Accessible information and optimal strategies for real symmetrical quantum sources

16 Pages·0.24 MB·English

by  Masahide Sasaki

Accessible information and optimal strategies
for real symmetrical quantum sources
Masahide Sasaki1, Stephen M. Barnett2, Richard Jozsa3, Masao Osaki4 and Osamu Hirota4
1Communications Research Laboratory, Ministry of Posts and Telecommunications
Koganei, Tokyo 184-8795, Japan
2Department of Physics and Applied Physics, University of Strathclyde,
Glasgow G4 0NG, Scotland
3School of Mathematics and Statistics, University of Plymouth,
Plymouth, Devon PL4 8AA, England
4Research Center for Quantum Communications, Tamagawa University
Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan
(15 Dec 1998)
9
9 WestudytheproblemofoptimizingtheShannonmutualinformationforsourcesofrealquantum
9 states i.e. sources for which there is a basis in which all the states have only real components. We
1 consider in detail the sources EM of M equiprobable qubit states lying symmetrically around the
great circle of real states on the Bloch sphere and give a variety of explicit optimal strategies.
n
We also consider general real group-covariant sources for which the group acts irreducibly on the
a
J subsetofallrealstatesandprovetheexistenceofarealgroup-covariantoptimalstrategy,extending
8 a theorem of Davies (E. B. Davies, IEEE. Inf. Theory IT-24, 596 (1978)). Finally we propose
1 an optical scheme to implement our optimal strategies, simple enough to be realized with present
technology.

3
v PACS numbers:03.65.Bz, 89.70.+c, 42.79.Sz, 89.80.+h, 32.80.-t
2
6
0
2 I. INTRODUCTION
1
8
9 Therearetwoprincipalmeasuresofqualityinthequantumdetectionproblemforagivenfinitenumberofquantum
/ stateswithfixedpriorprobabilities. OneistheminimizationofaspecifiedBayescost,andthe otheris themaximiza-
h
tion of the Shannon mutual information [1,2,3]. The former is useful if one has to reach a decision after performing
p
a single quantum measurement whereas the latter is more relevant for the problem of transmitting as much classical
-
t information as possible using the given ensemble of states. In this paper we will consider the problem of maximizing
n
the Shannon mutual information for a certain class of quantum ensembles.
a
u In a general communication setting, let xi X be input letters and let ξi be their prior probabilities. Let us
{ ∈ } { }
q denote output letters by y Y . Both the Bayes cost and the Shannon mutual information are defined in terms of
j
: the conditional probabilit{y P∈(j i)}of obtaining output y provided that the letter sent was x . The former is defined
v j i
|
i as
X
r B(X :Y)= C ξ P(j i), (1)
ij i
a |
ij
X
for a Bayes cost matrix [C ], while the latter is defined as
ij
P(j i)
I(X :Y)= ξ P(j i)log | . (2)
i
| ξ P(j k)
Xi Xj k |
k
X
(Since all the results in this paper are valid for any logarithm base, we shall specify the base only where necessary.)
In classicalinformationtheory, the channelmatrix [P(j i)] is givenand fixed, characterisingthe noise in the channel.
|
In contrast, in a quantum information theoretic context where signal carriers are to be quantum states transmitted
without noise, the channel matrix generally becomes a variable. This is because the act of quantum detection itself
generally has a probabilistic output so the channel matrix is dependent on the choice of quantum detection strategy.
More precisely, the input letters correspond to a set of positive trace class operators of trace one ρˆ on a Hilbert
i
{ }
space . A quantum detection strategy is described by a positive operator-valued measure (POVM) on . A
s s
H H
POVM is any set πˆ of hermitian positive operators forming a resolution of the identity:
j
{ }
1
πˆj† =πˆj, πˆj ≥0 ∀j, πˆj =Iˆ. (3)
j
X
The detection operator πˆ corresponds to the output letter y and the conditional probabilities are given by
j j
P(j i) = Tr(πˆ ρˆ).
j i
|
Thus in the quantum context the optimization of I(X : Y) is carried out with respect to the choice of POVM πˆ
j
{ }
for fixed ensemble = ρˆ;ξ (i.e. with fixed letter states ρˆ and fixed prior probabilities ξ ). The maximum value
i i i i
E { }
of I(X :Y) is called the accessible information of the ensemble .
E
The set of all POVM’s is a convex set and I(X :Y) enjoys the following fundamental property:
P
(CONV): For a fixed ensemble = ρˆ;ξ , I(X :Y) is a convex function on .
i i
E { } P
A proof of (CONV) is given in theorem 2.7.4 of [4]. Let I( : ) denote the mutual information obtained from the
E A
POVM applied to the ensemble . Then if is a convex combination of POVMs :
i
A E A A
= p + p .
