Optimal measurements to access classical correlations of two-qubit states

0.2 MB·English

by  Xiao-Ming Lu

Optimal measurements to access classical correlations of two-qubit states
Xiao-Ming Lu,1 Jian Ma,1 Zhengjun Xi,1,2 and Xiaoguang Wang1
1Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China
2College of Computer Science, Shaanxi Normal University, Xi’an 710062, China
Weanalyzetheoptimalmeasurementsaccessingclassicalcorrelationsinarbitrarytwo-qubitstates.
Two-qubitstates can betransformed intothecanonical forms vialocal unitaryoperations. Forthe
canonical forms, we investigate the probability distribution of the optimal measurements. The
probability distribution of the optimal measurement is found to be centralized in the vicinity of a
specific von Neumann measurement, which we call the maximal-correlation-direction measurement
(MCDM). We prove that for the states with zero-discord and maximally mixed marginals, the
MCDMistheveryoptimalmeasurement. Furthermore,wegiveanupperboundofquantumdiscord
based on theMCDM, and investigate its performance for approximating the quantumdiscord.
1
1
PACSnumbers: 03.67.-a,03.65.Ta
0
2

n I. INTRODUCTION spite recent progress, the optimization procedure over
a von Neumann measurements is still hard to be resolved
J
forgeneraltwo-qubitstates,andanalyticapproachesstill
6 Optimizationprocedures are involvedin many quanti-
lack. This motives us to systematically investigate the
ties inthe quantuminformationtheory. Oneofthe most optimal measurements and find an effective way to ap-
]
important examples is the entanglement of formation,
h proach the optimal measurement.
p which is defined as the leastexpected entanglementover
In the present article, we investigate the probability
- allensembles of pure states realizing the givenstates [1].
t distribution of the optimal measurements. Based on a
n Here we focus on the optimization procedure involvedin
generalanalysisonthosefactorswhichinfluencetheclas-
a the classicalcorrelation,whichisdefined asthe maximal
u sicalcorrelation,weintroducethecanonicalformsforthe
amount of information about the subsystem B that can
q two-qubit states and the maximal-correlation-direction
be obtained via performing measurement on the other
[ measurement (MCDM). For arbitrary two-qubit states,
subsystemA[2]. Meanwhile,thenon-classicalcorrelation
the optimalmeasurements arefound to be centralizedin
2 is measured by the so-called quantum discord [3], which
v is the difference of the total amount of correlations and the vicinity of the MCDM. We prove that for the states
6 with zero-discord and maximally mixed marginal, the
the classicalcorrelation. Over the pastdecade,quantum
7 MCDM is the very optimal measurement accessing clas-
discordhasreceivedalotofattention,includinganalytic
4 sical correlation. We also study its veracity for X-states
calculations for some special two-qubit sates [4, 5], de-
1
andarbitrarystates. ThenweproposetheMCDM-based
. tecting of quantum discord [6–13], quantum discord in
9 discord as an upper bound of quantum discord. It is
the concrete physical models [14–23], the dynamics of
0 demonstrated that the MCDM-based could be a good
quantumdiscord[24–35],quantumdiscordincontinuous
0 approximationof quantum discord.
1 variable system [36, 37], quantum discord of multipar-
: titestates[38],andexplorationinthelaboratory[39–41], This article is organized as follows. In Sec. II, we give
v etc. Most recently, operational interpretations of quan- abriefreviewonthemeasuresofthetotal,quantumand
i
X tumdiscordareproposedinRef.[42,43],wherequantum classical correlations. In Sec. III, first we give a general
r discord was shown to be a quantitative measure about analysisonthosefactorswhichinfluencetheclassicalcor-
a the performance in the quantum state merging. relations. Thenweintroducedtheconceptsofthecanon-
ical form and the MCDM. In Sec. IV, we investigate the
Toaccessthetotalclassicalcorrelationorgetthevalue
probabilitydistributionoftheoptimalmeasurementsand
of quantum discord, one has to find the corresponding
theperformanceoftheMCDM.InSec.V,weproposethe
optimal measurements. On the other hand, the sud-
MCDM-baseddiscordasanupperboundofquantumdis-
den change of the optimal measurements during evolu-
cord, and investigate its performance for approximating
tion is related to a novel phenomenon of the quantum
quantum discord. Section VI is the conclusion.
discord—the sudden change in decay rates [27–30, 40].
