Maximally discordant mixed states of two qubits

8.9 MB·English

by  Fernando Galve

Maximally discordant mixed states of two qubits
Fernando Galve, Gian Luca Giorgi, and Roberta Zambrini
IFISC (UIB-CSIC), Instituto de F´ısica Interdisciplinar y Sistemas Complejos,
UIB Campus, E-07122 Palma de Mallorca, Spain
(Dated: January 17, 2011)
Westudytherelativestrengthofclassicalandquantumcorrelations,asmeasuredbydiscord,for
two-qubit states. Quantum correlations appear only in the presence of classical correlations, while
thereverseisnotalwaystrue. Weidentifythefamilyofstatesthatmaximizethediscordforagiven
value of the classical correlations and show that the largest attainable discord for mixed states is
greater than for pure states. The difference between discord and entanglement is emphasized by
the remarkable fact that these states do not maximize entanglement and are, in some cases, even
separable. Finally, by random generation of density matrices uniformly distributed over the whole
Hilbert space, we quantify the frequency of the appearance of quantum and classical correlations
1 for different ranks.
1
PACSnumbers: 03.67.Mn,03.65.Ta
0
2

n I. INTRODUCTION the difference I((cid:37))−J((cid:37)):
a
J δ ((cid:37))= min (cid:2)S((cid:37) )−S((cid:37))+S(A|{ΠB})(cid:3), (1)
4 One of the most striking features of quantum mechan- A:B {ΠB} B i
i
1 ics is entanglement, first considered (although not by
that name) by Einstein, Podolsky, and Rosen in their that is, when measurement is performed in the basis

] seminal paper in 1935 [1]. This is an exclusively quan- which disturbs the state the least. A complementary ap-
h
p tum feature of composite states that can not be written proach was described in Ref. [5], defining classical corre-
- as mixtures of product states. Theoretical and experi- lations and showing that total correlations given by the
t
n mentalresearchactivitytocharacterizeentanglementhas mutual information are actually larger thAn the sum of
a beenparticularlyintenseinthelastdecade(seereview[2] the classical correlations and entanglement E [8]. As a
u andreferencestherein),beingpartofabroaderendeavor matter of fact, the quantum mutual information can be
q
to explore distinctive aspects of quantum versus classi- seen as the sum of quantum correlations [4] δ ((cid:37)) and
[ A:B
2 cpaulrppohsyessic[s3a].ndAnnoivmelproerstoaunrtceisssfuoercqounasnidtuemredinbfoyrmseavteiroanl tmhaerkcltahssaitc,ailnc[o4r],retlhaetidoinscso[r5d]imsdaxefi{ΠnBied}Jin(t(cid:37)e)r{mΠBjs}o.foWrtehorge--
v authors[4–6]istheexistenceofquantumcorrelationsbe-
onal(perfect)measurements. Evenifpossiblegeneraliza-
4 yond entanglement, in separable states. As a matter of
tionstopositive-operator-valuedmeasurements(POVM)
7
fact,examplesofimprovedquantumcomputingtasksnot
1 were considered at the end of that paper, as well as in
relying on entanglement have been reported [7].
2 [5],calculationsofdiscordintheliteraturegenerallycon-
. sider only orthogonal measurements (see, e.g., [4, 7, 9–
7
11, 16, 17]).
0
0 In contrast with state separability, this new paradigm
1 II. QUANTUM CORRELATIONS: THE of quantumness of correlations is measurement oriented,
: DISCORD considering an experiment where all information of a
v
i system A is extracted by measuring another system B.
