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Minimax quantum state discrimination
Giacomo Mauro D’Ariano∗ and Massimiliano Federico Sacchi†
QUIT Group of the INFM, Unita` di Pavia and
Universita` di Pavia, Dipartimento di Fisica “A. Volta”, via Bassi 6, I-27100 Pavia, Italy‡
Jonas Kahn§
Universite´ Paris-Sud 11, De´partement de Mathe´matiques, Baˆt 425, 91405 Orsay Cedex, France
(Dated: February 1, 2008)
We derive the optimal measurement for quantum state discrimination without a priori proba-
bilities, i. e. in a minimax strategy instead of the usually considered Bayesian one. We consider
both minimal-error and unambiguous discrimination problems, and provide the relation between
the optimal measurements according to the two schemes. We show that there are instances in which
the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature
of the minimax estimation strategy.
PACS numbers: 03.67.-a 03.65.Ta
I. INTRODUCTION
Since the pioneering work of Helstrom [1] on quan-
tum hypothesis testing, the problem of discriminating
nonorthogonal quantum states has received much atten-
tion [2], with some experimental verifications as well [3].
The most popular scenarios are the minimal-error prob-
ability discrimination [1], where each measurement out-
come selects one of the possible states and the error prob-
ability is minimized, and the optimal unambiguous dis-
crimination of linearly independent states[4], where un-
ambiguity is paid by the possibility of getting inconclu-
sive results from the measurement. The problem of dis-
crimination has been addressed also for bipartite quan-
tum states, with both global joint measurements and lo-
cal measurements with classical communication[5]. The
concept of distinguishability can be applied also to all
physically allowed transformations of quantum states,
and in fact, more recently, the problem of discrimination
has been considered for unitary transformations [6] and
more general quantum operations [7]. In all the above
mentioned discrimination problems, a Bayesian approach
has always been considered, with given a priori proba-
bility distribution for the states (or operations) to be
discriminated.
In this paper, we consider the problem of optimal dis-
crimination of quantum states in the minimax approach.
In this strategy no prior probabilities are given. The rel-
evance of this approach is both conceptual, since for a
frequentist statistician the a priori probabilities have no
meaning, and practical, because the prior probabilities
may be actually unknown, as in a noncooperative cryp-
tographic scenario. We will derive the optimal measure-
∗Electronic address: dariano@unipv.it
†Electronic address: msacchi@unipv.it
‡URL: http://www.qubit.it
§Electronic address: jokahn@clipper.ens.fr
ment for minimax state discrimination for both minimal-
error and unambiguous discrimination problems. We will
also provide the relation between the optimal measure-
ments according to the minimax and the Bayesian strate-
gies. We will show that, quite unexpectedly, there are in-
stances in which the minimum risk can be achieved only
by non orthogonal positive operator-valued measurement
(POVM), and this is a common feature of the minimax
estimation strategy.
The paper is organized as follows. In Sec. II we pose
the problem of discrimination of two quantum states in
the minimax scenario. Such an approach is equivalent
to a minimax problem, where one should maximise the
smallest of the two probabilities of correct detection over
all measurement schemes. For simplicity we will con-
sider equal weights (i.e. equal prices of misidentifying
the states), and we will provide the optimal measurement
for the minimax discrimination, along with the connec-
tion with the optimal Bayesian solution. As mentioned,
a striking result of this section is the existence of cou-
ples of mixed states for which the optimal minimax mea-
surement is unique and nonorthogonal. In Sec. III we
generalize the results for two-state discrimination to the
case of N > 2 states and arbitrary weights. First, we
consider the simplest situation of the covariant state dis-
crimination problem. Then, we address the problem in
generality, resorting to the related convex programming
method. In Sec. IV we provide the solution of the min-
imax discrimination problem in the scenario of unam-
biguous discrimination. The conclusions of the paper are
summarized in Sec. V.
II. OPTIMAL MINIMAX DISCRIMINATION OF
TWO QUANTUM STATES
We are given two states ρ1 and ρ2, generally mixed,
and we want to find the optimal measurement to discrim-
inate between them in a minimax strategy. The mea-
surement is described by a POVM with two outcomes,

2namely ~P ≡ (P1, P2), where Pi for i = 1, 2 are nonnega-
tive operators satisfying P1 + P2 = I.