So a study on the optimal measurements will help us
to understand the dynamical properties of the quan-
tum discord. The optimization involved is taken over
II. REVIEW OF QUANTUM DISCORD
general measurements, which is described by a positive-
operator-valued measure (POVM). In Ref. [44], Hamieh
etal. showedthatfortwo-qubitsystemtheoptimalmea- First,werecalltheconceptsofthetotalamountofcor-
surement over POVM is a projective measurement. In relations, the classical correlation and the quantum cor-
this paper, we only consider orthogonal projective mea- relation. GivenaquantumstateρinacompositeHilbert
surements , knows as von Neumann measurement. De- space = A B, the total amount of correlation is
H H ⊗H
2
quantified by the quantum mutual information [45] we consider the relative entropy as a measure of the dis-
tance, then the classicalcorrelationcanbe consideredas
(ρ)=S(ρA)+S(ρB) S(ρ), (1) maximal average distance between the remaining state
I −
ρB and the reduced state ρB. It seems that the optimal
k
wanhderρeAS(B(ρ))=≡T−rBT(rA[)ρ[ρlo]gi2sρt]hiesrtehdeuvcoedn Ndeenusmitaynmnaetnrtirxobpyy mmeaainsiunrgemsteanttestfeanrdaswtaoybfreomthethoenreedwuhciecdhstmaatek,easltthhoeurgeh-
tracing out system B(A). The total amount of correla- the probabilityp alsoplaysanimportantroleinthe ac-
k
tions can be splitted into the quantum and the classi- tual optimization problem.
cal parts [2, 3]. The classical correlation is seen as the Based on this tendency, we use the following Fano-
amount of information about the subsystem B that can Bloch decomposition of an arbitrary two-qubit state[46,
be obtained via performing a measurement on the other 47]:
subsystemA. Thenthe measureof the classicalcorrela-
tion is defined by
3
1
C(ρ)=S(ρB)−{mEikAn} k pkS(ρBk), (2) ρ=ρA⊗ρB + 4iXj=1ΛijσiA⊗σjB, (5)
X
where {EkA} is the POVM performed on A and ρBk := wtiohnerefunΛcitjion=wihtσhiAσOjBiρ :−=hTσriA[iρρOhσ]jBdieρfiniesd.thTehcisorfroerlma-
TrA EkA⊗11Bρ /pk istheremainingstateofB afterob- was used to investhigaitρe the dynamics of open quan-
taining the outcome k on A with the probability p :=
Tr E(cid:0)A 11Bρ .(cid:1) 11A(B) is the identity operator onkthe tum systems in the presence of initial correlation [48],
k ⊗ and the correlation functions were used to characterized
subsystemA(B). Meanwhile,the quantumcorrelationis
(cid:0) (cid:1) the correlations in a quantum state of a composite sys-
measured via the quantum discord defined by [2, 3]
tem [49, 50]. Because the quantum and classicalcorrela-
tions are both invariant under local unitary transforma-
(ρ)= (ρ) (ρ), (3)
D I −C tions, we will consider a special set of two-qubit states,
in which for arbitrary two-qubit state we can find an
which is the difference of the total amountof correlation
equivalent state up to local unitary transformations. In
(ρ) and the classical correlation (ρ).
I C Ref. [4], Luo showedthat the matrix Λ in Eq.(5) can be
The definition of the classical and quantum correla-
diagonalizedthroughlocalunitarytransformations. Fur-
tions involves an optimization process to minimize the
term p S(ρB), which is considered as the quantum thermore,the orderandthe signs ofthe eigenvaluesofΛ
k k k can be realigned through SO(3) transformations, which
versionofthe conditionalentropy[3]. For two-qubitsys-
P correspondto SU(2)transformationsonthe density ma-
tems, it is shownthat the optimalPOVMis a projective
trices. Hereweintroducethecanonicalformoftwo-qubit
measurement [44]. Hereafter, we will only consider the
states, which is defined by
orthogonalmeasurementprojectivemeasurement,knows
as von Neumann measurement. A von Neumann mea-
ssapuchrteeermrieze,endtthbr{yoΠuaA1g,huΠniA2t}veocftoartnwo=-q(unb1i,tns2y,snt3e)mT ocannthbeeBclhoacrh- ρ=ρA⊗ρB + 14 3 ΛiσiA⊗σiB (6)
i=1
X
ΠA1 = 12 11A+ 3 niσiA!, wTihtehsΛigin=ofhσΛiA3σisiBdiρet−erhmσiiAnieρdhσbiByitρheanddetΛer1m≥inaΛn2t≥|Λ||Λv3i|a.