X
Two complementary approaches on quantum correla- According to this measure, a state is classically corre-
r tions are receiving great attention [4, 5]. In Ref. [4], lated only when consecutive measurements of system B
a
quantum correlations (quantum discord) have been as- yield the same picture of the state of system A, which
sociated to the difference of two classically equivalent is achieved after decoherence into the pointer basis of
expressions for the mutual information, I and J. In B [4, 9]. For pure states, quantum discord is equiva-
particular, the quantum mutual information is defined lent to entanglement and actually has the same value of
as I((cid:37)) = S((cid:37) ) + S((cid:37) ) − S((cid:37)), where S stands for classicalcorrelations[5]. Ontheotherhand,whenmixed
A B
the von Neumann entropy and (cid:37) is the reduced den- statesareconsidered,entanglementdoessignificantlyde-
A(B)
sity matrix of each subsystem. The classically equiva- partfromthequantumdiscord,thedifferencebeingpos-
lentexpressionstemmingfromBayesruleisJ((cid:37)) = itive for some states and negative for others [10]. As the
{ΠB}
j definition of discord comes from a minimization over all
S((cid:37) )−S(A|{ΠB}),withtheconditionalentropydefined
as SA(A|{ΠB}) =j (cid:80) p S((cid:37) ), p = Tr (ΠB(cid:37)), and possible measurement basis, only a few general results
j i i A|ΠB i AB i have been reported. Analytic expressions are known for
i
where (cid:37)A|ΠB = ΠBi (cid:37)ΠBi /pi is the density matrix after a statesoftwoqubitswithmaximallymixedmarginals[10],
i
complete projective measurement ({ΠB}) has been per- for X-shaped states [11], and also for Gaussian states of
j
formed on B. Quantum discord is obtained minimizing continuous variable systems [12].
2
In this paper, we explore the whole Hilbert space of thus consider states
twoqubitstogaininsightontheircorrelationsformixed
states of different ranks. Our main goal is to discern (cid:37)=(cid:15)|Φ+(cid:105)(cid:104)Φ+|+(1−(cid:15))|01(cid:105)(cid:104)01| (2)
the proportion of quantum to classical correlations be- √
tweenthetwoqubits. Wefindthemostnonclassicaltwo- with |Φ+(cid:105) = (|00(cid:105)+|11(cid:105))/ 2 as usual. With local uni-
qubit states, i.e., the family with maximal quantum dis- tary operations, which leave discord invariant, we can
cord versus classical correlations, were formed by mixed obtain from this ansatz any combination of a Bell state
states of rank 2 and 3, which we name maximally dis- mixedwithacomputationalbasisstateofoppositeparity
cordant mixed states (MDMS). The analogous effort to (number of 1’s in the state). The expression of discord
identify largest deviations from classical states has led for states (2) is invariant under permutation of the indi-
to mixed states maximizing entanglement versus purity vidual labels A ↔ B. As a matter of fact, we find that
[13]. In contrast with maximally entangled mixed states states (2) maximize the symmetrized version of discord
(MEMS), where a geometrical property such as separa- (δA:B +δB:A)/2 for all rank-2 matrices.
bility could be considered with several constraints, the
MDMS are naturally defined and allow us to quantify
the relative strength of quantum and classical correla-
tions, which are related in a closed from. The MDMS
proposedheredonotmaximizeentanglementforagiven
amountofclassicalcorrelations;partofthemare,infact,
separable. Thispinpointsthefundamentaldifferencebe-
tween entanglement and discord for mixed states, in op-
position to their exact equivalence for pure states. Their
discordalsoimpliesthatquantumcorrelationsarealways
accompanied by classical correlations, while the reverse
is not always true. Furthermore, we study the proba-
bility of states with a given amount of discord in the
whole two-qubit Hilbert space, supporting the recent re-
sult that the closed set of purely classically correlated
states (δ = 0) has measure zero [14], and we com-
A:B
pare it with the probability for classical correlations and
entanglement. Only such states with no discord have
been shown to ensure a future non-negative evolution in FIG. 1: (Color online) Quantum discord (δA:B) versus clas-
sical correlations (J) for two-qubit states. The MDMS fam-
the presence of dissipation [15], while discord can not be
ily (continuous line) gives two segments for rank 3 ((cid:37)(R3))
made zero in a finite time by any Markovian map [14].
and one for rank 2 ((cid:37)(R2)). Layers of 108 random matrices
Furthermore, even in the presence of a noisy environ-
ofrank2(darkpoints),3(intermediatecolor),and4(lighter
ment, for some family of initial states, discord can be
color)aresuperimposed. Forpurestates(dotted-dashedline),
robust under decoherence for a finite time [16]. Experi-
δ =J =E.
mental results with polarization entangled photons also A:B
have been reported recently [17].