In the usually considered Bayesian approach to the dis-
crimination problem, the states are given with a priori
probability distribution ~a ≡ (a1, a2), respectively, and
one looks for the POVM that minimizes the average er-
ror probability
pE = a1Tr[ρ1P2] + a2Tr[ρ2P1]. (1)
The solution can then be achieved by taking the orthog-
onal POVM made by the projectors on the support of
the positive and negative part of the Hermitian operator
a1ρ1 − a2ρ2, and hence one has [1]
p(Bayes)E =
1
2
(1− ‖a1ρ1 − a2ρ2‖1) , (2)
where ‖A‖1 denotes the trace norm of A.
In the minimax problem, one does not have a priori
probabilities. However, one defines the error probability
εi(~P ) = Tr[ρi(I − Pi)] of failing to identify ρi. The opti-
mal minimax solution consists in finding the POVM that
achieves the minimax
ε = min
~P
max
i=1,2
εi(~P ), (3)
or equivalently, that maximizes the smallest of the prob-
abilities of correct detection
1− ε = max
~P
min
i=1,2
[1− εi(~P )] = max
~P
min
i=1,2
Tr[ρiPi]. (4)
The minimax and Bayesian strategies of discrimination
are connected by the following theorem.
Theorem 1 If there is an a priori probability ~a =
(a1, a2) for the states ρ1 and ρ2, and a measurement ~B
that achieves the optimal Bayesian average error for ~a,
with equal probabilities of correct detection, i.e.
Tr[ρ1B1] = Tr[ρ2B2], (5)
then ~B is also the solution of the minimax discrimination
problem.
Proof. In fact, suppose on the contrary that there
exists a POVM ~P such that mini=1,2 Tr[ρiPi] >
mini=1,2 Tr[ρiBi]. Due to assumption (5) one has
Tr[ρiPi] > Tr[ρiBi] for both i = 1, 2, whence
∑
i
ai Tr(ρiPi) >
∑
i
ai Tr(ρiBi) (6)
which contradicts the fact that ~B is optimal for ~a.
The existence of an optimal ~B as in Theorem 1 will be
shown in the following.
First, by labeling with ~P (a) an optimal POVM for the
Bayesian problem with prior probability distribution ~a =
(a, 1− a), and defining
χ(a, ~P ) .= aTr(ρ1P1) + (1− a)Tr(ρ2P2), (7)
we have the following lemma.
Lemma 1 The function f(a) .= Tr(ρ1P (a)1 )−Tr(ρ2P
(a)
2 )
is monotonically nondecreasing, with minimum value
f(0) 6 0, and maximum value f(1) > 0.
In fact, consider ~P (a) and ~P (b) for two values a and b
with a < b and define ~D = ~P (b) − ~P (a). Then
χ(a, ~P (b)) = χ(a, ~P (a)) + χ(a, ~D)
χ(b, ~P (a)) = χ(b, ~P (b))− χ(b, ~D).
(8)
Now, since χ(a, ~P (a)) is the optimal probability of correct
detection for prior a, and analogously χ(b, ~P (b)) for prior
b, then χ(a, ~D) 6 0 and χ(b, ~D) > 0, and hence
0 ≤ χ(b, ~D)− χ(a, ~D) = (b− a)[Tr(ρ1D1)− Tr(ρ2D2)].
It follows that Tr(ρ1D1) > Tr(ρ2D2), namely
Tr(ρ1P (b)1 )− Tr(ρ1P
(a)
1 ) > Tr(ρ2P
(b)
2 )− Tr(ρ2P
(a)
2 ) (9)
or, equivalently,
Tr(ρ1P (b)1 )−Tr(ρ2P
(b)
2 ) > Tr(ρ1P
(a)
1 )−Tr(ρ2P
(a)
2 ). (10)
Equation (10) states that the function f(a) is monoton-
ically nondecreasing. Moreover, for a = 0 the POVM
detects only the state ρ2, whence Tr(ρ2P (0)2 ) = 1, and
one has f(0) = −1 + Tr[ρ1P (0)1 ] 6 0. Similarly one can
see that f(1) > 0.
We can now prove the following theorem.
Theorem 2 An optimal ~B as in Theorem 1 always ex-
ists.
