Table Of ContentCalculation of quantum discord in arbitrary dimensions, especially for X- and other
specialized states
A. R. P. Rau∗
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
(Dated: January 24, 2017)
Quantum discord, a kind of quantum correlation based on entropic measures, is defined as the
difference between quantum mutual information and classical correlation in a bipartite system.
Procedures are available for analytical calculation of discord when at least one of the parties is a
qubitwithdimensiontwo. Weextendnowtosystemswhenbothpartiesareoflargerdimension,of
interest to qudit-quDit with d,D ≥ 3 or spin chains of spins ≥ 1. Unitary operations involved are
reducedtothemostcompactformsoastominimizethenumberofvariablesneededforextremizing
7
theclassical correlation that isthemost difficult part of thediscord calculation. Resultsare boiled
1
0 down to a simple recipe for the extremization that for some classes of density matrices even gives
2 trivially the final value without that extremization. A qutrit-qutrit (d = 3) system is discussed in
detail with specific applications to density matrices for whom results are available for an unrelated
n
butsimpler-to-evaluatecorrelationtermedgeometricdiscordandforwhomtheentropicdiscordhas
a
involved difficult numerics. Special attention is given to the so-called X-states and Werner and
J
pseudo-pure states when the calculations become particularly simple. Systematics for X-states of
0
arbitrary dimension are discussed in an appendix.
2
PACSnumbers: 03.65.Ta,03.67.-a
]
h
p
- I. INTRODUCTION X-states,theirsymmetryandnumber,inanydimension.
t
n
The arrangement of this paper is as follows. Section
a
Quantum correlations other than entanglement [1]
u II reviews the procedure for calculating discord when a
have increasingly been discussed in quantum informa-
q qubitisinvolved[6]. Thispresentationisfocussedonone
[ tion. Among them, quantum discord has attracted at- of our primary concerns, to reduce to a minimum the
tention and is calculated in terms of von Neumann en- number of parameters that need to be varied for max-
1
tropiesofdensitymatrices[2]. Thereby,itsdefinitionand
v imizing the classical correlation, so as to point to the
calculation is in principle available for bipartite systems
2 same for higher dimensions. Many-parameter variation
AB of arbitrary dimension whereas entanglement mea-
2 can be tedious so that reducing the number of them is
9 sures such as concurrence or negativity have not been very important. Section III presents such a procedure
5 establishedbeyond qubit-qubit and qubit-qutrit (dimen-
andillustrateswith the example ofa qutritofdimension
0 sion three) systems [3]. On the other hand, the tricky
3. Section IV applies the method to compute quantum
1. part of computing discord lies in considering all possible discord for some qutrit-qutrit density matrices, of both
0 projective measurements on one of the parties to get a
pure and mixed states, of interest. An alternative geo-
7 conditional entropy of the other, thereby accounting for
metric discord has been presented for some of these [9]
1 all classical correlations between the parties. Quantum
andwe use that and other work for comparison. We dis-
:
v discordisthendefinedastheresidualcorrelationleftbe- cuss especially X-states where, for arbitrary dimension,
i hind once the maximum classical correlation has been
X theformofthedensitymatrixispreservedmakingcalcu-
subtracted from the total.
lationssimpler;atthesametime,thesestatesencompass
r
a A convenient procedure for describing and computing muchphysics ofinterest. The appendix presentsagroup
such measurements in terms of parametrization of uni- symmetry of these states that makes transparent their
tary transformationsofa qubit has been given[4, 5] and generation and properties, again connecting to and gen-
reducedtoaverysimpleprescriptionapplicabletoqubit- eralizing what is available for qubits [6, 8, 10].