Itcanbeshownthat,whenwecomparewithanumer-
ical scan of Hilbert’s space, this state is too symmetric.
In fact, a better option in terms of discord is obtained if
III. MIXED STATES WITH LARGEST some amount of entanglement is sacrificed for the good
DISCORD of quantum correlations. This results from asymmetriz-
ing the maximally entangled (Bell) state leading to the
As pure states with maximum entanglement that are ansatz
alsomaximallydiscordant,Bellstatesareanaturalstart-
ing point to identify states with large discord; thus, by (cid:37)(R2) =(cid:15)|Φ˜+(cid:105)(cid:104)Φ˜+|+(1−(cid:15))|01(cid:105)(cid:104)01|, (3)
mixing these states with other components, it is quite √ √
plausible that we find states with a large proportion of with |Φ˜+(cid:105)= p|00(cid:105)+ 1−p|11(cid:105)), which coincides with
quantum versus classical correlations. We then consider a Bell state for p = 1/2. The increase of discord for
an example of a mixture of any Bell state |ψ(cid:105) with an- the states (3) with respect to (2) highlights the im-
other orthogonal pure state, i.e., (cid:37) = (cid:15)|ψ(cid:105)(cid:104)ψ| + (1 − portance of the asymmetric definition of quantum dis-
(cid:15))|φ(cid:105)(cid:104)φ|. If |φ(cid:105) is any other Bell state, then δ = E cord, based on the asymmetric operation of measuring
A:B
(with E being entanglement) and J = 1, and a worse B in order to know about A. The discord for this fam-
discordisfoundthaninthatofpurestateswiththesame ily can be written once we know the conditional en-
classical correlation J. In contrast, a mixture of a Bell tropies min√{ΠBi }S(A|{ΠBi }) = (xlog2 11−+xx − log2y)/2,
state and a state of the computational basis of the op- with x = 1−4y and y = (cid:15)(1 − p)(1 − (cid:15)), while for
posite parity sector, gives a huge amount of discord. We δ weneedtousey =(cid:15)p(1−(cid:15)). Thetotalandreduced
B:A
3
entropies are easy to calculate once we notice that the
ansatz is given in spectral decomposition, though we do
not give the whole expression for reasons of space.
The family of MDMS is obtained for an optimal func-
tion (cid:15) (p) through the use of Lagrange multipliers, as
opt.
detailed later in this paper. Once this optimal curve
(cid:15) (p) is used, Eq. (3) gives the family of states that
opt.
maximize the quantum part of correlations for a given
classical part , when all rank 2 (R2) states are consid-
ered. As shown in Fig. 1, these (cid:37)(R2) states are the
MDMS for a large range of classical correlations.
In order to find the states that maximize δ , we
A:B
also need to use rank-3 states. In this case, it can be
checked that asymmetrization of the Bell-state compo-
nent |φ(cid:105) does not help, and that the best choice for an
ansatz is
(cid:37)(R3) =(cid:15)|Φ+(cid:105)(cid:104)Φ+|+(1−(cid:15))(m|01(cid:105)(cid:104)01|+(1−m)|10(cid:105)(cid:104)10|). FIG.2: (Coloronline)Entanglement(E)vsclassicalcorrela-
(4) tions (J). Random matrices and lines for the MDMS family
Asbefore,anycombinationofBellstateplustwocompo- asinFig. 1. MDMSofrank3areseparable,while(cid:37)(R2)states
nents of opposite parity belonging to the computational showlargeentanglement,evenifnotmaximum,asseeninthe
inset.
basis, will do. The optimal (cid:15) (m) is discussed later
opt.
and leads to the family of states (cid:37)(R3) maximizing dis-
cord for small classical correlations. This optimal fam- [4]. The angles minimizing Eq. (1) for the states (3),
ily has the property of being separable (not entangled), and thus giving the correct discord, are θ =π/4+nπ/2
as shown in Fig. 2, while it maximizes quantum dis- and any value of φ. We then can choose φ = 0 and
cord, highlighting the inequivalence of these measures of hence the optimal projectors are ΠB = |+B(cid:105)(cid:104)+B| and
1 √
quantumness. It is actually found that, for these states, ΠB =|−B(cid:105)(cid:104)−B|, with |±(cid:105)=(|0(cid:105)±|1(cid:105))/ 2.