1 1 n n
A A ··· A
it follows from (CONV) that
I( : ) p I( : ) maxI( : ). (4)
i i i
E A ≤ E A ≤ i E A
i
X
The BayescostB(X :Y) is anaffine concavefunction onthe convexset . Therefore the Bayescostminimization
P
problemisakindoflinearprogrammingproblemandisexpectedtohaveauniquesolution. Anecessaryandsufficient
condition for specifying the optimum solution is known [1,2]. On the other hand, the Shannon mutual information
I(X :Y) is a nonlinear and convex function on . The maximization of this quantity is a much harder problem and
P
only a necessary condition for the optimum is known [1]. Thus the maximization of I(X : Y) with respect to the
detection strategy πˆ is a basic and open problem in quantum information theory.
j
{ }
In this problem, the number of outputs is not necessarily the same as the number of the inputs. The optimum
solutionisnotnecessarilyuniqueeither. Howeveritis knownthattheremustbeatleastoneoptimumsolutionwhich
corresponds to an extreme point of the convex set . This is due to the convexity of the function I(X :Y). Such an
P
extreme point is a set of rank one elements, which means that each πˆ has the form κ v v where v is a pure state
j
and 0 κ 1. The number of elements, N, can be bounded by d N d2 where d |isithhe|dimens|ioin of the Hilbert
≤ ≤ ≤ ≤
space of which the input state ensemble ρˆ is made [5]. I(X : Y) is also possibly maximized at some interior
s i
pointsHof as well. In that case the number o{f o}utcomes may exceed d2. Explicit examples of optimalsolutions have
P
been given for binary ensembles [6,7,8] and for the ensemble of four qubit states with tetrahedral symmetry [5]. The
latter is a specific example of a general result of Davies [5] characterising the form of an optimal strategy for any
symmetrical ensemble whose symmetry group acts irreducibly on the whole state space.
In this paper we will study the accessible information and corresponding optimal strategies for an ensemble of
M
E
M qubit states with symmetry group Z , the group of integers modulo M. Some of our results will also apply to
M
1 0
moregeneralensembles. maybe explicitlydescribedasfollows. Let , be the z-spineigenstatesand
EM { 0 1 }
(cid:18) (cid:19) (cid:18) (cid:19)
1
write ψ = . Let
| 0i 0
(cid:18) (cid:19)
π cos π sin π
Vˆ ≡exp(−iMσˆy)= sin Mπ −cos πM . (5)
(cid:18) M M (cid:19)
Then consists of the M states
M
E
coskπ
ψ =Vˆk ψ = M , k =0,...,M 1, (6)
| ki | 0i sinkπ −
(cid:18) M (cid:19)
takenwithequalpriorprobabilitiesξ = 1 . Notethatthesestates(inthez-spinbasis)involveonlyrealcomponents.
k M
On the Bloch sphere they are equally spaced around a great circle C in the x z plane consisting of all real states.
−
The antipodal points which have C as equator, are the two σˆ eigenstates. Thus is clearly symmetrical with
y M
respect to the group Z whose generator is represented by 2π rotation about the axiEs joining the σˆ eigenstates. At
M M y
the Hilbert space levelthe operatorsVˆk in Eq. (5) provide a projectiveunitary representationof Z (e.g. VˆM = I
M
−
and c.f. Eq. (7) later).
2
Thissymmetrygroupdoesnotactirreduciblyonthe wholestatespace. Indeedthe σˆ eigenstatesareleftinvariant
y
by the group action. (Irreducibility on the whole state space requires that the only invariant point is the maximally
mixed state 1Iˆ.) Hence we cannot apply Davies’ theorem [5] to provide an optimal strategy for . Nevertheless we
2 EM
willprovethatthe conclusionof Davies’theoremremainstrue in this casei.e. thatthere exists a pure state a such
0
| i
that the Z -symmetric POVM
M
2
= a a : k = 0,...,M 1 where a =Vˆk a ,
M k k k 0
A {M | ih | − } | i | i
is an optimal strategy for . Furthermore we will show that a may be taken to be the state orthogonal to ψ .
M 0 0
E | i | i
The case M = 3 is of particular interest. It is the so-called trine ensemble which has been much studied [9,10,11].
Holevo in 1973 [9] showed that no von Neumann measurement in can be an optimal strategy, demonstrating the
2
H
necessity of considering more general POVMs in quantum detection theory. Since that time it has been conjectured
that the strategy above is optimal for the trine source. Our results resolve this conjecture affirmatively.