i=1 Λ =Λ Λ Λ . Anarbitrarytwo-qubitstateisequivalent
X 1 2 3
| |
1 3 toacertainstateinthecanonicalformuptolocalunitary
ΠA2 = 2 11A− niσiA!. (4) transformations. So studying the states in the canonical
i=1 formisadequatetounderstandthequantumandclassical
X
correlationsofarbitrarytwo-qubitstates. Hence,wewill
So the optimization may be taken over half of the Bloch
only consider the states in the canonical form hereafter.
sphere,sinceexchangingΠA andΠA givesthesamemea-
1 2 After von Neumann measurement characterized by
surement.
Eq.(4) performedonA, we getthe remaining state ofB
with the outcome k =1,2 on A as follows
III. GENERAL ANALYSIS AND 1
ρB =ρB + ∆ (7)
MAXIMAL-CORRELATION-DIRECTION k p k
k
MEASUREMENT
where p = Tr ρAΠA and ∆
k k k ≡
The classical correlation (2) can be expressed as 1( 1)k+1 3 n Λ σB are both dependent on ΠA(n ).
CS=(ρBm) ax{TΠrAk(ρ}BlokgpρkBS)(ρiBks||tρhBe),relwathievreeeSnt(rρoBkp|y|ρB[2)]. :=If tN4hoe−tevotnhaNtPe∆uim2=a1=nni−m∆ie1ia.suFrerom(cid:2)menEtqin.(cid:3)fl(u7e)n,cwesetchaenqsueaekntthuiamt
− k − kP
3
conditional entropy through p and ∆ . The influence have R = Λ+abT, see the definitions of R, Λ, a and b.
k k
of p is complicated, since p not only impacts the Then the conditions (11) and (12) leads to
k k
remaining state ρB but also impacts the averaging of
k
S(ρB), while ∆ only impacts the remaining state. If Λ=nnTΛ. (13)
k k
we assume that the influence of ∆ to the quantum
k
conditional entropy is stronger than that of pk, the Considering Λ = diag{Λ1,Λ2,Λ3} is a diagonal matrix
optimal measurement will tend to be the one which with Λ1 Λ2 Λ3 , from Eq. (13), the first diagonal
≥ ≥ | |
element of the matrix Λ reads
maximizes ∆ Tr∆ ∆† (Remind ∆ = ∆ ).
| 1| ≡ 1 1 | 2| | 1|
After some algebras,qwe have Λ =(n )2Λ . (14)
1 1 1
3
1 1 Soweobtainn = 1,whichcorrespondstotheMCDM.
|∆1|2 = 16 Λ2in2i ≤ 16Λ21. (8) So we conclu1de±that for the zero-discord states, the
i=1
X MCDM is just the optimal measurement to access the
The maximum of ∆ 2 is achieved when n = (1,0,0)T. classical correlation.