2
the discord amounts to the weight of the Bell compo- ThemethodofLagrangemultipliersallowsustomaxi-
nent, and the simple relation δA:B = δB:A = (cid:15) holds. mizefirstthediscordfortherank-2familyδA:B((cid:37)(R2))≡
For completeness, the entanglement (as quantified by δ((cid:15),p) while keeping its classical correlations J((cid:37)(R2))≡
(cid:112)
the concurrence [8]) yields E((cid:37)(R2)) = 2(cid:15) p(1−p) and J ((cid:15),p) constant (and the same for the rank-3 family
0
E((cid:37)(R3)) = max[0,(cid:15)−2(1− (cid:15))(cid:112)m(1−m)]. Although (cid:37)(R3)). This is achieved through definition of the func-
MDMSofrank3areseparable,theMDMSofrank2have tion Λ((cid:15),p,λ) = δ((cid:15),p)+λ(J((cid:15),p)−J ) where λ is the
0
ahighamountofentanglement,eveniftheydonotmax- Lagrange multiplier, and J is an arbitrary but fixed
0
imize it (Fig. 2). As mentioned before, asymmetrization amount of classical correlation. The extremization pro-
of the Bell component increases quantum correlations at cedure is then simply the simultaneous solution of the
the expense of entanglement. three equations ∂ Λ=0, with µ=λ,(cid:15),p. From the first
µ
An intriguing state is that of the singular point for equation, J((cid:15),p) = J is obtained, as expected. From
0
(cid:37)(R3) shown in Fig. 1: (cid:37)cusp = (|Φ+(cid:105)(cid:104)Φ+|+|01(cid:105)(cid:104)01|+ the last two equations, we can isolate λ yielding the ex-
|10(cid:105)(cid:104)10|)/3 reaches the lowest possible purity for a rank- tremality condition
3 state and is separable, yet with a high level of discord.
∂ δ/∂ J =∂ δ/∂ J. (5)
Another important feature emerging from Fig. 1 is that (cid:15) (cid:15) p p
MDMShaveadiscordlargerthanpurestates(δ >J)
A:B We stress that this condition is equivalent to maximiza-
and satisfy J =0 only when δ =0, thus showing the
A:B tion of δ versus I, or minimization of J with respect to
lackofstateswithfinitequantumwithoutclassicalcorre-
I, due to the closed relation I = δ+J. These quanti-
lations. In other words, no state of two qubits is purely
ties present nontrivial trigonometric relations, leading to
quantum. Maybe less surprisingly, there are no states
a transcendental equation the solution of which can only
with finite entanglement and zero classical correlations
be given numerically.
(Fig. 2).
The same procedure is followed for the rank-3 family
(cid:37)(R3), with m playing the role of p. In this case, obtain-
ing (cid:15) (m) is a bit trickier, due to the fact that there
opt.
IV. DETAILED ANALYSIS
are two optimal angles (each of them good for different
ranges of (cid:15) and m), θ = 0,π/4; the angle φ is again
We first consider the commonly accepted definition non important. We can consider for the moment that
of discord, obtained expressing the measurement pro- projector maximization of discord has been simplified to
jectors as ΠBj = |ψjB(cid:105)(cid:104)ψjB|, j = 1,2, with |ψ1B(cid:105) = δA:B((cid:37)(R3))=min(δ0,δπ/4), the latter being functions of
cosθ|0(cid:105)+eiφsinθ|1(cid:105) and |ψB(cid:105)=−e−iφsinθ|0(cid:105)+cosθ|1(cid:105) (cid:15) and m. The goal is to find the zero(es) of the function
2
4
∂ δ/∂ J −∂ δ/∂ J, which of course needs the knowl- to lower ranks, we observe that rank 2 matrices yield
(cid:15) (cid:15) m m
edgeofwhentouseoneangleortheother. However,the ∼ 10.76%, and rank 3 yield ∼ 16.3%. Finally, as shown
problem is greatly reduced by noticing that the latter in the insets in Fig. 3, the border of MDMS seems to be
function is positive when using δ and negative when us- ratherimprobabletofindinthespaceoftwoqubitstates,
0
ingδ . Thismeansthatthezeroofsuchfunctionoccurs except for the middle branch in the cusp, meaning that
π/4
exactly (and conveniently) when δ ((cid:15),m) = δ ((cid:15),m). such extremely nonclassical states are quite rare.