qudit or N-qubit systems [6]. The largest number of
variables over which the classical correlation has to be
maximized is two, although simplifications often reduce
the work involved even further. We extend now these
prescriptions to cases when both parties are of dimen-
sion larger than that of a qubit. In particular, for a
qutrit or spin 1 with dimension three, the largest num-
ber of variables needed is four or six, and even fewer for II. PROCEDURE FOR A QUBIT
classes of density matrices of most interest. We discuss
X-states [7, 8] as well as Bell, Werner, and many other
states that are often discussed in a variety of situations. AbipartitesystemABwithAaqubitandBofpossibly
Anappendixalsoprovidesasystematicdefinitionofsuch larger dimension such as four with a density matrix
2
ρ11 0 0 ρ14 ρ15 0 0 ρ18 Ai(σ V~)Ai = (~z V~)Ai, (6)
0 ρ ρ 0 0 ρ ρ 0 · ± ·
22 23 26 27
0 ρ32 ρ33 0 0 ρ36 ρ37 0 whichgreatlysimplifiestheseeminglycomplicatedcalcu-
ρ41 0 0 ρ44 ρ45 0 0 ρ48 lation in Eq. (5). Here, ~z is a unit vector formed out of
ρ= ρ51 0 0 ρ54 ρ55 0 0 ρ58 (1) the four parameters in Eq. (4) as given in [4]
0 ρ62 ρ63 0 0 ρ66 ρ67 0
ρ0 ρ072 ρ073 ρ0 ρ0 ρ076 ρ077 ρ0 ~z ={2(−ty2+y1y3),2(ty1+y2y3),t2+y32−y12−y22}, (7)
81 84 85 88
and we have
serves to illustrate our previous results [5, 6]. Such a
so-called “extended-X” state [6] can be viewed as four
equally-sizedblocks, such a 2 2 block structure reflect- A =UΠ U† =(I ~σ ~z)/2. (8)
± ±z
× ± ·
ing sub-system A; within each block, the 4 4 denotes
the dimension of sub-system B, these blocks×having the Theaboveprocedure,firstgivenin[4],alreadycontains
structureoftheletter Xwithnon-zeroentriesonlyalong a subtlety at this point to which we will return in Sec.
the diagonal and anti-diagonal[7, 8]. III, namely, a reduction from the three parameters in
In calculating quantum discord, the quantum mutual Eq.(4)tojustthetwooftheunitvector~z inEq.(7). An
information in the full system AB, defined as [2, 4] alternative standard parametrization of the unit vector
inpolarangles,theusual‘Blochangles’[1],givesthe2 2
×
matrix in Eq. (8) as
(ρAB)=S(ρA)+S(ρB) S(ρAB), (2)
I −
wisheearseilSy(cρo)m=pu−tterd(,ρevloegn2aρn)ailsytthiceavlloyn. NNoetuemtahnant ienntdrooipnyg, A+ = 1scionsθ2(eθx/p2()iφ) 21sisninθ2e(xθp/(2−)iφ)!, (9)
so, for the eigenvalues required for these entropies, the 2
1-4-5-8 and 2-3-6-7 subspaces of Eq. (1) are decoupled,
andA its parity conjugate with (θ,φ) replacedby (π
−
simplifying the algebra involved. −
θ,π+φ), that is, with diagonalentries interchangedand
The second part of calculating discord is the comput-
a change in sign of the off-diagonal entries.
ing of classicalcorrelationbetween A and B by account-
The conditional density matrix [4] for sub-system B
ing for all possible projective measurements overA. The
because of measurements on A is given by
essential idea [4] is to use von Neumann projections, as
transformedby arbitraryunitarytransformations,to de-
S(ρ A )= p S(ρ ), (10)
i i i
scribe all possible measurements on qubit A. Starting |{ }
i
with two orthogonal projectors, Π = i i, i = z for X
i
| ih | ±
sub-system A, and unitary operators U SU(2), a gen- and its quantum mutual information, a measure of total
∈
eral measurement for A can be written as [4] correlation[11], by
A =UΠ U†. (3)
i i
(ρ A )=S(ρB) S(ρ A ). (11)
i i
U may be parametrized in terms of Pauli spin matrices I |{ } − |{ }
of A as [4] Ameasureofthe resultingclassicalcorrelationsthenfol-
lows [2, 4] as
U =tI +i~y ~σ, (4)
·
(ρ)= sup (ρ A ). (12)
with t,y ,y ,y R and t2 +y2 +y2 +y2 = 1. Only C I |{ i}
1 2 3 ∈ 1 2 3 {Ai}
threeoftheseparametersareindependent,assumingval-
ues t,y [ 1,1] for i=1,2,3. Finally,thequantumdiscordisobtainedasthedifference
i
∈ −
The conditional density operator ρ associated with a between Eq. (2) and Eq. (12),
i
measurement of sub-system A is [4]
1 (ρ)= (ρ) (ρ). (13)
ρi = (Ai I)ρ(Ai I), (5) Q I −C
p ⊗ ⊗
i
Aswehaveshownbefore[6],theentirecalculationcan
wherethe probabilityp equalstr[(A I)ρ(A I)]. Be- be condensed into the form of a recipe or a simple pre-
i i i
cause of the projection operators in⊗it, we ha⊗ve A2 = scriptionforanysuchsystemABwithAaqubit,thatall
i
A , A = I, and with the choice of measurement di- elements of each jk -block of the four blocks of Eq. (1)
i i i { }
