Table Of ContentMaximal Holevo quantity based on weak measurements
Yao-KunWang1,2, Shao-MingFei3,4, Zhi-XiWang3, Jun-Peng Cao1,5 andHeng Fan1,5,⋆
1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese
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AcademyofSciences,Beijing100190,China
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2CollegeofMathematics,TonghuaNormalUniversity,Tonghua,Jilin134001,China
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J 3SchoolofMathematicalSciences,CapitalNormalUniversity,Beijing100048,China
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4Max-Planck-InstituteforMathematicsintheSciences,04103Leipzig,Germany
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p 5CollaborativeInnovativeCenterofQuantumMatter,Beijing100190,China
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⋆e-mail:[email protected].
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The Holevo bound is a keystone in many applications of quantum information theory. We
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3 propose “weak maximal Holevoquantity” with weak measurements as the generalization of
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2 the standard Holevo quantity which is defined as the optimal projective measurements. The
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scenarios that weak measurements is necessary are that only the weak measurements can
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1 be performed because for example the system is macroscopic or that one intentionally tries
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X to do so such that the disturbance on the measured system can be controlled for example in
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quantum key distribution protocols. We evaluate systematically the weak maximal Holevo
quantity forBell-diagonal statesand find aseries ofresults. Furthermore, wefind that weak
measurements can berealized by noiseandproject measurements.
Weak measurements was introduced by Aharonov, Albert, and Vaidman (AAV) 1 in 1988.
The standard measurements can be realized as a sequence of weak measurements which result
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in small changes to the quantum state for all outcomes 2. Weak measurements realized by some
experimentsis alsovery usefulforhigh-precisionmeasurements3–7.
The quantum correlations of quantum states include entanglement and other kinds of non-
classical correlations. It is well known that the quantum correlations are more general than the
well-studied entanglement 8,9. Quantum discord, a quantum correlation measure differing from
entanglement,is introducedby Oliverand Zurek10 and independentlyby Henderson and Vedral11.
It quantifies the difference between the mutual information and maximum classical mutual infor-
mation,i.e.,itisameasureofthedifferencebetweentotalcorrelationandtheclassicalcorrelation.
Significant developments have been achieved in studying properties and applications of quantum
discord. In particular, there are some analytical expressions for quantum discord for two-qubit
states, such as for the X states 12–17. Besides, researches on the dynamics of quantum discord
in various noisy environments have revealed many attractive features 23–25. It is demonstrated
that discord is more robust than entanglement for both Markovian and non-Markovian dissipative
processes. As with projection measurements, weak measurements are also applied to study the
quantificationof quantumcorrelation. Forexample,thesuper quantumcorrelation based on weak
measurementshas attractedmuch attention18–22.
In general, maximum classical mutual information is called classical correlation which rep-
resents the difference in Von Neumann entropy before and after the measurements11. A similarly
defined quantity is the Holevo bound which measures the capacity of quantum states for classical
communication26,27. The Holevo bound is an exceedingly useful upper bound on the accessible
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informationthat plays an importantrole in many applicationsofquantum informationtheory28. It
isakeystoneintheproofofmanyresultsinquantuminformationtheory29–33.
The maximal Holevo quantity and classical correlation are both classical and based on von
Neumann measurements. Due to the fundamental role of weak measurements, it is interesting
to know how those classical correlations will be if weak measurements are taken into account.
Recently, it is shown that weak measurements performed on one of the subsystems can lead to
“super quantum discord” which is always larger than the normal quantum discord captured by
projective measurements 18. It is natural to ask whether weak measurements can also capture
more classical correlations. In this article, we shall give the definition of “super classical corre-
lation” by weak measurements as the generalization of classical correlation defined for standard
projectivemeasurements. AsthegeneralizationofthemaximalHolevoquantitydefinedforprojec-
tivemeasurements, we propose “weak maximal Holevo quantity” according weak measurements.
Interestingly, by tuning continuously from strong measurements to weak measurements, the dis-
crepancy between the weak maximal Holevo quantity and the maximal Holevo quantity becomes
larger. Such phenomenon also exits between super classical correlation and classical correlation.