3
A
The strategy has M elements. However, as noted above, for ensembles in d=2 dimensions there is always an
M
A
optimal strategy with at most d2 = 4 elements (which does not increase with M). We will show that the ensembles
alwayshave an optimal strategy with at most 3 elements and explicit strategiesof this form will be described for
M
Eall M. If M is even then consists of M pairs of orthogonal states. Let ξ , η be any one of these pairs. We
EM 2 {| i | i}
will show that the two-element POVM ξ ξ , η η (a regular von Neumann measurement) is always an optimal
{| ih | | ih |}
strategy when M is even. We will also describe further optimal K-element POVMs where K lies between 3 and M.
II. A GROUP-THEORETIC APPROACH
We begin by setting up a group-theoretic formalism for symmetric ensembles, leading to a main result (theorem
1) which applies to symmetric ensembles of real states in any dimension d 2. An essential requirement in many of
≥
our results will be that various states andunitary operatorsbe real. The requirementthat a state oroperatorbe real
has of course, no intrinsic physical meaning. When we speak of real states and real operators we will always mean
simply that there exists a basis of the Hilbert space relative to which all the required objects simultaneously have real
components or real matrix elements.
A projective unitary representation of a group G is an assignment of a unitary operation Uˆ to each member of G
g
satisfying
Uˆ Uˆ =eiφ(g1,g2)Uˆ , (7)
g1 g2 g1g2
where the phases φ(g ,g ) may be chosen arbitrarily. A finite ensemble of equiprobable (generally mixed) states is
1 2
E
said to be symmetric with respect to the group G, or G-covariant, if the following condition is satisfied: there is a
projective unitary representation Uˆ of G such that for all g, Uˆ ρˆUˆ is in whenever ρˆis in . We write
{ g} g g† E E
gρˆ=Uˆ ρˆUˆ , (8)
g g†
for the action of g on the state ρˆ. The phases φ(g ,g ) do not appear in Eq. (8) and g (g (ρˆ)) = (g g )(ρˆ). Note
1 2 1 2 1 2
that, in contrast to Davies [5] we do not require that G parameterises i.e. G need not act transitively on the set of
E
states of . For example, is Z -covariant and the action is transitive, but is also Z - and Z -covariant via
M M 2N 2 N
E E E
non-transitive actions.
A G-covariant POVM (for the projective unitary representation Uˆ ) is a POVM such that Uˆ AˆUˆ is in
A { g} g g† A
whenever Aˆ is in . We write
A
gAˆ Uˆ AˆUˆ , (9)
≡ g g†
for the action of g on a POVM element Aˆ. From Eqs. (8) and (9) we see that Tr(Aˆρˆ) = Tr(gAˆ gρˆ) i.e. the
·
probability of outcome Aˆ on state ρˆis G-invariant. Hence
Tr(gAˆ ρˆ)= Tr(Aˆ g 1ρˆ), (10)
−
· ·
so that the set of probabilities of the G-shifted outputs gAˆ on a fixed input ρˆare obtained as a permutation of the
set of probabilities of the unshifted output Aˆ acting on suitably shifted inputs.
3
Let be a G-covariant ensemble with projective unitary representation Uˆ . We aim to find conditions on Uˆ
g g
E { } { }
whichwill guaranteethe existence of aG-covariantPOVM = Aˆ :g G with elements parameterisedby G, and
g
A { ∈ }
having group action gAˆ =Aˆ . Thus if e is the identity of G we have
h gh
Aˆ =Uˆ Aˆ Uˆ , (11)
g g e g†
and we require
Mˆ Aˆ =Iˆ. (12)
g
≡
g G
X∈
(Later we will take the elements of to be rank1 and considerthe question ofwhen is an optimalstrategyfor .)
A A E
From Eq. (11) we see that Mˆ commutes with all the Uˆ ’s:
g
Uˆ Mˆ =MˆUˆ . (13)
g g
Thus if the set Uˆ acts irreducibly on the state space (i.e. there is no proper invariant subspace) the Schur’s
g
{ }
lemma will guarantee that Eq. (12) holds. This fact is used by Davies [5] to characterise an optimal strategy for any
G-covariantensemblewhosesymmetrygroupactsirreduciblyonthe wholestatespace. Howeverthisconditionoffull
irreducibility on the whole state space is not necessary for Eq. (12) to hold. We will use the following more general
form of Schur’s lemma:
Lemma 1: Let Mˆ be any set of non-singular d by d matrices over some field F which acts irreducibly on the
g
{ }
vector space V = Fd (i.e. there is no proper subspace mapped to itself by all the Mˆ ’s). Suppose that Kˆ is any
g
matrix that commutes with all the Mˆ ’s:
g
KˆMˆ =Mˆ Kˆ. (14)
g g
Then:
(a) either Kˆ =0 or Kˆ is non-singular,
(b) If Kˆ has a non-zero eigenvalue λ in F, then Kˆ =λIˆ.
