Table Of ContentTight lower bound to the geometric measure of quantum discord
Ali Saif M. Hassan,1, Behzad Lari,2, and Pramod S. Joag2,
∗ † ‡
1Department of Physics, University of Amran, Amran, Yemen
2Department of Physics, University of Pune, Pune, India-411007.
(Dated: January 31, 2012)
Dakic,VedralandBrukner[PhysicalReviewLetters105,190502(2010)]gaveageometricmeasure
ofquantumdiscordinabipartitequantumstateasthedistanceofthestatefromtheclosestclassical
quantum (or zero discord) state and derived an explicit formula for a two qubit state. Further,
S.Luo and S.Fu [Physical Review A 82, 034302 (2010)] obtained a generic form of this geometric
measure for a general bipartite state and established a lower bound. In this brief report we obtain
a rigorous lower bound to the geometric measure of quantum discord in a general bipartite state
2
which dominates that obtained byS.Luo and S.Fu.
1
0 PACS numbers: 03.67.-a, 03.65.Ta
2
n Understanding quantum correlations in a multipartite quantum state is a fundamental open prob-
a
lem. Over the last two decades quantum correlations are studied in the entanglement-separability
J
0 scenario [1] leading to important insights in quantum computing [2], quantum communication pro-
3 tocols like teleportation [3, 4], superdense coding [5], cryptography [6] etc. However, recently it was
shown that even some separable states contain nonclassical correlation and can be used to accom-
]
h plish information processing tasks which cannot be achieved classically [7-13]. These nonclassical
p
correlations of bipartite states are measured by quantum discord [14, 15]-the discrepancy between
-
t
n quantum versions of two classically equivalent expressions for mutual information. Quantum dis-
a cord became a subject of intense research in different contexts [16-44]. As the evaluation of quantum
u
q discord involves optimization procedure, analytical results are known only in a few families of two-
[ qubit states [18, 29, 44]. Recently, a necessary and sufficient condition for the existence of non-zero
3 quantum discord was obtained and a geometrical way of quantifying quantum discord was proposed
v [45]. Geometric measure of quantum discord, introduced by Dakic et al [45], is given by
0
2
D(ρ) = min ρ χ 2, (1)
9
1 χ∈Ω0|| − ||
.
0 where Ω denotes the set of zero-discord states and ρ χ 2 := tr(ρ χ)2 is the square norm in the
1 0 || − || −
Hilbert-Schmidt space. A state χ Ha Hb is of zero discord if and only if it is a classical-quantum
0
∈ ⊗
1 state [14, 18], which can be represented as
:
v
m
i
X χ = p k k ρ , (2)
k k
| ih |⊗
r
a Xk=1
where p is a probability distribution, k is an arbitrary orthonormal basis in Ha and ρ is a
k k
{ } {| i}
set of arbitrary states (density operators) acting on Hb.
The quantum discord of a bipartite state ρ on a system Ha Hb with marginals ρa and ρb can be
⊗
expressed as
Q(ρ) = min I(ρ) I(Πa(ρ)) . (3)
Πa { − }
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
Here the minimum is over von Neumann measurements (one dimensional orthogonal projectors
summing up to the identity) Πa = Πa on subsystem a, and
{ k}
Πa(ρ) = (Πa Ib)ρ(Πa Ib)
k ⊗ k ⊗
k
X
is the resulting state after the measurement. I(ρ) = S(ρa) + S(ρb) S(ρ) is the quantum mutual
−
information, S(ρ) = trρlnρ is the von Neumann entropy, and Ib is the identity operator on Hb.
−
Intuitively, quantum discordmaythusbeinterpreted astheminimal lossofcorrelations(asmeasured
by the quantum mutual information) due to measurement. This formulation of quantum discord is
equivalent to the original definition of quantum discord by Ollivier and Zurek [14].