In comparison with super quantum discord which is larger than the standard discord, the weak
maximal Holevo quantity and super classical correlation becomes less when weak measurements
are applied, while they are completely the same for projective measurements. In this sense, weak
measurements do not capture more classical correlations. It depends on the specified measure of
correlations. We calculate the maximal Holevoquantity for Bell-diagonal states, and compare the
results with classical correlation. We give super classical correlation and weak maximal Holevo
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quantity for Bell-diagonal states and compare the relations among super quantum correlations,
quantum correlations, classical correlation and super classical correlation and entanglement. The
dynamicbehaviorofweak maximalHolevoquantityunderdecoherence isalso investigated.
Results
Maximal Holevo quantity and weak maximal Holevo quantity. The quantum discord for a
bipartitequantum state ρ with the projection measurements ΠB performed on the subsystem
AB { i }
B isthedifferencebetweenthemutualinformationI(ρ )34 andclassicalcorrelationJ (ρ )11:
AB B AB
D(ρ ) = I(ρ ) J (ρ ),
AB AB B AB
−
where
I(ρ ) = S(ρ )+S(ρ ) S(ρ ),
AB A B AB
−
J (ρ ) = sup S(ρ ) p S(ρ )
B AB A i A|i
{ − }
{Bk} i
X
= S(ρ ) min p S(ρ )
A i A|i
−{ΠBi } i
X
withtheminimizationgoingoverallprojectionmeasurements ΠB ,whereS(ρ) = tr(ρlog ρ)
{ i } − 2
is the von Neumann entropy ofa quantumstate ρ, ρ , ρ are thereduced density matrices of ρ
A B AB
and
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p = tr [(I ΠB)ρ (I ΠB)], ρ = tr [(I ΠB)ρ (I ΠB)].
i AB A ⊗ i AB A ⊗ i A|i p B A ⊗ i AB A ⊗ i
i
The Holevo quantity of the ensemble p ;ρ 33 that is prepared for A by B via its local
i A|i
{ }
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measurementsisgivenby
χ ρ ΠB = χ p ;ρ S( p ρ ) p S(ρ ). (1)
{ AB|{ i }} { i A|i} ≡ i A|i − i A|i
i i
X X
It denotes the upper bound of A’s accessible information about B’s measurement result when B
projects its system by the projection operaters ΠB . The classical correlation in the state ρ is
{ i } AB
defined as themaximalHolevoquantity33 overall localprojectivemeasurementson B’s system:
C (ρ ) maxχ ρ ΠB . (2)
1 AB ≡ {ΠBi } { AB| i }}
Theweak measurementoperators aregivenby2
(1 tanhx) (1+tanhx)
P(x) = − Π + Π ,
0 1
2 2
r r
(1+tanhx) (1 tanhx)
P( x) = Π + − Π ,
0 1
− 2 2
r r
(3)
where x is the measurement strength parameter, Π and Π are two orthogonal projectors with
0 1
Π + Π = I. The weak measurements operators satisfy: (i) P†(x)P(x) + P†( x)P( x) = I,
0 1
− −
(ii)lim P(x) = Π and lim P( x) = Π .