Proof: (a) Let Kˆ(V) denote the image of V under the map Kˆ and similarly for Mˆ (V). Since Mˆ is non-singular
g g
we have Mˆ (V)=V. By Eq. (14) we have Mˆ Kˆ(V)=KˆMˆ (V)=Kˆ(V) i.e. Kˆ(V) is an invariant subspace. Hence
g g g
either Kˆ(V)=0 (in which case Kˆ =0) or else Kˆ(V)=V (in which case Kˆ is non-singular).
(b) If Kˆ has eigenvalue λ in F then Bˆ = Kˆ λIˆis singular. Also BˆMˆ = Mˆ Bˆ for all g. Hence by (a), Bˆ must be
g g
−
zero i.e. Kˆ =λIˆ.
We will apply this lemma with F = IR to obtain useful results about G-covariant ensembles of real states whose
group G acts irreducibly only on the restricted set IRd of real states (but not necessarily irreducibly on the full state
space). This is the case for our ensembles . Let G denote the size of G and let d= TrIˆbe the dimension of the
M
E | |
Hilbert space.
Lemma 2: Suppose that Uˆ is a projective unitary representation of G such that Uˆ are all real matrices and
g g
{ }
Uˆ acts irreducibly on IRd. Let v IRd be any real state. Write
g
{ } | i∈
d
Aˆ = Uˆ v v Uˆ .
g G g| ih | g†
| |
Then Aˆ :g G is a G-covariantPOVM i.e. Aˆ =Iˆ.
{ g ∈ } g G g
Proof: Let Mˆ = Aˆ . Then Mˆ is a real m∈atrix and MˆUˆ = Uˆ Mˆ for all g G. Also Mˆ is a hermitian
g G g P g g ∈
positive matrix (being a∈sum of projectors with positive coefficients) so it has a real positive eigenvalue λ > 0. By
P
the previous lemma, Mˆ =λIˆ. Since TrAˆ = d for all g, we get TrMˆ =d= TrIˆso λ=1.
g G
Theorem 1: Let beanyensembleofequip|r|obablerealstatesindimensiond. Supposethat isG-covariantwith
respect to a projectiEve unitary representation Uˆ of real matrices which acts irreducibly on IREd. Then there exists
g
{ }
a real pure state v such that the G-covariant POVM = Dˆ :g G defined by
g
| i D { ∈ }
d
Dˆ = Uˆ v v Uˆ ,
g G g| ih | g†
| |
is an optimal strategy for .
E
4
Proof: Wewillworkinthebasiswithrespecttowhichthestatesof andthematricesUˆ haverealentries. Let =
g
E A
Aˆ ,...,Aˆ beanyoptimalPOVMfor . Wewilltransmogrify intotherequiredformwhilepreservingoptimality.
1 n
{ } E A
First strip off all imaginary parts of the entries of the matrices Aˆ . Let A˜ = Re(Aˆ ) and ˜= A˜ ,...,A˜ . Then
k k k 1 n
A { }
˜is again a POVM and has real symmetric matrices as elements. (To see that A˜ is a positive matrix note that A
k k
A
positive implies that the complex conjugateAˆ is positive so A˜ = 1(Aˆ +Aˆ ) must be positive. Also Aˆ =Iˆand
∗k k 2 k ∗k k
Iˆis real so A˜ =I too.) Next note that TrAˆ ρˆ= TrA˜ ρˆfor any real state ρˆ(since Im(Aˆ ) is antisymmetric) so
k k k k P
˜remains an optimal strategy.
AIn generaPl ˜will nothave rank1 elements evenif hadrank 1 elements. Thus decomposeeachA˜ into its rank 1
k
A A
eigenprojectors(multiplied by the corresponding eigenvalues) which are necessarily real as the eigenvalues/vectorsof
any real symmetric matrix are real. Then form the larger POVM = Bˆ ,...,Bˆ comprising all the scaled rank 1
1 m
B { }
eigenprojectorsabove. Such a refinement of a POVM can never decrease the mutual information so with real rank
B
1 elements, is still optimal.
Now look at
1
Cˆ = gBˆ for g G and k =1,...,m. (15)
kg k
G ∈
| |
Note that Cˆ =Iˆsince Bˆ =Iˆand gIˆ=Iˆfor all g. Let = Cˆ be the corresponding POVM with Gm
kg kg k C { kg} | |
elements. Thus is G-covariant but the action of G is not transitive. We finally aim to cut down to a smaller
P C P C
optimal G-covariant POVM with elements labeled by G.