1
| |
So the corresponding measurement is
1 1 B. states with maximally mixed marginals
ΠA = (11+σA), ΠA = (11 σA) . (9)
1 2 1 2 2 − 1
(cid:26) (cid:27)
The states with maximally mixed marginals are the
BecauseΛi arecorrelationfunctionsandΛ1 isthelargest ones satisfying ρA = 11A/dA and ρB = 11B/dB, where
one of them, we call Eq. (9) the maximal-correlation- d = dim( A(B)) is the dimension of the Hilbert-
A(B)
directionmeasurement(MCDM)performedonAforthe space A(B). HFor two-qubit systems, The canonical
two-qubit states in the canonical form. For an arbitrary H
forms of states with maximally mixed marginals must
two-qubitstate,wecanalwaysfindalocalunitarytrans-
be Bell-diagonal states, while an arbitrary Bell-diagonal
formation U1⊗U2 such that ρ˜=U1⊗U2ρU1†⊗U2† is in state need not be in the canonical form. An arbitrary
the canonical form. So the MCDM performed on A for Bell-diagonalstate reads
an arbitrary two-qubit state is given by
3
1 1
21 11+U1†σ1AU1 ,21 11−U1†σ1AU1 . (10) ρ= 411A⊗11B + 4 i=1ciσiA⊗σiB (15)
(cid:26) (cid:16) (cid:17) (cid:16) (cid:17)(cid:27) X
with c = σA σB . Only when c c c , the
i h i ⊗ i i i ≥ 2 ≥ | 3|
IV. PERFORMANCE OF THE MCDM Bell-diagonal state is in the canonical form. For Bell-
diagonalstates,Luogottheanalyticalresultsofquantum
discord [4]. There it was shown that the optimal mea-
Inthefollowing,wewillinvestigatetheperformanceof
surement is given by the unit vector n = (n ,n ,n )T
the MCDM to access the classical correlation measured 1 2 3
by the quantum discord. which maximizes 3 c2n2. So the optimal measure-
i=1 i i
mentsofBell-diagqonalstatesmustbe inthe universal fi-
P
nite set 1(11 σA) i=1,2,3 . Here universal means
A. Zero-discord states this set{o{f2von±Neium}a|nn measu}rements is independent
on the given states. For the Bell-diagonal states in the
The zero-discord states are the states satisfying ρ = canonical form, we have c c c . Then the op-
1 2 3
ΠA 11BρΠA 11B, where ΠA is just the optimal timal measurement is explici≥tly gi≥ven| b|y n = (1,0,0)T,
k k ⊗ k ⊗ { k}
von Neumann measurement to access the classical cor- whichis consistentwith the MCDM (9). This is because
Prelation, see Ref. [3]. In Appendix A, we show that a Λ = c due to σA = 0 (maximal mixed margins),
i i h i iρ
sufficient and necessary condition of zero-discord is the then the maximization of 3 c2n2 is equivalent to
existence of such a unit vector n satisfying the following i=1 i i
the maximization of ∆ 2,qsee Eq. (8).
equations | 1| P
So we conclude that for the states with maximally
nnTa = a, (11) mixed marginals, the MCDM is just the optimal mea-
nnTR = R, (12) surement to access the classical correlation.
where a = Tr(σAρA) is the polarization vector of the
i i
reduceddensity matrix ρA , R is a 3 3 matrix with the C. X-states
elementsR =Tr(σA σBρ),andni×sthecolumnvector
ij i ⊗ j
characterizing the optimal von Neumann measurement In the following, we consider the so-called X-states,
via Eq. (4). For the states in the canonical form, we named because of the visual appearance of the density
4
matrix θ φ Percentage
π/2 0 99.40%
a 0 0 w∗
π/2 −π/2 0.60%
0 b z∗ 0
ρ= . (16)
 0 z c 0  TABLEI:Distributionoftheoptimalmeasurementstoaccess
w 0 0 d the classical correlation, for random density matrices in the
 
  canonicalformandequivalenttoX-statesuptolocalunitary
The X-states,including maximally entangledBell states transformations. The total number of the random states is
and Werner states, are a class of typical quantum states 10000.
in the field of quantum information. An algebraic char-
acterization of X-states is presented in Ref. [51]. The
an explicit example:
Fano-Blochrepresentationmatrix τ =Tr(σA σBρ) of
ij i ⊗ j
X-states is also of X-type
0.0783 0 0 0
0 0.1250 0.1000 0
1 0 0 b ρ= . (18)
3  0 0.1000 0.1250 0 
0 R R 0
τ = 11 12 . (17) 0 0 0 0.6717
 0 R R 0   
21 22  
a 0 0 R
 3 33  This state is in the canonical form with Λ1 = Λ2 = 0.2
 
and Λ = 0.1479. The optimal measurements are char-
An important property of the class of the X-states 3
acterized by two angle θ and φ, via Eq. (4) and
is that an X-state after the local unitary operations
exp(iσ3Aϕ1/2)⊗exp(iσ3Bϕ2/2) is also an X-state. This n=(sinθcosφ,sinθsinφ,cosθ)T (19)
leads to the following theorem:
with θ [0,π) and φ [ π/2, π/2). It can be directly
Theorem. For the X-states, it is impossible to exist ∈ ∈ − −
verified that the state (18) is invariant under the opera-
a universal finite set of von Neumann measurements
tionexp(iϕ σA) exp(iϕ σB),sotheoptimizationisonly
among which the optimal measurement must be. 1 3 ⊗ 2 3
relevant to θ, see Ref. [30]. Following the optimization
strategyofRef. [5],the value ofθ canonlybe either0 or
Proof. The proof is divided into two steps. First, we
π/2. However, the optimal measurement of this state is
assume that for the class of X-state, such a univer-
numerically found at θ 0.155π.