Proof. Consider the value a0 of a where f(a) changes
its sign from negative to positive, and there take the left
and right limits
~P (∓) = lim
a→a∓0
~P (a). (11)
For f(a+0 ) = f(a−0 ) = 0 just define ~B = ~P (a0). For
f(a+0 ) > f(a−0 ) define the POVM ~B
~B = f(a
+
0 )~P (−) − f(a−0 )~P (+)
f(a+0 )− f(a−0 )
. (12)
In fact, one has
Tr[ρ1B1]− Tr[ρ2B2] = [f(a+0 )− f(a−0 )]−1×
{Tr[ρ1P (−)1 − ρ2P
(−)
2 ]f(a+0 )−
Tr[ρ1P (+)1 − ρ2P
(+)
2 ]f(a−0 )} = 0 ,
(13)
namely Eq. (5) holds. 
Notice that the value a0 is generally not unique, since
the function f(a) can be locally constant. However,
on the Hilbert space Supp(ρ1) ∪ Supp(ρ2), the optimal
POVM for the minimax problem is unique, apart from

3the very degenerate case in which D = a0ρ1 − (1− a0)ρ2
has at least two-dimensional kernel. In fact, upon denot-
ing by Π+ and K the projector on the strictly positive
part and the kernel of D, respectively, any Bayes opti-
mal POVM is written (B1 = Π+ + K ′, B2 = I − B1),
with K ′ 6 K. Since for the optimal minimax POVM we
need Tr[ρ1B1] = Tr[ρ2B2], one obtains Tr[(ρ1 +ρ2)K ′] =
1−Tr[(ρ1+ρ2)Π+], which has a unique solutionK ′ = αK
if K is a one-dimensional projector.
Remark 1 For two pure states the optimal POVM for
the minimax discrimination is orthogonal and unique (up
to trivial completion of Span{|ψi〉}i=1,2 to the full Hilbert
space of the quantum system).
In fact, on the space Span{|ψi〉}i=1,2 the optimal Bayes
measurement is always orthogonal and unique for any
prior probability, hence there exists an optimal POVM
for the minimax discrimination that coincides with the
optimal Bayesian one, which is orthogonal. Uniqueness
of the minimax optimal POVM follows from the consid-
erations after the proof of Theorem 2 when restricting to
the subspace spanned by the two states.
Remark 2 There are couples of mixed states for which
the optimal minimax POVM is unique and nonorthogo-
nal.
For example, consider the following states in dimension
two
ρ1 =
[
1 0
0 0
]
, ρ2 =
[ 1
2 0
0 12
]
. (14)
Then an optimal minimax POVM is given by
P1 =
[ 2
3 0
0 0
]
, P2 =
[ 1
3 0
0 1
]
. (15)
In fact, clearly there is an optimal POVM of the diagonal
form. We need to maximize mini=1,2 Tr[ρiPi], whence,
according to Theorem 2, we need to maximize Tr[ρ1P1]
with the constraints Tr[ρ1P1] = Tr[ρ2P2] and P2 = I −
P1. Such an optimal POVM is unique, otherwise there
would exists a convex combination a0ρ1−(1−a0)ρ2 with
kernel at least two-dimensional, which is impossible in
the present example (see comments after the proof of
Theorem 2).
Notice that when the optimal POVM for the mini-
max strategy is unique and nonorthogonal, then there
is a prior probability distribution ~a for which the opti-
mal POVM for the Bayes problem is not unique, and the
nonorthogonal POVM that optimizes the minimax prob-
lem is also optimal for the Bayes’ one. In the example of
Remark 2 the optimal POVM (15) is also optimal for the
Bayes problem with ~a = (13 ,
2
3 ) as one can easily check.
However, in the Bayes case one can always choose an op-
timal orthogonal POVM, whereas in the minimax case
you may have to choose a non-orthogonal POVM.
Finally, notice that, unlike in the Bayesian case, the
optimal POVM for the minimax strategy may also be
not extremal.
III. OPTIMAL MINIMAX DISCRIMINATION
OF N > 2 QUANTUM STATES
We now consider the easiest case of discrimination with
more than two states, namely the discrimination among
a covariant set. In a fully covariant state discrimination,
one has a set of states {ρi} with ρi = Uiρ0U †i ∀i, for
fixed ρ0 and {Ui} a (projective) unitary representation
of a group. In the Bayesian case full covariance requires
that the prior probability distribution {ai} is uniform.
Then, one can easily prove (see, for example, Ref. [8])
that also the optimal POVM is covariant, namely it is of
the form Pi = UiKU †i , for suitable fixed operator K > 0.