Let I( : ) denote the mutual information obtained from any POVM applied to any ensemble . First we
E A A E
show that I( : ) = I( : ) so that remains optimal. Let us label the inputs by i and denote conditional
E C E B C ∈ I
probabilities for by P(kg i). Denote the conditional probabilities for by P (k i) and let ξ be the constant prior
B
C | B |
input probability. Then
1
P(kg i) = TrCˆ ρˆ = TrgBˆ ρˆ.
kg i k i
| G ·
| |
According to Eq. (10), for each fixed g and k the resulting set of probabilities labelled by i , will be just a
permutation of the set P (k i), rescaled by 1 . Thus ∈ I
B | G
| |
1
ξP(kg l) = ξP (k l),
B
| G |
l | | l
X X
will be independent of g and also
P(kg i) 1 P (k i)
B
P(kg i)log | = P (k i)log | ,
B
| ξ P(kg l) G | ξP (k l)
i l | | | i l B |
X X
P P
will be independent of g. The mutual informations I( : ) and I( : ) are given by (c.f. Eq. (2)):
E C E B
P(kg i)
I( : ) = ξ P(kg i)log | ,
E C | ξ P(kg l)
i kg l |
X X
P
P (k i)
B
I( : ) = ξ P (k i)log | .
B
E B | ξ P (k l)
i k l B |
X X
P
On substituting the above G-invariant expressions into I( : ) we readily get I( : )=I( : ). (Our argument is
E C E C E B
actually an explicit example of the claim in lemma 5 of [5]). Hence remains optimal.
C
Finally note that for each i, Bˆ /(TrBˆ ) is a real pure state so by lemma 2,
i i
d gBˆ
i
= :g G ,
Di (|G| TrBˆi ∈ )
is a POVM for each i. Now TrBˆi = 1 gBˆ :g G so is a convex combination
d Di { G i ∈ } C
| |
5
m TrBˆ
i
= .
i
C d D
i=1
X
Hence by Eq. (4)
I( : ) maxI( : ).
i
E C ≤ i E D
Since wasoptimalit followsthat atleastoneof the ’s is optimal. This givesanoptimalG-covariantPOVMwith
i
C D
real rank 1 elements, parameterised by G, completing the proof.
III. OPTIMAL STRATEGIES FOR EM
We now return to the Z -covariantensemble in 2 dimensions, comprising the states
M M
E
coskπ
ψ = M , k = 0,...,M 1,
| ki sinkπ −
(cid:18) M (cid:19)
with equal prior probabilities 1 . According to theorem 1, there must exist an optimal Z -covariant POVM =
M M A
Aˆ ,...,Aˆ with M real rank 1 elements. The elements will have the form Aˆ = a a with
0 M 1 j j j
{ − } | ih |
2 cos(θ+ jπ)
a =Vˆj a = M , j =0,...,M 1, (16)
| ji | 0i M sin(θ+ jπ) −
r (cid:18) M (cid:19)
andVˆ is giveninEq. (5). The conditionalprobabilitiesp(j k)= a ψ 2 maybe readily computedand after some
j k
| |h | i|
rearrangementwe obtain the mutual information I(θ) explicitly as
M 1
1 − 2kπ 2kπ
I(θ)= (1+cos(2θ ))log(1+cos(2θ )). (17)
M − M − M
k=0
X
In this section, the base of the logarithm is taken as e. (For this base the numerical value of Eq. (17) is the amount
of information in nats rather than bits.) From the symmetry, I(θ) is a periodic function with period π . Figure 1
M
shows numerical plots of I(θ) for M =2,3,4 and 5 and illustrates the following basic property:
Lemma 3: For each M, I(θ) has a global maximum at θ = π.
2
Proof: Since cos(2θ 2kπ) <1, I(θ) can be expanded by using the formula
| − M |
∞ ( 1)n
(1+x)ln(1+x)=x+ − xn, x <1, (18)
n(n 1) | |
n=2 −
X
We get
1 M−1 2kπ ∞ ( 1)n 2kπ
I(θ)= cos(2θ )+ − cosn(2θ )
M − M n(n 1) − M
Xk=0 h nX=2 − i
1 ∞ ( 1)n M−1 2kπ
= − cosn(2θ ).
