Discriminating States: The Quantum Chernoff Bound
K. M. R. Audenaert
Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, London SW7 2PG, United Kingdom
J. Calsamiglia, R. Mun
˜
oz-Tapia, and E. Bagan
Grup de Fı
´
sica Teo
`
rica, Universitat Auto
`
noma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Ll. Masanes
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
A. Acin
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
F. Verstraete
Fakulta
¨
tfu
¨
r Physik, Universita
¨
t Wien, Boltzmanngasse 5, 1090 Wien, Austria
(Received 18 October 2006; published 17 April 2007)
We consider the problem of discriminating two different quantum states in the setting of asymptotically
many copies, and determine the minimal probability of error. This leads to the identification of the
quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the
classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff
bound is the natural symmetric distance measure between quantum states because of its clear operational
meaning and because it does not seem to share some of the undesirable features of other distance
measures.
DOI: 10.1103/PhysRevLett.98.160501 PACS numbers: 03.67.a
One of the most basic tasks in information theory is the
discrimination of two different probability distributions:
given a source that outputs variables following one out of
two possible probability distributions, determine which
one it is with the minimal possible error. In a seminal
paper, Chernoff [1] solved this problem in the asymptotic
regime and showed that the probability of error P
e
in
discriminating two probability distributions decreases ex-
ponentially in the number of tests n that one can perform:
P
e
expn
CB
. The optimal exponent
CB
arising in
the asymptotic limit is called the Chernoff bound [2]. One
of the virtues of the Chernoff bound is that it yields a very
natural distance measure between probability distributions;
it is essentially the unique distance measure in the ubiq-
uitous situation of independent and identically distributed
(i.i.d.) random variables.
A quantum generalization of this result is highly desired.
Indeed, the concept of randomness is much more elemen-
tary in the field of quantum mechanics than in classical
physics. Given the large amount of experimental effort in
the context of quantum information processing to prepare
and measure quantum states, it is of fundamental impor-
tance to have a theory that allows one to discriminate
different quantum states. Despite considerable effort, this
quantum generalization of the Chernoff bound has until
now remained unsolved. The problem is to discriminate
two sources that output many identical copies of one out of
two different quantum states and , and the question is to
identify the exponent arising asymptotically when per-
forming the optimal test to discriminate them. This task
is so fundamental that it was probably the first problem
ever considered in the field of quantum information theory;
it was solved in the one-copy case more than 30 years ago
[3,4]. In this Letter, we finally identify the asymptotic error
exponent when the optimal strategy for discriminating the
states is used. A nice feature of such a result is its universal-
ity, as it identifies the unique metric quantifying the dis-
tance of quantum states in the i.i.d. setting. Note that the
related question of the optimal error exponents in asym-
metric hypothesis testing in the sense of Stein’s lemma was
already solved a long time ago [5], leading to the opera-
tional meaning of quantum relative entropy. The quantum
Chernoff bound can therefore be understood as the sym-
metric version of the quantum relative entropy.
Distance measures between quantum states have been
used in a wide variety of applications in quantum infor-
mation theory. The most popular such measure seems
to be Uhlmann’s fidelity [6], which happens to coincide
with the quantum Chernoff bound when one of the states is
pure. The trace distance has a more natural operational
meaning, but lacks monotonicity under taking tensor
powers of its arguments. The problem is that one can easily
find states , ,
0
,
0
such that Trj j < Trj
0
0
j
but Trj
2
2
j > Trj
02
02
j. The quantum
Chernoff bound exactly characterizes the exponent arising
in the asymptotic behavior of the trace distance in the case
of many identical copies, and therefore does not suffer
from this problem. Note that a similar situation happens
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in the case of one-copy entanglement versus the asymp-
totic entanglement entropy.
In this Letter, we give an upper bound for the probability
of error for discriminating two arbitrary states. In the
particular case of a large number of identical copies, this
result nicely complements the recent work of Nussbaum
and Szkoła [7], where a lower bound for the asymptotic
error exponent was found and hence a lower bound for the
probability of error. These respective upper and lower
bounds coincide in the asymptotic limit and hence give
the exact expression for the error exponent. The conjecture
of Ogawa and Hayashi concerning the quantum Chernoff
bound raised in [8] is thus solved.
Our Letter is organized as follows. After the mathemati-
cal formulation of the problem, we prove a nontrivial and
fundamental inequality relating the trace distance to the
quantum Chernoff bound. Finally, we discuss some inter-
esting properties of the quantum Chernoff bound.
The optimal error probability of discriminating two
quantum states
0
and
1
has been identified a long time
ago by Helstrom [3]. We consider the two hypotheses H
0
and H
1
that a given quantum system is prepared either in
the state
0
or in the state
1
, respectively. Since the
(quantum) Chernoff bound arises in a Bayesian setting,
we supply the prior probabilities
0
and
1
, which are
positive quantities summing up to 1 (the degenerate cases
0
0 or
1
0 are excluded).