0 π/4
Again, the solution to this transcendental equation can
only be given numerically.
Finally, the MDMS are a family of states
(cid:37)MDMS =(cid:15)|Φ˜+(cid:105)(cid:104)Φ˜+|+(1−(cid:15))(m|01(cid:105)(cid:104)01|+(1−m)|10(cid:105)(cid:104)10|),
(6)
where the optimum choice of parameters gives the three
curves in Fig. 1, two of them rank 3 and the other rank
2. The first curve, going from zero discord up to the
cusp, is the rank 3 family (cid:37)(R3) with (cid:15) (m) given by
opt.
the solution of δ = δ . It is restricted to the domain
0 π/4
m ∈ [0,1], (cid:15) ∈ [0,1/3]. The second branch of MDMS is
givenby(cid:37)(R3)withm=1/2withdomain(cid:15)∈[1/3,0.385],
approximately. These two curves correspond to separa-
ble states, as noted above (Fig. 2). The remaining curve
of MDMS is the rank 2 family (cid:37)(R2) when the optimal
function (cid:15) (p) given by Eq. (5) is used, and for (cid:15) ap-
opt.
proximately in the interval [0.408,1]. One might wonder
how thepicture changes ifmore general (nonorthogonal)
measurements are considered. It has been shown that,
fortwoqubits,discordisextremizedexclusivelybyrank-
1 POVMs with a maximum of four elements [21, 22].
Perfect orthogonal measurements correspond to the case FIG. 3: (Color online) Probability (density) to find a two-
qubitstatewithagivenamountofquantumdiscordδ,classi-
with the two elements considered above. Considering
calcorrelationsJ,andentanglementErespectively,forranks
POVMs with measurement operators E , the measured
i 2 (dashed), 3 (dot-dashed), and 4 (dot-dot-dashed). The in-
densitymatrixtakestheform(cid:37) =E (cid:37)/p withprob-
A|Ei i i sets show these probabilities (larger for light color) for the
abilityp =Tr (E (cid:37)). Evenusingthegeneralmeasure-
i AB i quantityunderstudy(y axis)againstclassicalcorrelationsJ
mentgivenbyPOVMsoffourelements,wefindthesame (x axis), for different ranks.
discord for the MDMS, meaning that they represent the
absolute border of maximally nonclassically correlated
states of two qubits. A detailed analysis about the full
Hilbert space will be presented elsewhere [23].
VI. CONCLUSIONS
Theuniquefamilyoftwo-qubitmixedstateswithmax-
V. STATISTICS imal proportion of quantum discord versus classical cor-
relations,theMDMSinEq.(6),hasbeenidentified. Part
Sinceourrandomgenerationofdensitymatricesisuni- of them have rank 2 and are highly, although not max-
form in the Hilbert space, preserving the Haar measure imally, entangled, while the other part has rank 3 and
[18,19],wecanmeasurethefrequencyoftheappearance is separable, thus providing another evidence of the in-
of states with different properties, as shown in Fig. 3 for equivalence of these two measures of quantumness. We
different ranks. Some main features arise for all quan- have shown that the presence of discord is a sufficient
tities investigated: (i) Zero correlations (be they quan- but not necessary condition to have nonvanishing classi-
tum,classical,ormutualinformation)havezeroprobabil- cal correlations. The uniform generation of states (ran-
ity. Notably, entanglement is the exception (consistently dom states preserving Haar measure) allowed us to find
withRef.[19]),whereonlyrank-2stateshaveaprobabil- theprobabilitiesandtypicalvaluesofclassicalandquan-
ity zero of being separable. (ii) The lower the rank, the tum correlations, as well as entanglement. We verified
higher the typical amount of correlations. This is quite that completely (either quantum or classical) uncorre-
understandable, since higher ranks describe more mixed latedstatesareveryrare,aswellasextremenonclassical
states. (iii) It is more probable to find states with more states. The identification of MDMS, together with the
abundanceofclassicalratherthanquantumcorrelations. ability to experimentally generate [20] and characterize
We note that only ∼ 7.45% of the two-qubit states has [17]thesestates, isakeytooltoestablishthefundamen-
greater discord than classical correlations. If restricted tal difference in performance of quantum versus classical
5
information [7]. Juan de la Cierva program are acknowledged.