M n(n 1) − M
n=2 − k=0
X X
since Mk=−01cos(2θ − 2Mkπ) = 0. Next we separate out the even and odd parts of the series and replace powers of
cosines by multiple angle cosines to get:
P
1 ∞ ( 1)2n M−1 2kπ
I(θ)= − cos2n(2θ )
M 2n(2n 1) − M
n=1 − k=0
X X
1 ∞ ( 1)2n+1 M−1 2kπ
+ − cos2n+1(2θ )
M (2n+1)2n − M
n=1 k=0
X X
6
1 ∞ ( 1)2n M−1 1 1 2n n−1 2n 2kπ
= − + cos (2n 2l)(2θ )
M 2n(2n 1) 22n 1 2 n l − − M
nX=1 − Xk=0 − n (cid:18) (cid:19) Xl=0(cid:18) (cid:19) (cid:20) (cid:21)o
1 ∞ ( 1)2n+1 M−1 1 n 2n+1 2kπ
+ − cos (2n+1 2l)(2θ ) , (19)
M (2n+1)2n 22n l − − M
nX=1 Xk=0 nXl=0(cid:18) (cid:19) (cid:20) (cid:21)o
Then recall that
M 1
− 2kπ M for L/M =q (integer)
cos L = . (20)
M 0 for L/M = integer
k=0 (cid:18) (cid:19) (cid:26) 6
X
Applying this to Eq. (19) with L=2n 2l and L=2n+1 2l in the even and odd series, we get:
− −
∞ ( 1)2n 1 2n n−1 ∞ 2n
I(θ)= − + cos(2θqM)δ
nX=12n(2n−1)22n−1h2(cid:18) n (cid:19) Xl=0Xq=0(cid:18) l (cid:19) 2n−2l,qMi
∞ ( 1)2n+1 n ∞ 2n+1
+ − cos(2θqM)δ
(2n+1)2n22n l 2n+1−2l,qM
nX=1 hXl=0Xq=0(cid:18) (cid:19) i
∞ 1 2n ∞
= + f(qM)( 1)qMcos(2θqM), (21)
2n(2n 1)22n n −
n=1 − (cid:18) (cid:19) q=0
X X
where
2l+qM
∞ n−1 l
f(qM)= (cid:18) (cid:19) (δ +δ ). (22)
n=1l=0 (2l+qM)(2l+qM −1)22l+qM−1 2n−2l,qM 2n+1−2l,qM
XX
Since f(qM)>0, I(θ) is maximized when ( 1)qMcos(2θqM)=1, that is, θ = π for all M.
− 2
M =2
] 0.6
s M =3
t
a
n M =4
0.4
[

)
0.2

θ
M =5
(
I
0
π π
0 /2
θ
FIG.1. The Shannon mutualinformation I(θ) in natsversus theoptimization parameter θ for M =2, 3, 4 and 5.
Hence in general an optimal strategy for consists of choosing a real rank 1 POVM with elements Aˆ lying in
M k
directions orthogonal to the input states ψE . This POVM will be denoted by . The output Aˆ signifies with
k M k
| i A
certainty that the input was not ψ but leaves a residual uncertainty in the remaining signal states.
k
| i
For a given ensemble the optimal strategy is not unique and in practice it may be of interest to find optimal
E
POVMs with the minimum number of elements. The G-covariant optimal POVM above has M elements and we
note here some ways of reducing this number using the group theoretic approach. In the next section, by different
methods, we will show that 3 elements always suffice for any real qubit source, and develop corresponding strategies
for the ’s.
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Lemma 4: Supposethatk =1divides M exactly. Thenthereis aZ -covariantoptimalPOVMfor withk real
k M
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rank 1 elements.
Proof: Since k divides M,Z hasasubgroupisomorphicto Z andso is Z -covariant. Since k =1,the action
M k M k
of Z contains a non-trivial rotation so it acts irreducibly on IR2. ThusEtheorem 1 immediately give6 s the required
k
result.
Remark: Lemma 4 may also be obtained by a convexity argumentas follows. We will illustrate the idea with the
specific example of M =15 and k =3. The general case is a straightforwardgeneralisation. Z = 0,1,...,14 has
15
{ }
the subgroup 0,5,10 isomorphic to Z . Let = Aˆ ,Aˆ ,...,Aˆ be the optimal strategy given by theorem 1
3 15 0 1 14
{ } A { }
and lemma 3, with the direction of Aˆ being orthogonal to the kth state of . According to lemma 2, the three
k 15
directions 0,5,10 corresponding to the subgroup, may be used to define a POVEM. We just need to rescale Aˆ ,Aˆ and
0 5
Aˆ so that they add upto Iˆ. The scalingfactor is M =5. Thus = 5Aˆ ,5Aˆ ,5Aˆ is a POVM.NowIˆis always
10 k B0 { 0 5 10}
G invariant so we can apply the group elements l=1,2,3 and 4 of Z to to obtain POVMs
15 0
B
l = 5Aˆ,5Aˆ ,5Aˆ for l=0,1,2,3,4.
l 0 l l+5 l+10
B ≡ B { }
Note that the ’s have elements parameterised by the cosets of Z in Z . Also by symmetry of the construction,
l 3 15
B
I( : ) is independent of l. Furthermore is a uniform convex combination of the ’s
15 l 15 l
E B A B
4
1
= ,
15 l
A 5B
l=0
X
so by Eq. (4):
I( : ) maxI( : ).