sal finite set exists and is denoted by := i
i ≃
M {M | ∈ In the following, we numerically investigate the prob-
, where is a von Neumann measurement and
i
I} M ability distribution of the optimal measurement for the
is a finite set of the indexes. Let R (ϕ ,ϕ ) :=
z 1 2
RIA(ϕ ) RB(ϕ ) with RA(B)(ϕ) := exp(iσA(B)ϕ/2). If classofX-states. Wefirstgenerate100,000randomden-
z 1 ⊗ z 2 z 3 sity matrices according to Hilbert-Schmidt measure [52]
is the optimalmeasurementfor a givenstate ρ, then
M˜i(ϕ ) = R (ϕ ) R†(ϕ ) is the optimal measure- and then project them into the X-state subspace via
mMeint 1for the zstat1eMρ˜(ϕi 1,zϕ21) = Rz(ϕ1,ϕ2)ρRz†(ϕ1,ϕ2). ρX = k=1,2EkρEk† with E1 = diag{1,0,0,1} and
E = diag 0,1,1,0 . These random X-states was later
Because ρ˜(ϕ1,ϕ2) is also an X-state, we must have 2 P{ }
˜ (ϕ ) . Remind that is a finite set, the only transformed into the canonical forms via local unitary
Mi 1 ∈ M M transformations. We numerically find optimal measure-
possibility is that all the are invariant under the op-
i
M mentstominimizethequantumconditionalentropy,uti-
erationR (ϕ )witharbitraryϕ ,i.e.,theonlypossibility
z 1 1 lizing the general expression of the quantum conditional
of Mi is {(11±σ3A)/2}. In the second step, we disprove entropy, see Eq. (B9) in Appendix B. If there are more
this only possibility. From Sec. IVB, we already know
than one optimal measurement, we chose the one clos-
that for Bell-diagonal states, there are three possibility
est to the MCDM, because we concern on how to find
oftheoptimalmeasurement: (11 σA)/2 fori=1,2,3.
{ ± i } anoptimal measurementbut notall of the optimal mea-
These three measurements constitute the minimal uni-
surements. Our numerical results in Table I show the
versal finite set of possible optimal measurement
MBD probability that the MCDM would be the optimal mea-
forBell-diagonalstates. BecauseBell-diagonalstatesare
surement is about 99.40%, and the second preference of
a subclass of the X-states, must be a subset of
MBD the optimal measurement is given by n=(0,1,0)T.
, but actually it is not. This disproves the only possi-
M In Table I, it seems that the value of θ for the optimal
bility derived in the first step. So the above theorem is
measurement is always π/2, however, a counterexample
proved.
does exist and is already given in the Eq. (18).
The above theorem implies for the entire class of X-
states, the optimization procedure involved in the clas-
sical correlation must be state-dependent. This result is D. arbitrary two-qubit states
opposite to that of Ref. [5], where the authors gave a
universal finite set of candidates and sought the optimal For arbitrary two-qubit states, the optimization pro-
measurement in this set. In Ref. [54], we show the con- cedure involved in the quantum discord is unreachable
straint missed in Ref. [5]. To elucidate this, we also give up to now. The MCDM solves this optimization for
5
the states with zero-discord and with maximally mixed
marginals. Besides, the MCDM hits the optimal mea- 11
surements for the most of the X-states. So what about DD˜((ρρ))
the performance of the MCDM for arbitrary two-qubit 00..88
DD˜((ρρ))
states?
00..66
0.5 0.5
0.4 0.4
00..44
0.3 0.3
0.2 0.2 00..22
0.1 0.1
00
0 0 00 00..22 00..44 00..66 00..88 11
0 0.5 1 −0.5 0 0.5 qq
θ/π φ/π
FIG. 2: MCDM-based discord and quantum discord for the
FIG.1: Probabilitydistributionoftheoptimalmeasurements statesρ(q)=(1−q)|ψ0ihψ0|+q|ψ1ihψ1|,where|ψ0iand|ψ1i
for the 100,000 random states according to Hilbert-Schmidt are given byEq. (21).
measure [52],in the canonical form. Here we discretized the
domain of θ and φ into 100 sections.