In order to obtain the desired lower bound for the quantum discord in an arbitrary bi-partite state,
we set up the following scenario. Consider a bipartite system Ha Hb with dim Ha = m and dim
⊗
Hb = n. Let L(Ha) be the space consisting of all linear operators on Ha. This is a Hilbert space
with the Hilbert-Schmidt inner product
X Y = trX Y.
†
h | i
The Hilbert spaces L(Hb) and L(Ha Hb) are defined similarly. Let X : i = 1,2, ,m2 and
i
⊗ { ··· }
Y : j = 1,2, ,n2 be sets of Hermitian operators which constitute orthonormal bases for L(Ha)
j
{ ··· }
and L(Hb), respectively. Then
trX X = δ , trY Y = δ .
i i′ ii′ j j′ jj′
X Y constitutes an orthonormal (product) basis for L(Ha Hb) (linear operators on Ha Hb).
i j
{ ⊗ } ⊗ ⊗
In particular, any bipartite state ρ on Ha Hb can be expanded as
⊗
ρ = c X Y , (4)
ij i j
⊗
ij
X
with c = tr(ρX Y ).
ij i j
⊗
S. Luo and S. Fu introduced the following form of geometric measure of quantum discord [46]
(Here and throughout this article, the superscript t denotes transpose of vectors or matrices).
D(ρ) = tr(CCt) maxtr(AˆCCtAˆt), (5)
− Aˆ
where C = [c ] (Eq.(4)) is an m2 n2 matrix and the maximum is taken over all m m2-dimensional
ij
× ×
isometric (see below) matrices Aˆ = [a ] such that
ki
a = tr( k k X ) = k X k , k = 1,2,...,m
ki i i
| ih | h | | i
i = 1,2,...,m2, (6)
and k is any orthonormal basis in Ha. We can expand the operator k k in this basis as
{| i} | ih |
k k = a X , k = 1,2,...,m. (7)
ki i
| ih |
i
X
Aˆ = [a ] is an isometry in the sense that AˆAˆt = Ia and the row vectors ~a of the matrix Aˆ satisfy
ki k
m2
~a 2 = a2 = 1. (8)
|| k|| ki
i=1
X
3
Further, using their definition, it immediately follows that
m
a = trX . (9)
ki i
k=1
X
We can represent the density operators acting on a bipartite system Ha Hb, with dimHa = m
⊗
and dimHb = n, as [47, 48]
1
ρ = (I I + x λ˜ I + y I λ˜
m n i i n j m j
mn ⊗ ⊗ ⊗
i j
X X
+ t λ˜ λ˜ ), (10)
ij i j
⊗
ij
X
where λ˜ ,i = 1,...,m2 1 and λ˜ ,j = 1,...,n2 1 are the generators of SU(m) and SU(n)
i j
respectively, satisfying tr−(λ˜ λ˜ ) = 2δ [47]. Notice th−at ~x Rm2 1 and ~y Rn2 1 are the coherence
i j ij − −
∈ ∈
vectors of the subsystems A and B, so that they can be determined locally. These are given by
[49, 50]
m m
x = tr(ρλ˜ I ) = tr(ρ λ˜)
i i n A i
2 ⊗ 2
n n
y = tr(ρI λ˜ ) = tr(ρ λ˜),
j m j B j
2 ⊗ 2
where ρ = tr (ρ) and ρ = tr (ρ) are the reduced density matrices. The correlation matrix
A B B A
T = [t ] is given by
ij
mn
T = [t ] = [tr(ρλ˜ λ˜)].
ij i j
4 ⊗
In this article, we find the lower bound of geometric measure of quantum discord which dominates
the lower bound in ref. [46].
Theorem 1. Let ρ be a bipartite state defined by Eq. (10); then
m 1
2 2 −
D(ρ) ( ~x 2 + T 2 η ), (11)
j
≥ m2n || || n|| || −
j=1
X
where η , j = 1,2, ,m2 1 are the eigenvalues of the matrix (~x~xt + 2TTt) arranged in non-
j ··· − n
increasing order (counting multiplicity).
We prove this theorem for arbitrary (finite) m and n.