Physically discriminating between these hypotheses cor-
responds to performing a generalized (POVM) measure-
ment on the quantum system with two outcomes, 0 and 1.
This POVM consists of the two elements fE
0
;E
1
g, where
E
0
E
1
1, E
i
0. The symmetric distinguishability
problem consists in finding those E
0
and E
1
that minimize
the total error probability P
e
, which is given by P
e
0
TrE
1
0
1
TrE
0
1
1
TrE
1
1
1
0
0
.
This problem can be solved using some basic linear alge-
bra. Let us first introduce some basic notations. Abusing
terminology, we will use the term ‘‘positive‘‘ for ‘‘positive
semidefinite‘‘ (denoted A 0). We employ the positive
semidefinite ordering throughout, A B iff A B 0.
The absolute value jAj is defined as jAj
:
A
A
1=2
. The
Jordan decomposition of a self-adjoint operator A is given
by A A
A
, where A
and A
are the positive and
negative part of A, respectively, and are defined by A
:
jAjA=2 and A
:
jAjA=2. Both parts are posi-
tive by definition, and A
A
0. The error probability P
e
has to be minimized over all operators E
1
that satisfy 0
E
1
1. The result is that E
1
has to be the projector on the
range of the positive part of (
1
1
0
0
), leading to
P
e;min
1
2
1 k
1
1
0
0
k
1
;
where kAk
1
TrjAj is the trace norm.
The basic problem to be solved now is to identify how
the error probability P
e
behaves in the asymptotic limit,
i.e., when one has to discriminate between the hypotheses
H
0
and H
1
corresponding to either n copies of
0
having
been produced or n copies of
1
. To do so, we need to study
the quantity P
e;min;n
:
1 k
1
n
1
0
n
0
k
1
=2, and
it turns out that it exponentially decreases in n:
P
e;min;n
expn
QCB
:
We will prove that the exponent
QCB
is given by the
following quantity, which can therefore be called the quan-
tum Chernoff bound:
QCB
lim
n!1
logP
e;min;n
n
(1)
log min
0s1
Tr
s
1s
: (2)
Note that the quantity Tr
s
1s
is well-defined and
guaranteed to be positive. As should be, this expression
for the quantum Chernoff bound reduces to the usual
definition of the classical Chernoff bound
CB
when
and commute: for classical distributions p
0
and p
1
,
CB
log min
0s1
X
i
p
0
i
s
p
1
i
1s
: (3)
The fact that
QCB
is lower bounded by the expression
on the right hand side of (2) was proven very recently in
[7]. The fact that this is also an upper bound can be inferred
from the following theorem:
Theorem 1.—Let A and B be positive operators, then for
all 0 s 1,
Tr A
s
B
1s
TrA B jA Bj=2: (4)
Indeed, let A
1
n
1
and B
0
n
0
, then the upper
bound trivially follows from the fact that the logarithm of
the left hand side of the inequality (4) becomes
log
s
0
1s
1
n logTr
s
0
1s
1
. Upon dividing by n
and taking the limit n !1, we obtain the quantum
Chernoff bound
QCB
, independently of the priors
0
,
1
(as long as the priors are not degenerate).
Note that it was already known that P
e;min;n
is upper
bounded by exp n logTr
1=2
1=2
([13], Lemma 3.2).
Inequality (4) is also very interesting from a purely
matrix analytic point of view, as it relates the trace norm
to a multiplicative quantity that is highly nontrivial and
very useful. Note that the optimal measurement to dis-
criminate the two sources enforces the use of joint mea-
surements. As pointed out by A. Harrow, the particular
permutational symmetry of N-copy states guarantees that
the optimal collective measurement can be implemented
efficiently (with a polynomial-size circuit) [10], and hence
that the minimum probability of error is achievable with
reasonable resources.
Let us now move on to prove Theorem 1. The proof that
we present here goes through in infinite dimensions.
The proof relies on the following Lemma:
Lemma 1. —Let A, B 0. Let 0 t 1, and let P be
the projector on the range of A B
. Then
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Tr PBA
t
B
t
0: (5)
Proof.—We exploit the integral representation [12]
a
t
sint
Z
1
0
dx
ax
t1
a x
;a 0; 0 t 1;
(6)
which can be extended to positive operators in the usual
way. Under the integral sign, A
t
B
t
contributes a factor
AA x
1
BB x
1
xfB x
1
A x
1
g:
We next use the obvious relations
1
b
1
a
Z
1
0
dy
d
dy
1
b a by
;
d
dy
1
c
1
c
dc
dy
1
c
;
which hold for arbitrary invertible operators a, b, and c.By
introducing the notation yAB and V B
x
1
, we can write A
t
B
t
1
sint
R
1
0
dxx
t
R
1
0
dyy
1
VV. Hence, to prove the lemma, we just need
to show that TrPBVV TrPV
1
xVV 0
for 0 x and 0 y 1.