Theorem 3 For a fully covariant state discrimination
problem, there is an optimal measurement for the min-
imax strategy that is covariant, and coincides with an
optimal Bayesian measurement.
Proof. A covariant POVM {Pi} gives a probability
p = Tr[ρiPi] independent of i. Moreover, there always
exists an optimal Bayesian POVM that is covariant and
maximizes p, which then is also the maximum over all
POVM’s of the average probability of correct estimation
Tr[ρiPi] for uniform prior distribution [8]. Now, suppose
by contradiction that there exists an optimal minimax
POVM {P ′i} maximizing p′ = mini Tr[ρiP ′i ], for which
p′ > p. Then, one has p < p′ 6 Tr[ρiP ′i ], contradict-
ing the assertion that an optimal Bayesian POVM max-
imizes Tr[ρiPi] over all POVM’s. Therefore, p = p′, and
the covariant Bayesian POVM also solves the minimax
problem.
Notice that in the covariant case also for any optimal
minimax POVM {Pi} one has Tr[ρiPi] independent of
i, since the average probability of correct estimation is
equal to the minimum one.
In the following we generalize Theorem 1 for two states
to the case of N > 2 states and arbitrary weights. We
have
Theorem 4 For any set of states {ρi}26i6N and any
set of weights wij (price of misidentifying i with j) the
solution of the minimax problem
r = inf
~P
sup
i
∑
j
wij Tr[ρiPj ] (16)
is equivalent to the solution of the problem
r = max
~a
rB(a), (17)
where rB(~a) is the Bayesian risk
rB(~a) .= max
~P
∑
i
ai
∑
j
wij Tr[ρiPj ]. (18)
Proof. The minimax problem in Eq. (16) is equivalent
to look for the minimum of the real function δ = f(~P )

4over ~P , with the constraints
∑
j wij Tr[ρiPj ] 6 δ, ∀i
Pj > 0, ∀j
∑
j Pj = I. (19)
Upon introducing the Lagrange multipliers:
µi ∈ R+ , ∀i
0 ≤ Zi ∈ Md(C), ∀i
Y † = Y ∈ Md(C),
(20)
Md(C) denoting the d× d matrices on the complex field,
the problem is equivalent to
r = inf
~P ,δ
sup
~µ,~Z,Y
′ l(~P , δ, ~µ, ~Z, Y ),
l(~P , δ, ~µ, ~Z, Y ) .= δ +
∑
i
[µi(
∑
j
wij Tr[ρiPj ]− δ)]
−
∑
i
Tr[ZiPi] + Tr[Y (I −
∑
i
Pi)], (21)
where sup′ denotes the supremum over the set defined
in Eqs. (20). The problem is convex [namely both the
function δ and the constraints (19) are convex] and meets
Slater’s conditions [9] (namely one can find values of ~P
and δ such that the constraints are satisfied with strict
inequalities), and hence in Eq. (21) one has
inf
~P ,δ
sup
~µ,~Z,Y
′ l(~P , δ, ~µ, ~Z, Y ) = max
~µ,~Z,Y
′ inf
~P ,δ
l(~P , δ, ~µ, ~Z, Y ).
(22)
It follows that
r = max
~µ,~Z,Y
′ Tr Y (23)
under the additional constraints
∑
i
µi = 1 ,
∑
i
wijµiρi − Zj − Y = 0 , ∀j. (24)
The constraints can be rewritten as
µi > 0 ,
∑
i
µi = 1 ,
Y 6
∑
i
wijµiρi , ∀j. (25)
Now, notice that for the Bayesian problem with prior
~a, along the same reasoning, one writes the equivalent
problem
rB(~a) = max
Y
′ Tr Y, (26)
with the constraint
∑
i
wijaiρi − Zj − Y = 0 , ∀j (27)
ai > 0 ,
∑
i
ai = 1 ,
Y 6
∑
i
wijaiρi , ∀j, (28)
which is the same as the minimax problem, with the role
of the Lagrange multipliers {µi} now played by the prior
probability distribution {ai}.
Clearly, a POVM that attains r in the minimax
problem (16) actually exists, being the infimum over a
(weakly) compact set—the POVM convex set—of the
(weakly) continuous function supi
∑
j wij Tr[ρiPj ].