We generate 100,000 random density matrices of two
qubit according to Hilbert-Schmidt measure [52] and
transform them into the canonical form. Then we nu-
merically investigate the probability distribution of the
optimalmeasurementcharacterizedbytwoangleθ andφ
viaEq.(4). InFig.1,weshowthattheMCDMisindeed
thepreferenceoftheoptimalmeasurements. Meanwhile,
the optimal measurements are centralizedin the vicinity
oftheMCDM.ThismotivatesustoconsidertheMCDM
as an alternative of the optimal measurement accessing
the classical correlation.
V. MCDM-BASED DISCORD FIG. 3: MCDM-based discord D˜ versus D, for 100,000 ran-
dom density matrices according tothe Hilbert-Schmidtmea-
The centralization of the optimal measurements in sure[52]. Thevarianceavg[(D˜−D)2]≃3.7433×10−5,where
the vicinity of the MCDM motivates us to introduce a avg[·] means the average taken over all the random density
MCDM-baseddiscordforanarbitrarytwo-qubitstateas matrices.
follows
D˜(ρ):=S(ρA)−S(ρ)+S{Π˜Ak}(B|A), (20) awiptrhotdhuectprsotabtaeb|iψlit0yi1andqaamndaxqimreaspleecnttiavneglyl.edSpsteactifiec|aψl1lyi,,
−
we choose ψ and ψ as follows
where S{Π˜Ak}(B|A) is the quantum conditional entropy | 0i | 1i
p S(ρB) based on the MCDM Π˜A given by 1
Eqk. (k10). kBecause the MCDM is in the{sekt}of von Neu- |ψ0i = √2(|00i+|10i),
P
mannmeasurementoverwhichtheoptimizationinvolved
1
in the quantum discord is taken, ˜(ρ) will be not less ψ1 = (01 + 10 ). (21)
D | i √2 | i | i
than the quantum discord (ρ). In other words, ˜(ρ) is
D D
an upper bound of the quantum discord (ρ). If ˜(ρ) InFig.2,weshowthatthe MCDM-baseddiscordisvery
D D
is close enough to (ρ), the MCDM-based discord will close to the quantum discord, which suggest that it can
D
be a goodapproximationofthe quantumdiscord. Inthe be taken as a good approximation to the quantum dis-
following, we investigate how close is the MCDM-based cord.
discord to the quantum discord. For more convincing arguments, we investigate the
First, for simplicity, we consider a family of states MCDM-based discord and the quantum discord for ran-
ρ(q)=(1 q)ψ ψ +q ψ ψ , whicharemixturesof dom states. It is shown that for the most of the ran-
0 0 1 1
− | ih | | ih |
6
dom states, the MCDM-based discord is very close to are the ones which can be written in the form
the quantum discord, see Fig. 3. This demonstrate the
efficientness of the MCDM-based discord. ρ= pkΠAk ⊗ρBk (A4)
k
X
with p =Tr(ΠAρ) and ρB =Tr (ΠAρ)/p , see Ref. [3].
VI. CONCLUSION k k k A k k
With Eq. (A2), p and ρB can be obtained via
k k
Inconclusion,wehaveinvestigatedtheprobabilitydis-
tribution of the optimal measurement accessing classical 3
1
correlationand show that the optimal measurements for p ρB = (α τ )σB,
1 1 4 i ij j
arbitrarytwo-qubitstatearecentralizedinthevicinityof i,j=0
X
theMCDM.WehaveprovedthattheMCDMisthe very
3
1
optimalmeasurementforthestateswithzero-discordand p ρB = (β τ )σB. (A5)
the ones with maximally mixed marginals. Besides, we 2 2 4 i ij j
i,j=0
X
have proposed the MCDM-based discord as an upper
bound of quantum discord, and demonstrated that the Substituting Eqs. (A2) and (A5) into the above form of
MCDM-based discordcould be a good approximationof the zero-discordstate (A4), we get
quantum discord.