x→∞ 0 x→∞ 1
−
Recently, super quantum discord for bipartite quantum state ρ with weak measurements
AB
on the subsystemB has been proposed 18. Similarly to thedefinition ofquantum discord, wegive
the another form of definition of super quantum discord. We define super classical correlation
Jw(ρ ) for bipartite quantum state ρ with the weak measurements PB( x) performed on
B AB AB { ± }
the subsystem B as follow. The super quantum discord denoted by D (ρ ) is the difference
w AB
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between themutualinformationI(ρ ) and superclassical correlationJw(ρ ), i.e.,
AB B AB
D (ρ ) = I(ρ ) Jw(ρ ),
w AB AB − B AB
where
I(ρ ) = S(ρ )+S(ρ ) S(ρ ),
AB A B AB
−
Jw(ρ ) = sup S(ρ ) S (A PB(x) )
B AB { A − w |{ } }
{Bk}
= S(ρ ) min p(x)S(ρ )+p( x)S(ρ ) , (4)
A A|PB(x) A|PB(−x)
−{P(±x)}{ − }
withtheminimizationgoingoverallprojectionmeasurements ΠB ,
{ i }
S (A PB(x) ) = p(x)S(ρ )+p( x)S(ρ ),
w A|PB(x) A|PB(−x)
|{ } −
p( x) = tr [(I PB( x))ρ (I PB( x))], (5)
AB AB
± ⊗ ± ⊗ ±
tr [(I PB( x))ρ (I PB( x))]
B AB
ρA|PB(±x) = tr [(I⊗ PB(± x))ρ (I⊗ PB(± x))], (6)
AB AB
⊗ ± ⊗ ±
PB(x) isweak measurementoperators performed onthesubsystemB.
{ }
Now, let us define the weak Holevo quantity of the ensemble p( x);ρ based on
A|PB(±x)
{ ± }
weak measurementson thesubsystemB,
χw ρ P( x) = χ p( x);ρ
AB A|PB(±x)
{ |{ ± }} { ± }
= S p( x)ρ p( x)S ρ . (7)
A|PB(±x) A|PB(±x)
± − ±
!
±x ±x
X X (cid:0) (cid:1)
It denotes the upper bound of A’s accessible information about B’s measurements results when B
projects the system with the weak measurements operaters P( x) . The weak maximal Holevo
{ ± }
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quantityoverall local weak measurementsonB’s systemisgivenby:
Cw(ρ ) = max χw ρ P( x) . (8)
1 AB {P(±x)} { AB|{ ± }}
Next,weconsiderthemaximalHolevoquantityandweakmaximalHolevoquantityfortwo-
qubitBell-diagonalstates,
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ρ = (I I + c σ σ ), (9)
AB i i i
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i=1
X
whereI istheidentitymatrix, 1 c 1. Themarginalstatesofρ are ρ = ρ = I.
− ≤ i ≤ AB A B 2
ThemaximalHolevoquantityforBell-diagonalstatesis givenas
C (ρ ) = maxχ ρ ΠB .
1 AB {ΠBi } { AB| i }}
1 C 1+C
= − log(1 C)+ log(1+C), (10)
2 − 2
where C = max c , c , c . We find that themaximal Holevoquantity C (ρ ) equals to the
1 2 3 1 AB
{| | | | | |}
classicalcorrelation J (ρ ),
B AB
C (ρ ) = J (ρ ). (11)
1 AB B AB
Theweak maximalHolevoquantityoftwo-qubitBell-diagonalstates isgivenby
Cw(ρ ) = max χw ρ P( x)
1 AB {P(±x)} { AB|{ ± }}
1 Ctanhx 1+Ctanhx
= − log(1 Ctanhx)+ log(1+Ctanhx). (12)
2 − 2
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Thesuperclassicalcorrelationoftwo-qubitBell-diagonalstatesis givenby
Jw(ρ ) = sup S(ρ ) S (A PB(x) )
B AB { A − w |{ } }
{Bk}
1 Ctanhx 1+Ctanhx
= − log(1 Ctanhx)+ log(1+Ctanhx). (13)
2 − 2
TheweakmaximalHolevoquantityCw(ρ )equalstothesuperclassicalcorrelationJw(ρ ),
1 AB B AB
i.e.,
Cw(ρ ) = Jw(ρ ). (14)
1 AB B AB
Next, we compare the weak maximal Holevo quantity(superclassical correlation), the max-
imal Holevo quantity(classical correlation), super quantum discord, quantum discord, and entan-
glementofformation. Forsimplicity,wechooseWernerstates, c = c = c = z,
1 2 3
−
(1 z)
ρ = z Ψ− Ψ− + − I,z [0,1], (15)
AB
| ih | 4 ∈
where Ψ− = ( 01 10 )/√2. Set z = α . TheWerner stateshavetheform
| i | i−| i 2−α
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ρ = (I αP), (16)
w
2(2 α) −
−
where 1 α 1, I is the identity operator in the 4-dimensional Hilbert space, and P =
− ≤ ≤
2 i j j i is the operator that exchanges A and B. The entanglement of formation
i,j=1| ih | ⊗ | ih |
P
E for the Werner states is given as E (ρ ) = h 1(1+ 1 [max(0, 2α−1)]2) , by h(x)
f f w 2 − 2−α ≡
(cid:16) q (cid:17)
xlog x (1 x)log (1 x).