IV. OPTIMAL MINIMAX UNAMBIGUOUS
DISCRIMINATION
In this section we consider the so-called unambiguous
discrimination of states [4], namely with no error, but
possibly with an inconclusive outcome of the measure-
ment. We focus attention on a set of N pure states
{ψi}i∈S. In such a case, it is possible to have unambigu-
ous discrimination only if the states of the set S are lin-
early independent, whence there exists a biorthogonal set
of vectors {|ωi〉}i∈S, with 〈ωi|ψj〉 = δij , ∀i, j ∈ S. We will
conveniently restrict our attention to Span{|ψi〉}i∈S ≡ H
(otherwise one can trivially complete the optimal POVM
for the subspace as a POVM for the full Hilbert space
of the quantum system). While in the Bayes problem
the probability of inconclusive outcome is minimized,
in the minimax unambiguous discrimination we need to
maximize mini〈ψi|Pi|ψi〉 over the set of POVM’s with
〈ψi|Pj |ψi〉 = 0 for i 6= j ∈ S, and the POVM element
that pertains to the inconclusive outcome will be given
by PN+1 = I −
∑
i∈S Pi. We have the following theorem.
Theorem 5 The optimal minimax unambiguous dis-
crimination of N pure states {ψi}i∈S is achieved by the
POVM
Pi =κ|ωi〉〈ωi|, i ∈ S ,
PN+1 =I −
∑
i∈S
Pi , (29)
where κ is given by
κ−1 = max eigenvalue of
∑
i∈S
|ωi〉〈ωi| . (30)
Proof. We need to maximize mini〈ψi|Pi|ψi〉 over the set
of POVM’s with 〈ψi|Pj |ψi〉 = 0 for i 6= j ∈ S, whence
clearly Pj = κj|ωj〉〈ωj |. Then the problem is to maxi-
mize mini∈S κi. This can be obtained by taking κi = κ

5independent of i and then maximizing κ. In fact, if there
is a κi > κj for some i, j, then we can replace κi with κj ,
and iteratively we get κi = κ independently of i. Finally,
the maximum κ giving PN+1 ≥ 0 is the one given in the
statement of the theorem. 
As regards the unicity of the optimal POVM, we can
show the following.
Theorem 6 The optimal POVM of Theorem 5 is non-
unique if and only if |ωi〉 ∈ Supp(PN+1) for some i ∈ S.
Proof. In fact, if there exists an i ∈ S such that |ωi〉 ∈
Supp(PN+1), this means that there exists ε > 0 such that
ε|ωi〉〈ωi| ≤ PN+1. Then the following is a POVM
Qj = Pj , for j 6= i
Qi = Pi + ε|ωi〉〈ωi|,
QN+1 = PN+1 − ε|ωi〉〈ωi|,
(31)
and is optimal as well. Conversely, if there exists another
equivalently optimal POVM {Qj}, then there exists an
i ∈ S such that Qi > Pi (since both are proportional to
|ωi〉〈ωi|, and mini〈ψi|Qi|ψi〉 has to be maximized). Then
|ωi〉 ∈ Supp(PN+1).
When the optimal POVM according to Theorem 6 is
not unique, one can refine the optimality criterion in the
following way. Define the set S1 ⊂ S for which one has
|ωi〉 ∈ Supp(PN+1). Denote by P1 the set of POVM’s
that are equivalently optimal to those of Theorem 5.
Then define the set of POVM’s P2 ⊂ P1 that maximizes
mini∈S1〈ωi|Pi|ωi〉. In this way one iteratively reach a
unique optimal POVM, which is just the one given in
Eqs. (29) and (30).
V. CONCLUSIONS
In conclusion, we have considered the problem of op-
timal discrimination of quantum states in the minimax
strategy. This corresponds to maximising the smallest
of the probabilities of correct detection over all measure-
ment schemes. We have derived the optimal measure-
ment both in the minimal-error and in the unambigu-
ous discrimination problem for any number of quantum
states. The relation between the optimal measurement
and the optimal Bayesian solutions has been given. Dif-
ferently from the Bayesian scenario, we have shown that
there are instances in which the minimum risk cannot be
achieved by an orthogonal measurement. Finally, in the
unambiguous discrimination problem, we have shown a
refinement of the minimax problem that leads always to
a unique optimal minimax measurement.
Acknowledgments
We thank G. Chiribella for correcting the original proof
of Theorem 3. Support from INFM through the project
PRA-2002-CLON, and from EC and MIUR through the
cosponsored ATESIT project IST-2000-29681 and Cofi-
nanziamento 2003 is acknowledged.
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