In Eq.(5) giving the quantum discord D(ρ), the maximum in the second term is taken over the
m m2 isometric matrices Aˆ which also satisfy Eq.s(6) and Eq.(7). In other words, the row vectors
×
of Aˆ are required to be the coherent vectors of states forming an orthonormal basis in Ha. If we
ignore this constraint while maximizing tr(ACCtAt) and maximize over the isometric matrices A
defined below via Eq.(18,19), the resulting maximum will be greater than or equal to the required
maximum of tr(AˆCCtAˆt) over the matrices Aˆ satisfying Eq.s(6,7). Since all the terms are positive,
this leads to
D(ρ) tr(CCt) maxtr(ACCtAt). (12)
≥ − A
4
We proceed to obtain the maximum in the above inequality (Eq.(12)). We choose the orthonormal
bases X and Y in Eq.(4) as the generators of SU(m) and SU(n) respectively [47].
i j
{ } { }
1 1
X = I ,Y = I
1 √m m 1 √n n
and
1
X = λ˜ , i = 2,3,...,m2
i i 1
√2 −
1
Y = λ˜ , j = 2,3,...,n2.
j j 1
√2 −
Since trλ˜ = 0; i = 1,2, ,m2 1, we have, via Eq.(9),
i
··· −
m
a = trX = trλ˜ = 0,i = 2,...,m2.
ki i i
k=1
X
Therefore,
m 1
−
a = a , i = 2,3, ,m2. (13)
mi ki
− ···
k=1
X
We now proceed to construct the m m2 matrix A defined via Eq.(6-9). We will use Eq.(13). The
×
row vectors of A are
~ak = (ak1,ak2, ,akm2);k = 1,2,...,m.
···
Next we define
m
eˆk = m 1(ak2,ak3,...,akm2), k = 1,2,...,m−1 (14)
r −
and using Eq.(13), we get
m 1
−
eˆ = eˆ . (15)
m k
−
k=1
X
We can prove
eˆ 2 = 1 k = 1,2,...,m 1 (16)
k
|| || −
using the condition ~a 2 = m2 a2 = 1 (Eq.(8)) and using Eq.(9) with i = 1, namely, a =
|| k|| i=1 ki k1
tr( k k X ) = 1 . Further, isometry of the A matrix (AAt = I) implies
| ih | 1 √m P
1
eˆeˆt = − ,j = i = 1,2,...,m 1. (17)
i j m 1 6 −
−
We can now construct the row vectors of m m2 matrix A, using Eq.(14) and Eq.(15),
×
1
~a = (1,√m 1eˆ ), k = 1,2, ,m 1 (18)
k k
√m − ··· −
5
m 1
1 −
~a = (1, √m 1 eˆ ) (19)
m k
√m − −
k=1
X
defining matrix A.
We get the elements of C = [c ] = [tr(ρX Y )] using the definitions of the bases X and Y
ij i j i j
⊗ { } { }
given above, in terms of the generators of SU(m) and SU(n). This gives
1 √2 ~yt
C = √mn n√m ,
√2 ~x 2 T
!
m√n mn
and
1 2 2
tr(CCt) = ( + ~y 2 + ~x 2
mn n2m|| || m2n|| ||
4
+ T 2). (20)
n2m2|| ||
where ~x, ~y and T are the coherent vectors and the correlation matrix respectively, defined in Eq.(10).