15 15 15 l
E A ≤ l E B
Since was optimal we see that is optimal for each l. This gives the result of lemma 4 and also identifies the
15 l
A B
directions of the k element POVM as being any chosen symmetrical set of k directions orthogonal to corresponding
states of .
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An immediate special case is:
Corollary: If M is even then is made up of M pairs of orthogonal states. The von Neumann measurement
EM 2
defined by any one of these orthogonal pairs is an optimal strategy for .
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Thus if M is composite we can significantly reduce the number of elements in our optimal strategy but if M is
prime then this number remains large. In the next section we give a different approach to reducing the number of
elements, showing that just 3 elements always suffices for any ensemble of real qubit states.
IV. OPTIMAL POVMS WITH 3 ELEMENTS
Davies[5]hasshownthatanyensembleinddimensionshasanoptimalstrategywithN elementswhered N d2.
≤ ≤
Thisisdirectlybasedon(CONV),thatis,I(X :Y)isaconvexfunctionontheconvexset ofallPOVMs. Because
P
of this, I(X : Y) will always take its maximum value at an extreme point of the convex set (and also possibly at
P
some interior points as well). Each extreme point of consists of N rank 1 elements bounded by d N d2. If we
P ≤ ≤
restrict attention to only real ensembles then this upper bound on N can be improved as follows [12].
Lemma 5: Let be any ensemble of real states in d dimensions. Then the Shannon mutual information can be
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maximized by a POVM with N elements where d N d(d+1)/2.
≤ ≤
Proof: The proof proceeds along the same lines as the original one in ref. [5] with a slight replacement. For any
POVM πˆ write πˆ =µ dπ¯ where Trπ¯ =1 so
j j j j j
{ }
µ π¯ =Iˆ/d, µ =1. (23)
j j j
j j
X X
Let be the (compact convex) set of all positive hermitian operators with trace 1 (such as the π¯ ’s). Since I( : )
j
X E A
is a convex function on the set of all POVMs its maximum is attained at an extreme point of . The essential
P P
point of the original proof in ref. [5] is that every extreme point of has D+1 rank 1 elements where D is the real
dimension of . In the case of general ensembles D = d2 1. In oPur case of real ensembles the members of and
X − X
can be restricted to real matrices so comprises real symmetric trace 1 matrices and D = d(d+1) 1. Hence the
P X 2 −
extreme points of have d(d+1) elements.
P ≤ 2
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Thusforthe realensembles withd=2,POVMswiththreerealelementssuffice toprovideanoptimalstrategy.
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To describe such a POVM, we first introduce the three real (un-normalised) vectors
1
ω =c , (24a)
| 0i 0
(cid:18) (cid:19)
cosϕ
ω =a a , (24b)
| 1i sinϕa
(cid:18) (cid:19)
cosϕ
ω =b b , (24c)
| 2i sinϕb
(cid:18) (cid:19)
where the first vector lies along the first basis direction and the remaining two are in general position. Imposing the
condition ω ω =Iˆwe get
j| jih j|
P c= 2 a2 b2, (25a)
− −
cosϕ
a2 =p b , (25b)
sinϕ sin(ϕ ϕ )
a a b
−
cosϕ
b2 = a , (25c)
sinϕ sin(ϕ ϕ )
b b a
−
and
0 a2+b2 2. (25d)
≤ ≤
Once the angles ϕ and ϕ have been chosen, a, b and c are fixed. Finally we rotate these vectors around the y-axis
a b
through an angle θ to make the general POVM with three real rank 1 elements:
ωˆ (θ) ω (θ) ω (θ), (26a)
j j j
≡| ih |
ω (θ) =Vˆ(θ)ω , Vˆ(θ) exp( iθσˆ ). (26b)
j j y
| i | i ≡ −
This gives the most general POVM ωˆ (θ),ωˆ (θ),ωˆ (θ) in terms of three independent parameters ϕ , ϕ and θ.
0 1 2 a b
{ }
WearenowinapositiontomaximizetheShannonmutualinformationof with(atmost)three-elementPOVMs.
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We first give a useful preliminary lemma.
