d.
The situation is different for MUBs. In dimension d, it is known that there exist d+ 1
sets of d vectors of type |aα〉 in Cd satisfying Eq. (19) when d is a prime or the power of
a prime. This shows that Eq. (35) can be solved for d prime or power of a prime. For d
prime, it is possible to find an explicit solution of Eq. (19). In fact, we have [26, 27]
|aα〉 =
2j + 1
ω(j+m)(j−m+1)a/2+(j+m)α|j,m〉, (36)
ω = exp
2j + 1
, j =
(d− 1) (37)
for a, α ∈ {0, 1, · · · , 2j} while
|aα〉 = |j,m〉 (38)
for a = 2j + 1 and α = j +m = 0, 1, · · · , 2j. Then, Eq. (26) yields
dkq(aα) =
2k + 1
2j + 1
m′=−j
ωθ(m,m
′)(−1)j−m
j k j
−m q m′
, (39)
θ(m,m′) = (m−m′)
(1−m−m′)a+ α
for a, α ∈ {0, 1, · · · , 2j} while
dkq(aα) = δq,0(2k + 1)(−1)j−m
j k j
−m 0 m
for a = 2j + 1 and α = j + m = 0, 1, · · · , 2j. It can be shown that Eqs. (40) and (41)
are in agreement with the results of Sec. 3. We thus have a solution of the equations for
the results of Sec. 3 when d is prime. As an open problem, it would be worthwhile to find
an explicit solution for the coefficients dkq(aα) when d = 2j + 1 is any positive power
of a prime. Finally, note that to prove (or disprove) the conjecture according to which a
complete set of MUBs in dimension d exists only if d is a prime or the power of a prime
is equivalent to prove (or disprove) that Eq. (35) has a solution only if d is a prime or the
power of a prime.
APPENDIX: WIGNER-RACAH ALGEBRA OF SU(2) ⊃
We limit ourselves to those basic formulas for the Wigner-Racah algebra of the chain
SU(2) ⊃ U(1) which are necessary to derive the results of this paper. The summations
in this appendix have to be extended to the allowed values for the involved magnetic and
angular momentum quantum numbers.
The definition (9) of the components u
q of the Racah unit tensor u(k) yields
〈j,m|u(k)q |j,m′〉 = (−1)j−m
j k j
−m q m′
, (42)
from which we easily obtain the Hermitian conjugation property
u(k)q
= (−1)qu(k)−q . (43)
The 3–jm Wigner symbol in Eq. (42) satisfies the orthogonality relations
j j′ k
m m′ q
j j′ ℓ
m m′ p
2k + 1
δk,ℓδq,p∆(j, j
′, k) (44)
(2k + 1)
j j′ k
m m′ q
j j′ k
M M ′ q
= δm,Mδm′,M ′. (45)
The trace relation on the space ε(j)
u(k)q
u(ℓ)p
2k + 1
δk,ℓδq,p∆(j, j, k) (46)
easily follows by combining Eqs. (42) and (44). Furthermore, by introducing
j j′ 0
m −m′ 0
= δj,j′δm,m′(−1)j−m
2j + 1
in Eq. (44), we obtain the sum rule
(−1)j−m
j k j
−m q m
2j + 1δk,0δq,0∆(j, k, j), (48)
known in spectroscopy as the barycenter theorem.
There are several relations involving 3–jm and 6–j symbols. In particular, we have
(−1)j−M
j k j
−m q M
j ℓ j
−M p m′
j K j
−m Q m′
= (−1)2j−Q
k ℓ K
−q −p Q
k ℓ K
j j j
, (49)
where {· · ·} denotes a 6–j Wigner symbol (or W Racah coefficient). Note that the intro-
duction of
k ℓ 0
j j J
= δk,ℓ(−1)j+k+J
(2k + 1)(2j + 1)
in Eq. (49) gives back Eq. (44). Equation (49) is central in the derivation of the coupling
relation
u(k)q u
(−1)2j−Q(2K + 1)
k ℓ K
−q −p Q
k ℓ K
j j j
Q . (51)
Equation (51) makes it possible to calculate the commutator [u
q , u
p ] which shows that
the set {u(k)q : k = 0, 1, · · · , 2j; q = −k,−k + 1, · · · , k} can be used to span the Lie
algebra of the unitary group U(2j + 1). The latter result is at the root of the expansions
(17) and (32).
Note added in version 3
After the submission of the present paper for publication in Journal of Russian Laser
Research, a pre-print dealing with the existence of SIC-POVMs was posted on arXiv [32].
The main result in [32] is that SIC-POVMs exist in all dimensions. As a corollary of this
result, Eq. (34) admits solutions in any dimension.
Acknowledgements
This work was presented at the International Conference on Squeezed States and Un-
certainty Relations, University of Bradford, England (ICSSUR’07). The authors wish to
thank the organizer A. Vourdas and are grateful to D. M. Appleby, V. I. Man’ko and M.
Planat for interesting comments.
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http://arxiv.org/abs/quant-ph/0205155
http://arxiv.org/abs/quant-ph/0502031
	INTRODUCTION
	SIC-POVMs
	MUBs
	CONCLUSIONS