− 2 − − 2 −
ThemaximalHolevoquantityforwernerstatesis givenby,seeEq. (33) insectionMethod,
1 z 1+z
C (ρ ) = − log(1 z)+ log(1+z). (17)
1 AB
2 − 2
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The weak maximal Holevo quantity for werner states is given by, see Eq. (41) in section
Method,
1 ztanhx 1+ztanhx
Cw(ρ ) = − log(1 ztanhx)+ log(1+ztanhx). (18)
1 AB 2 − 2
Thequantumdiscord forWernerstates isgivenby 12
1 z 1+z 1+3z
D(ρ ) = − log(1 z) log(1+z)+ log(1+3z). (19)
AB
4 − − 2 4
Andthesuperquantumdiscord forWernerstates isgivenby 18
3(1 z) 1 z (1+3z) 1+3z
D (ρ ) = − log − + log
w AB
4 4 4 4
(cid:18) (cid:19) (cid:18) (cid:19)
(1 ztanhx) 1 ztanhx
+1 [ − log −
− 2 2
(cid:18) (cid:19)
(1+ztanhx) 1+ztanhx
+ log ]. (20)
2 2
(cid:18) (cid:19)
In Fig.1 we plot the weak maximal Holevo quantity, the maximal Holevo quantity, super
quantum discord, quantum discord, and entanglement of formation for the Werner state. We find
that super quantum discord , quantum discord, the maximal Holevo quantity and the weak max-
imal Holevo quantity have the relations, D D > J (C ) Jw(Cw). For the case of pro-
w ≥ B 1 ≥ B 1
jection measurements, limx , we have D = D,J (C ) = Jw(Cw). The weak maximal
→ ∞ w B 1 B 1
Holevo quantity approaches to zero for smaller values of x. The weak maximal Holevo quantity
approaches to the maximal Holevo quantity and super quantum discord approaches to quantum
discord forlarger valuesofx. Theweak maximalHolevoquantityand themaximalHolevoquan-
tity are larger than the entanglement of formation for small z and smaller than the entanglement
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of formation for big z. It shows that the weak maximal Holevo quantity and the maximal Holevo
quantitycan not always capture morecorrelation than theentanglementas superquantum discord
and quantumdiscord do.
As a natural generalizationof theclassical mutualinformation,theclassical correlation rep-
resents thedifferencein VonNeumann entropybeforeand afterprojectionmeasurements,i.e.,
J (ρ ) = S(ρ ) min p S(ρ ).
B AB A i A|i
−{ΠBi } i
X
Similarly, the super classical correlation represents the difference in Von Neumann entropy
beforeand after weak measurements,i.e.,
Jw(ρ ) = S(ρ ) min p(x)S(ρ )+p( x)S(ρ ) .
B AB A −{P(±x)}{ A|PB(x) − A|PB(−x) }
As weak measurements disturb the subsystem of a composite system weakly, the infor-
mation is less lost and destroyed by weak measurements on the subsystem alone. That is the
physical interpretation that the super classical correlation is smaller than the classical correla-
tion, Jw(Cw) J (C ). According this fact, we can infer that weak measurements can cap-
B 1 ≤ B 1
ture more quantum correlation than the projection measurements. In fact, the super quantum
correlation D (ρ ) = I(ρ ) Jw(ρ ) is lager than the quantum correlation D(ρ ) =
w AB AB − B AB AB
I(ρ ) J (ρ ). And there is a similarity to the Holevo quantity which measures the capacity
AB B AB
−
ofquantumstates forclassicalcommunication.
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