Acknowledgments
Funding from FISICOS (FIS2007-60327), ECuSCo
(200850I047), and CoQuSys (200450E566) projects and
[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 042105 (2010).
777 (1935). [12] P. Giorda and M. G. A. Paris, Phys. Rev. Lett. 105,
[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. 020503(2010);G.AdessoandA.Datta,ibid.105,030501
Horodecki, Rev. Mod. Phys. 81, 865 (2009). (2010).
[3] M. A. Nielsen and I. L. Chuang, Quantum Computation [13] S. Ishizaka and T. Hiroshima, Phys. Rev. A 62, 022310
andQuantumInformation(CambridgeUniversityPress, (2000); F. Verstraete, K. Audenaert, and B. De Moor,
Cambridge, U.K., 2000). ibid. 64, 012316 (2001).
[4] H.OllivierandW.H.Zurek,Phys.Rev.Lett.88,017901 [14] A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti,
(2001). and A. Ac´ın, Phys. Rev. A 81, 052318 (2010).
[5] L.HendersonandV.Vedral,J.Phys.A34,6899(2001). [15] C.A. Rodr´ıguez-Rosario, K. Modi, A. Kuah, A. Shaji,
[6] M.Horodecki,K.Horodecki,P.Horodecki,R.Horodecki, andE.C.G.Sudarshan,J.Phys.A41,205301(2008);A.
J.Oppenheim,A.Sen(De),andU.Sen,Phys.Rev.Lett. Shabani and D. A. Lidar, Phys. Rev. Lett. 102, 100402
90, 100402 (2003). (2009).
[7] E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672 [16] L.Mazzola,J.Piilo,andS.Maniscalco,Phys.Rev.Lett.
(1998); S. L. Braunstein, C. M. Caves, R. Jozsa, N. Lin- 104, 200401 (2010).
den,S.Popescu,andR.Schack,ibid.83,1054(1999);D. [17] J.-S. Xu, X.-Y. Xu, C.-F. Li, C.-J. Zhang, X.-B. Zou,
A. Meyer, ibid. 85, 2014 (2000); E. Biham, G. Brassard, G.-C. Guo, Nature Commun. 1, 7 (2010).
D.Kenigsberg,andT.Mor,Theor.Comput.Sci.320,15 [18] K. Zyczkowski and M. Kus, J. Phys. A 27, 4235 (1994).
(2004);A.Datta,A.Shaji,andC.M.Caves,Phys.Rev. [19] K.Zyczkowski,P.Horodecki,A.Sanpera,andM.Lewen-
Lett.100,050502(2008);B.P.Lanyon,M.Barbieri,M. stein,, Phys. Rev. A 58, 883 (1998).
P. Almeida, and A. G. White, ibid. 101, 200501 (2008); [20] F.F.Fanchini,L.K.Castelano,andA.O.Caldeira,New
V. Vedral, Found. Phys. 40, 1141 (2010). J. Phys. 12, 073009 (2010).
[8] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [21] A. Datta, PhD thesis, arXiv:0807.4490.
[9] J. Maziero, L. C. C`eleri, R. M. Serra, and V. Vedral, [22] G. M. D’Ariano, P. Perinotti and P. Lo Presti, J. Phys.
Phys. Rev. A 80, 044102 (2009). A 38, 5979 (2005).
[10] S. Luo, Phys. Rev. A 77, 042303 (2008). [23] F. Galve, G. L. Giorgi and R. Zambrini, in preparation.
[11] M. Ali, A. R. P. Rau, and G. Alber, Phys. Rev. A 81,