Having constructed the matrices A and C, we get, for tr(ACCtAt),
m 1 m 1m 1
1 1 2 2(m 1) − − −
tr(ACCtAt) = + ~y 2 + − eˆ Geˆt + eˆGeˆt , (21)
m n n2|| || m2n j j i j
( " #)
j=1 i=1 j=1
X X X
where
2TTt
G = ~x~xt + (22)
n
is the (m2 1) (m2 1) real symmetric matrix. The eigenvectors of G span Rm2 1 and form a
−
orthonorma−l bas×is of R−m2−1. Let η1,η2,...,ηm2 1 be the eigenvalues of G arranged in non-increasing
order (counting multiplicity). Let ( fˆ1 , fˆ2 ,..−., fˆm2 1 ) be the corresponding orthonormal eigen-
vectors of G. To maximize the right|haind| siide of|Eq.(−21i), we choose
eˆ = fˆ (23)
1 1
and expand eˆ m 1 in the eigenbasis of G. Thus,
{ }j=−2
m2 1
eˆ = − ǫ(j)fˆ j = 2,3,...,m 1. (24)
j k k −
k=1
X
Eq.(16) with eˆ replaced by eˆ gives us
i 1
ǫ(1) = 1
1
1
ǫ(j) = − ,j = 2,3,...,m 1. (25)
1 m 1 −
−
Since eˆ = fˆ (Eq.(23)) and eˆ 2 = 1 for all j (Eq.(16)) we get
1 1 j
|| ||
m2 1
− ǫ(j)2 = 1. (26)
k
k=1
X
We substitute Eq.(24) for eˆ in the expression for tr(ACCtAt) (Eq.(21)) and use Eq.(25) to get
j
6
m 1m2 1
tr(ACCtAt) = 1 1 + 2 ~y 2 + 2(m−1) m η + 2 − − ǫ(j)2η
m n n2|| || m2n m 1 1 k k
( "
− j=2 k=2
X X
m 2m 1m2 1
+ 2 − − − ǫ(i)ǫ(j)η . (27)
k k k
#)
i=2 j>i k=2
X X X
We have to choose vectors eˆ j = 1,...,m 1, that is, the expansion coefficients ǫ(j) ; j =
{ j} − { k }
1,...,m 1 : k = 1,...,m2 1 in Eq.(24), consistent with Eq.s(25,26), so as to maximize
− −
tr(ACCtAt). First we note that we must use at least (m 1) eigenvectors fˆ in Eq.(24) to ex-
k
−
pand all of eˆ j = 1,...,m 1, because otherwise eˆ becomes a set of m 1 linearly dependent
j j
{ } − { } −
vectors in a subspace of dimension less than m 1, in which case the row vectors of matrix A cease
−
to be mutually orthogonal. Thus, for m = 3 a choice like eˆ = ǫ(2)fˆ yields ~a ~a = 0. With eigen-
2 1 1 1 · 2 6
values η k = 1,...,m2 1 arranged in non-increasing order, we now see that, for every choice of
k
{ } −
ǫ(j) ; j = 1,...,m 1; k = 1,...,m2 1 we get the following upper bound on tr(ACCtAt)
{ k } − −
m 1 j 1 m2 1
tr(ACCtAt) 1 1 + 2 ~y 2 + 2(m−1) m η +2 − − ǫ(j)2η +η − ǫ(j)2
≤ m n n2|| || m2n m 1 1 k k j k
( " !
− j=2 k=2 k=j
X X X
m 2m 1 i 1 m2 1
+2 − − − ǫ(i)ǫ(j)η +η − ǫ(i)ǫ(j) . (28)
k k k i k k
!#)
i=2 j>i k=2 k=i
X X X X
The maximum value of RHS is then obtained by choosing ǫ(j) = 0 for all k > j, that is,
k
j
eˆ = ǫ(j)fˆ j = 2,3,...,m 1, (29)
j k k −
k=1
X
which leads, using the fact that ǫ(j) = 0 if k > j, to
k
m 1 m 1
max[tr(ACCtAt)] = 1 1 + 2 ~y 2 + 2(m−1) m η +2 − η − ǫ(j)2
A m (n n2|| || m2n "m 1 1 k" k
− k=2 j=k
X X
m 2m 1
+ − − ǫ(i)ǫ(j) . (30)
k k
##)
i=k j>i
X X
The remaining task is to find the coefficients ǫ(j), k = 1,2,...,j in the expansion of eˆ in the
k j
eigenbasis fˆ , k = 1,2,...,j which satisfy Eq.s(15,16,29) and the consequential equations (25,26).