Lemma 6: Let = λ2 a a be any POVM with rank 1 elements labelled by a where 0<λ 1 is real and
A { a| ih |} a ≤
cosθ
a = a
| i sinθa
(cid:18) (cid:19)
in the z-spin basis. Then the mutual information for is given by
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λ2
I( : )= aI(θ ) (27)
M a
E A 2
a
X
where I(θ) is the function given in Eq. (17).
Proof: The states ψ of given in eq. (6) lead to the conditional probabilities
k M
| i E
λ2 2kπ
P(ak) = λ2 ψ a 2 = a(1+cos(2θ ))
| a|h k| i| 2 a− M
Substituting these into eq. (2) readily yields the formula eq. (27) after a little algebra.
Theorem 2: The Shannon mutual information of (for M > 2) is maximized by the POVM = ωˆ =
EM W { j∗
ω ω :j =0,1,2 where
| j∗ih j∗| }
0
ω = , (28a)
| 0∗i √2 a2 b2
(cid:18) − − (cid:19)
sin(mπ)
|ω1∗i=a −cos(mMπ) , (28b)
(cid:18) M (cid:19)
sin(nπ)
ω =b M , (28c)
| 2∗i cos(nπ)
(cid:18) M (cid:19)
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and
cos(nπ)
a2 = M 0, (29a)
sin(mπ)sin((m+n)π) ≥
M M
cos(mπ)
b2 = M 0. (29b)
sin(nπ)sin((m+n)π) ≥
M M
Here m and n are any positive integers satisfying
0 a2+b2 2. (29c)
≤ ≤
In some cases one of a, b and √2 a2 b2 is zero and the POVM has only two elements.
− −
Proof: For the three element POVM (θ,ϕ ,ϕ )= ωˆ (θ),ωˆ (θ),ωˆ (θ) with rank 1 elements, lemma 6 immedi-
a b 0 1 2
W { }
ately gives
a2 b2 a2 b2
I( : )=(1 )I(θ)+ I(θ+ϕ )+ I(θ+ϕ )
M a b
E W − 2 − 2 2 2
Hence I( : ) max I(θ). By lemma 3 this maximum is I(π), the accessible information of . Furthermore
I(θ) is peErMiodicWin≤θ withθperiod π . Hence we can achieve I( :2 )=I(π) by setting θ = π andEMchoosing φ and
φ to beanyintegermultiples of Mπ . ThisgivesEqs. (28). EqsE.M(29W)arejust2the conditionfor 2ωˆ tobe aPOVaM.
b M { j∗}
Fromthis theoremwecandevelopvariouskindsofoptimalstrategies. We notedpreviouslyincorollary1thatifM
iseven,thenthereexistsanoptimalstrategybasedonapairoforthogonaldirections. Thisalsofollowsfromtheorem
2: if M =4L 2 with L=1,2,... then we may take n=2L 1 giving a=0 and a 2-element POVM based on the
− −
0 1
directions and . If M = 4L with L = 1,2,..., we may take m = n = L giving √2 a2 b2 = 0 and
1 0 − −
(cid:18) (cid:19) (cid:18) (cid:19)
1 1
an optimal POVM based on the directions − and . In both cases the pair of directions coincides with an
1 1
(cid:18) (cid:19) (cid:18) (cid:19)
orthogonalpair of states of .
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If M is odd, at least3 outputs are required. In the case of M =3 we get anoptimum strategywith three elements
ofequalnorm. This coincideswithourpreviousresult oftheorem1andlemma3. ThecasesofM =5andM =7
3
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aremoreinteresting. Inbothcases,the optimumstrategiesconsistofthe threeelements withthe twodifferentnorms
(in contrast to the Z -covariant strategies of theorem 1). A solution for M = 5 is shown in Fig. 2. The POVM
M
elements are represented by the thick solid lines and the dashed lines represent the input states. (Note that, for ease
of presentation these dashed lines representing the states of – symmetrically distributed around a whole circle –
M
correspondto the vectors ( 1)k ψ rather than the originalEvectors in Eq. (6)). According to choices of parameters
k
− | i
(m,n) in theorem2,there canbe severalconfigurationsof the POVMdirections. But by the symmetry of they all
5
lie in the same position relative to the ensemble as a whole, characterized by a2 =b2 =1/(2sin22π) as shoEwn in Fig.
5
2.
|ω〉 |ρ〉
0 1
|ρ〉
2
|ρ〉
0
|ω〉 |ω〉
1
2
|ρ〉
3
|ρ〉
4
FIG. 2. The optimal POVM directions (thick solid lines) given by theorem 2 in the case of M = 5. The input states are
represented as (−1)k|ψ i by the dashed lines whose lengths correspond to a unit state vector. The lengths of the thick solid
k
lines are scaled according to thenormalization factors of the corresponding POVMelements.
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