Let have the Jordan decomposition
.
Thus P is the projector on the range of
. We choose the
basis in which is diagonal, and hence , P, and V can be
partitioned as
0
0
;P
10
00
;V
V
V
V
V
:
For ease of notation, the subscript ij will henceforth refer
to the (ij)-th block of an operator valued expression. We
can rewrite TrPBVV as
Tr PBVV Tr
V
xVV
:
By the fact that V
and V
are each other’s adjoint, the
latter expression is positive, which finally proves the state-
ment of the Lemma. 䊐
Proof of theorem.—We apply Lemma 1 to the case t
s=1 s, A a
1s
, and B b
1s
, where a, b are posi-
tive operators and 0 s 1=2. With P the projector on
the range of a
1s
b
1s
, this yields
Tr Pb
1s
a
s
b
s
0:
Subtracting both sides from TrPa b then yields
Tr a
s
Pa
1s
b
1s
TrPa b:
Since P is the projector on the range of the positive part of
(a
1s
b
1s
), the LHS can be rewritten as Tra
s
a
1s
b
1s
. Because a
s
0, this is lower bounded by
Tra
s
a
1s
b
1s
Tra a
s
b
1s
.
On the other hand, the RHS is upper bounded by Tra
b
; this is because for any self-adjoint H, TrH
is the
maximum of TrQH over all self-adjoint projectors Q.We
thus have
Tr a a
s
b
1s
Tra b
Tra bja bj=2:
Subtracting both sides from Tra finally yields (4) for 0
s 1=2. The remaining case 1=2 s 1 obviously fol-
lows by interchanging the roles of a and b. 䊐
In the remainder of this Letter, we discuss the main
properties of the nonlogarithmic variety of the quantum
Chernoff bound, which we denote here by Q;
:
min
0s1
Tr
s
1s
.
The following upper and lower bounds on Q in terms of
the trace norm distance T;
:
jj jj
1
=2 exist [9]:
1 Q T
1 Q
2
q
: (9)
Based on these bounds, the following properties of the
Q-quantity and the Chernoff bound can be derived:
Inverted measure.—The maximum value Q can attain is
1, and this is reached when . This follows, for
example, from the upper bound Q
2
T
2
1. The mini-
mal value is 0, and this is only attained for pairs of
orthogonal states, i.e., states such that 0. This im-
plies that the Chernoff bound is infinite if the states are
orthogonal; this has to be contrasted with the asymmetric
error exponents occurring in the context of relative entropy,
where infinite values are obtained whenever the states have
a different support.
Convexity in s.—The function to be minimized in Q is
s 哫 Tr
s
1s
. It is important to realize that this function
is convex in s 20; 1 because that means that the mini-
mization has only one local minimum, and therefore this
local minimum is automatically the global minimum. This
is an important benefit in actual calculations.
Indeed, the function s 哫 x
s
y
1s
is convex for positive
scalars x and y, as one easily confirms by calculating the
second derivative x
s
y
1s
logx logy
2
, which is non-
negative. Consider then a basis in which is diagonal
and given by Diag
1
;
2
; .... Let the eigenvalue
decomposition of (in that basis) be given by
UDiag
1
;
2
; ...U
, where U is a unitary. Then
Tr
s
1s
P
i;j
s
i
1s
j
jU
ij
j
2
. As this is a sum with
positive weights of convex terms
s
i
1s
j
, the sum itself
is also convex.
Joint concavity in (, ).—By Lieb’s theorem [11],
Tr
s
1s
is jointly concave in (, ). Since the quantum
Chernoff bound is the point-wise minimum of Tr
s
1s
(over a fixed set, namely, over s 20; 1), it is itself jointly
concave as well. The Chernoff bound is therefore jointly
convex, just like the relative entropy.
Monotonicity under CPT maps.—From the joint con-
cavity, one easily derives the following monotonicity prop-
erty: for any completely positive trace preserving (CPT)
map ,
Q; Q; : (10)
For a proof, see [13].
Continuity.—By the lower bound Q T 1, 1 Q is
continuous in the sense that states that are close in trace
norm distance are also close in 1 Q distance: 0 1
Q T.
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Relation to fidelity.—If one of the states is pure, then Q
equals the Uhlmann fidelity. Indeed, assume that
1
j ih j is pure, then the minimum of the expression
Tr
s
1
1s
2
is obtained for s 0 and reduces to
h j
2
j i. From inequality (9), the fidelity is always an
upper bound to Q.