k
{ }
Using Eq.s(16,17,25,26,29) the problem can be reduced to the coupled pair of equations
j 1
ǫ(j)2 = m(m−2) − ǫ(j)2 ;j > 1 (31)
j (m 1)2 − k
− k=2
X
i 1
ǫ(j) = 1 −m − ǫ(i)ǫ(j) ;j > i > 1. (32)
i ǫ(i) "(m 1)2 − k k #
i − k=2
X
7
which can be solved iteratively, starting from j = 2. The result is
ǫ(j)2 = m m−j , (33)
j (m 1) m j +1
− (cid:20) − (cid:21)
and
√m
ǫ(j) = − ;j > i (34)
i
(m 1)(m i+1)(m i)
− − −
Using Eq.s(33,34) the last term in Epq.(30) can be evaluated. We have,
m 1 m 2m 1
− ǫ(j)2 + − − ǫ(i)ǫ(j) = m . (35)
k k k 2(m 1)
j=k i=k j>i −
X X X
Eq.(35) simplifies Eq.(30) to
m 1
1 2 2 −
max[tr(ACCtAt)] = + ~y 2 + η (36)
j
A mn n2m|| || m2n
j=1
X
Finally, Eq.(20), Eq.(36) and Eq.(12) together imply
m 1
2 2 −
D(ρ) ( ~x 2 + T 2 η ),
j
≥ m2n || || n|| || −
j=1
X
which completes the proof of the theorem.
Now, we prove that
m 1 m
2 2 −
D(ρ) ≥ m2n(||~x||2 + n||T||2− ηj↓) ≥ tr(CCt)− λ↓i (37)
j=1 i=1
X X
where λ↓i are the eigenvalues of CCt listed in decreasing order (counting multiplicity) and ηj↓ are
the eigenvalues of G = ~x~xt + 2TTt listed in decreasing order (counting multiplicity). The last lower
n
bound in Eq.(37) is proved in ref. [46] where the inequality maxAˆtr(AˆCCtAˆt) ≤ mi=1λ↓i is derived.
Since we have proved maxAˆtr(AˆCCtAˆt) ≤ m1n + n22m||~y||2 + m22n mi=−11ηi↓ = a +Pm22n mi=−11ηi↓, to
prove Eq.(37), it is enough to prove
P P
m m 1
2 −
λ↓i ≥ a+ m2n ηi↓. (38)
i=1 i=1
X X
Let us rewrite CCt as
a ~ut
CCt = ,
~u 2 G
(cid:18) m2n (cid:19)
where a = 1 + 2 ~y 2 , G = ~x~xt+ 2TTt and ~u = √2 ~x+ 2√2 T~y. Let eigenvalues of 2 G be
mn n2m|| || n mn√m mn2√m m2n
η = 2 η , where η are the eigenvalues of G. Then
{ j′} m2n{ j} { j}
m2 1 m2
2 −
tr(CCt) = a+ η = λ . (39)
j i
m2n
j=1 i=1
X X
8
Let λ↑1 ≤ λ↑2 ≤ ... ≤ λ↑m2 and η1′↑ ≤ η2′↑ ≤ ... ≤ ηm′↑2 1. From theorem (4.3.8) in [51], we have
−
λ η λ η ... η λ ,
↑1 ≤ 1′↑ ≤ ↑2 ≤ 2′↑ ≤ ≤ m′↑2 1 ≤ ↑m2
−
so that
m2 m m2 m
− −
λ↑i ≤ ηj′↑. (40)
i=1 j=1
X X
Now, we use Eq.(39) to get
m2 m m2 m2 m m2 1
− − −
λ↑i + λ↑i = a+ ηj′↑ + ηj′↑. (41)
i=1 i=m2 m+1 j=1 j=m2 m+1
X X− X X−
Replacing the first term on the left hand side of Eq.(41) by mj=21−mηj′↑ we get, using Eq.(40),
m2 m2 1 P
−
λ↑i ≥ ηj′↑ +a, (42)
i=m2 m+1 j=m2 m+1
X− X−
Finally, we use λ↑j = λ↓m2 j+1 and ηj′↑ = ηm′↓2 j [52], to get
− −
m m 1
−
λ↓i ≥ ηj′↓ +a.