3 3
1
τ σA σB = (α α +β β )τ σA σB. (A6)
ij i ⊗ j 2 i k i k kj i ⊗ j
Acknowledgments ij=0 ijk=0
X X
Because σA σB is a set of orthogonalbasis of opera-
X. Wang is supported by National Natural Science { i ⊗ j }
tors on A B, we get
FoundationofChina(NSFC)withGrantsNo. 11025527,
H ⊗H
No. 10874151, and No. 10935010. Z.-J. Xi is supported
bytheSuperiorDissertationFoundationofShaanxiNor-
1
mal University (S2009YB03). τ = (ααT +ββT)τ. (A7)
2
where α = (α ,α ,α ,α )T and β = (β ,β ,β ,β )T.
0 1 2 3 0 1 2 3
Appendix A: zero-discord state in Fano-Bloch The matrix τ can be further decomposed into
representation
Here, we derive the condition of zero-discord states in τ = 1 bT , (A8)
the Fano-Bloch representation. An arbitrary two-qubit a R
(cid:20) (cid:21)
state can be written in the Fano-Blochrepresentationas
where a, b are column vectors and R are 3 3 matrix.
follows [53]:
×
CombiningEqs.(A3),(A7)and(A8),weobtainthecon-
dition of zero-discord states with the Fano-Bloch repre-
3 sentation as follows
1
ρ= τ σA σB (A1)
4 ij i ⊗ j
ij=0
X
nnTa = a, (A9)
whereσA(B) =11A(B) isthe2 2identityoperator,σA(B) nnTR = R, (A10)
0 × 1,2,3
is Pauli matrices. Meanwhile, a von Neumann measure-
ment performed on A is characterized by a set of von where n is a unit column vector. The existence of such
Neumann operators n satisfying the above equations is a sufficient and nec-
essary condition for the zero-discord.
3 3
1 1
ΠA = α σA, ΠA = β σA. (A2)
1 2 k k 2 2 k k Appendix B: a general expression of the quantum
Xk=0 kX=0 conditional entropy
The coefficients α and β are given by
k k
In the following, we give a general expression of the
quantum conditional entropy in the Fano-Bloch repre-
α =β =1, α = β =n for k =1,2,3, (A3) sentation. For the states (A1) and the von Neumann
0 0 k k k
−
measurement (A2), the remaining state of the system B
where n is the k-th component of the unit vector n = with measurement result k = 1,2 of A and the corre-
k
(n ,n ,n )T on Bloch sphere. The zero-discord states sponding probability are given by Eq. (A5).
1 2 3
7
To obtainthe eigenvaluesofρB, wefirstgetthe eigen- where X 2 =Tr(XX†)istheHilbert-Schmidtnorm. By
values of p ρB. From Eq. (A5),kwe have introdu|cin|g a new set of parameter as follows
k k
f(n) = aTn, (B7)
2
1 3 1 3 3 g (n) = b RTn , (B8)
eig(p1ρB1)= 4(i=0αiτi0)± 4vuuj=1 i=0αiτij! , (B1) ± (cid:12) ± (cid:12)
X uX X we get the quantum conditio(cid:12)nal entro(cid:12)py [3] as follows
t
2
3 3 3
1 1
eig(p2ρB2)= 4(Xi=0βiτi0)± 4vuuutXj=1 Xi=0βiτij! . (B2) pkS(ρBk)= 1+2 fh 1g++f + 1−2 fh 1g−f
Withthedecompositionform(A8),weobtainthefollow- Xk (cid:18) (cid:19) (cid:18) − (cid:19)
(B9)
ing results
with h(x) := 1+xlog 1+x 1−xlog 1−x. Then the
1 b+RTn classical correl−atio2n a2nd2th−e qu2antum2 d2iscord can
eig(ρ ) = 1 , (B3) C D
1 2 ± (cid:12)1+aTn(cid:12)! be obtained via
(cid:12) (cid:12)
eig(ρ2) = 21 1± (cid:12)(cid:12)b1−−RaTTnn(cid:12)(cid:12)!, (B4) C = S(ρB)−|mn|=in1Xk pkS(ρBk), (B10)
p1 = 21(1+aTn), (B5) D = S(ρA)−S(ρ)+|mn|=in1 k pkS(ρBk). (B11)
X
1
p = (1 aTn), (B6)
2
2 −
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