Relation to the relative entropy.—Just as in the classical
case, there is a nice connection between the quantum
relative entropy and the Chernoff bound. By differentiating
the expression Tr
s
1s
with relation to s, one observes
that the minimum (which is unique due to convexity) is
obtained when
Tr
s
1s
logTr
s
1s
log:
One easily verifies that this is equivalent to the condition
that
S
s
kS
s
k
with SAkB the quantum relative entropy TrA logA
A logB and
s
defined as
s
s
1s
Tr
s
1s
: (11)
Note that
s
is not a state because it is not even self-adjoint
(except in the commuting case). Nevertheless, as it is
basically the product of two positive operators, it has
positive spectrum, and its entropy and the relative entropies
used in (11) are well-defined. The value of s for which both
relative entropies coincide is the optimal value s
. This
s
can be considered the quantum generalization of the
Hellinger arc and interpolates between two different quan-
tum states, albeit in a rather special (unphysical) way.
Metric.—The quantum Chernoff bound (or its nonlogar-
ithmic variety) between two infinitesimally close states
and d induces a monotone metric that gives a geo-
metrical structure to the state space [14]. The metric is
given by [9]
ds
2
1 min
0s1
Tr
s
d
1s
1
2
X
ij
jhijdjjij
2
i
p
j
p
2
where
P
i
i
jiihij is the eigenvalue decomposition of
.
In conclusion, we have identified the exact expression of
the quantum generalization of the Chernoff bound, which
allows us to quantify the asymptotic behavior of the error
in the context of Bayesian discrimination of different
sources of quantum states. This resolves a long-standing
open question. Our main theorem (Theorem 1), which
gives a computable lower bound to the trace norm differ-
ence of two states in the many-copy regime, may also find
other relevant applications in and outside the field of state
discrimination [15].
F. V. and K. A. thank the hospitality of the Max Planck
Institute for Quantum Optics where part of this work was
done. K. A. was supported by The Leverhulme Trust (Grant
No. F/07 058/U), by the QIP-IRC (www.qipirc.org) sup-
ported by EPSRC (No. GR/S82176/0), by EU Integrated
Project QAP, and by the Institute of Mathematical
Sciences, Imperial College London. We acknowledge fi-
nancial support from CIRIT, Project No. SGR-00185, and
from the Spanish MEC under Ramo
´
n y Cajal program
(A. A. and J. C.); Projects Nos. FIS2004-05639-C02-02
and FIS2005-01369; and Consolider-Ingenio 2010,
Project ‘‘QOIT.’’ We are grateful to Montserrat Casas,
Juli Ce
´
spedes, Alex Monra
`
s, Sandu Popescu, and
Andreas Winter for discussions, and to A. Harrow for
pointing out that the optimal measurement can be imple-
mented efficiently.
[1] H. Chernoff, Ann. Math. Stat. 23, 493 (1952).
[2] The quantity which in this paper is referred to as the
‘‘Chernoff bound’’ also goes under the alternative names
of Chernoff distance, Chernoff divergence, and Chernoff
information. While the term ‘‘Chernoff bound’’ is also
used for a bound on tail probabilities for statistical dis-
tributions (of which, actually, the Chernoff information is
a derived quantity), we decided, against common sense, to
follow common terminology.
[3] C. W. Helstrom, Quantum Detection and Estimation
Theory (Academic Press, New York, 1976).
[4] A. S. Holevo, Theory Probab. Appl. 23, 411 (1979).
[5] F. Hiai and D. Petz, Commun. Math. Phys. 143, 99 (1991);
T. Ogawa and H. Nagaoka, IEEE Trans. Inf. Theory 46,
2428 (2000); T. Ogawa and H. Nagaoka, IEEE Trans. Inf.
Theory 46, 2428 (2000).
[6] A. Uhlmann, Rep. Math. Phys. 9, 273 (1976).
[7] M. Nussbaum and A. Szkoła, quant-ph/0607216.
[8] T. Ogawa and M. Hayashi, IEEE Trans. Inf. Theory 50,
1368 (2004).
[9] K. Audenaert et al., quant-ph/0610027.
[10] D. Bacon, I. Chuang, and A. Harrow, Phys. Rev. Lett. 97,
170502 (2006).
[11] E. H. Lieb, Advances in Mathematics 11, 267 (1973).
[12] R. Bhatia, Matrix Analysis (Springer, Berlin, 1997).
[13] M. Hayashi, Quantum Information: An Introduction
(Springer Verlag, Berlin, 2006).
[14] D. Petz, Linear Algebra Appl. 244, 81 (1996).
[15] And, indeed, after the appearance of the first draft of the
manuscript, our main inequality has been used by Hayashi
to prove achievability of the quantum Hoeffding bound;
see M. Hayashi, quant-ph/0611013.
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