i=1 j=1
X X
Examples
(1) We consider the two qutrit state
I
ρ = p e e +(1 p) (43)
| ih | − 9
where e = 1 ( 2 2 + 3 3 + 2 1 + 1 2 + 1 3 + 3 1 ), I is the identity
| i √6 | i⊗| i | i⊗| i | i⊗| i | i⊗| i | i⊗| i | i⊗| i
operator and i ;i = 1,2,3 is the standard basis in C3. Fig. 1 shows the variation of lower bound
(given in Eq.({1|1)i) and the lo}wer bound on D(ρ), as given in [46], namely, tr(CCt) m λ (where
− i=1 i
λ are the eigenvalues of CCt listed in the decreasing order, counting multiplicity), with p.
i
{ } P
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
p
Fig. 1 Lower bound of quantum discord (Eq.(11), solid line) and lower bound (Eq.(6) in [46], dashed line) as a function of p
(Eq.(43)).
9
We see that the lower bound in Eq.(11) dominates this lower bound for p > 0.2.
(2) We consider the two qutrit state
ρ = p e e +(1 p) e e (44)
1 1 2 2
| ih | − | ih |
where e = 1 11 + 1 22 + 1 33 and e = 1 ( 2 2 + 3 3 + 2 1 + 1 2 +
| 1i 2| i 2| i √2| i | 2i √6 | i ⊗ | i | i ⊗ | i | i ⊗ | i | i ⊗ | i
1 3 + 3 1 ). Fig. 2 shows the variation of D(ρ) (given in Eq.(11) and the lower bound on
|Di(ρ⊗),|ais giv|ein⊗in| [i46], namely, tr(CCt) m λ (where λ are the eigenvalues of CCt listed in
− i=1 i { i}
the decreasing order, counting multiplicity), with p.
P
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0.2 0.4 0.6 0.8 1
p
Fig. 2 Quantum discord (Eq.(11), solid line) and lower bound (Eq.(6) in [46], dashed line) as a function of p (Eq.(44)).
We see that the lower bound in Eq.(11) dominates this lower bound.
To the best of our knowledge, only the lower bounds on the quantum discord in an arbitrary
bipartite state are presently available. This seems quite surprising as detecting quantum discord
is not a NP-hard problem such as the separability problem and one expects an exact computable
expression for quantum discord in all dimensions. This expectation is further augmented by the fact
that the set of zero discord states is of measure zero. In order to get such an exact computable
expression for quantum discord in all dimensions, we have to maximize the second term in Eq.(5)
over the set of m m2 matrices A which satisfy (in addition to other conditions) the requirement
×
that each row vector of A must be a coherent vector of a orthonormal basis state in Ha. For the
two qubit case, this requirement becomes redundant as every unit vector in R3 is a coherent vector
of some single qubit pure state. If dim(Ha) > 2, not every unit vector in Rd2 1 is a coherent vector
−
of some pure state in Ha. Hence whenever dim(Ha) > 2, the above requirement is to be included
as an independent constraint in the constrained optimization of the second term in Eq.(5). The
resulting constrined optimization problem is very difficult because the set of coherent vectors (for
dim(Ha) > 2) do not have some simple geometric structure like Bloch sphere. This is the reason why
an exact computable expression for quantum discord for all dimensions still eludes us. We note that
the lower bound in Eq.(11) becomes exact for a 2 d system (with measurement on the qubit) since
×
the above constraint is relaxed in this case. Finally, our lower bound on discord can be meaningfully
used to compare discordant states and check on the possible monogamy property of dicord [53].
Acknowledgments :
This work was supported by the BCUD research grant RG-13. ASMH thanks Pune University for
the hospitality during his visit when this work was initiated. We thank the anonymous referee whose
suggestions have contributed towards the improvement of this paper.
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