rections i= z, we get the identity, are multiplied by the kj-th element of Eq. (9) and the
P ±
3
four blocks added together to get p ρ in Eq. (5) after qubit it is at most two, not the expected 3 of SU(2) uni-
i i
whicheigenvaluesandentropiesinEq.(10)andEq.(11) tarymatrices. Understandingthisreductionprovidesthe
areeasilycalculated. ThesupremuminEq.(12)requires clue to the generalizationto the higher SU(d ). For this
A
varying the two angle parameters in Eq. (9). Because of purpose, the casting of a unitary U as a product U U
1 2
the paritysymmetrythathasbeennoted[5], the expres- with the first term describing a “base manifold” and the
sionisalwaysanevenfunctionofcosθ. Further,because second a “fiber” is useful [15]. For SU(2), this latter is
of the way φ occurs in Eq. (9), for many density matri- a pure phase and is the exponential of σ . Since it com-
z
ces,thisparameterdisappearsfromtheeigenvaluecalcu- mutes with the projectorsin Eq. (8) that also involveσ
z
lationandthe extremizationreduces to just onevariable only, it disappears at this stage of the calculation, leav-
θ. Evenfurther,althoughouroriginalconclusion[5]that ing the A matrices in that equation dependent only on
the extremum is reached at the value of θ = π/2 is only U and, therefore, on that base manifold’s two dimen-
1
true when the function has only one extremum and an sions only. It is conveniently regarded [16] as a complex
actual computation is necessary in general, that conclu- number z or throughan inverse stereographicprojection
sion seems to hold for all but a tiny fraction of density as the unit vector of the Bloch sphere or its polar an-
matrices [6, 12, 13], thereby making the calculationeven gles as in Eq. (9). (There is here an interesting parallel
simpler in most instances. between the complex number z and the unit vector ~z of
We also note that when it is B that is the qubit and Eq.(7).) Thispictureofunitaryoperationsintermsofa
A of possibly any dimension, the equivalent recipe for basemanifoldanda fiber,the latter notoccurringinthe
discordcalculationwithmeasurementsoverthequbitend extremizationandtheformersimplyrelatedtotheBloch
is to take each of the 2 2 blocks in the qudit-qubit ρ, sphere,givesareadyexplanationforthereductiontotwo
×
multiply the jk-th element by the kj-th element of A variationalparametersandanaturalgeometricalcasting
i
in Eq. (9) and add all four to give the d d conditional of them. It is the key to our simplification compared to
×
density matrix. This same prescription applies to the other results on qutrit systems in the recent literature
more general considerations in Sec. III, blocks added [17–19]. Therefore, we will now extend this argument to
togetherwhenmeasurementsmadeontheleftAendbut qutrits and higher-dimensional systems.
all within a block added when the measured end is B on Inrepeatingthecalculationofclassicalcorrelationand
theright. Ofcourse,forsymmetricsystemssuchasthose thereby quantum discord, we use for concreteness the
considered in Sec. IV, the entropic discord either way is value d =3 of a qutrit to illustrate the procedure that,
A
the same. however, applies more generally. For the 3 3 case, the
×
question is how many parameters will be involved in the
extremizationinanalogyto the previoussection’sreduc-
III. DISCORD CALCULATION FOR SPIN tiontotwofromthree. Thatitwillagainnottakethefull
LARGER THAN 1/2 eightofSU(3)symmetryasmaybeexpectedatfirstsight
can be seen as follows. In establishing results analogous
TheaboveprocedurewhenAisaqubitis,inprinciple, to Eq. (6) and Eq. (8), we use a similar decomposition
easilyextendedtowhenit(andB)isoflargerdimension, of U as U1U2 that has been provided in [16]. Although
say dA 3. The density matrix of AB would, as in U1 in Eq.(28)of that paper is a linear combinationofall
Eq.(1),≥be viewedintermsofd d blocks,eachblock eight λ, it depends only on four parameters, viewed ei-
A A
a matrix of dB dB. With dA pr×ojectors Πi in Eq. (3), ther as two complex numbers (z1,z2) or four real angles
a natural param×etrization of the unitary matrix in that (θ1,θ2,ǫ1,ǫ2). The second unitary matrix U2 is block-
equation would involve (d2 1) independent operators diagonal with an upper 2 2 SU(2) block and a lower
andcoefficientsinEq.(4).AT−hus,forAaspin-1orqutrit 1 1 single entry. It is cl×early a linear combination of
×
with dA = 3, eight Gell-Mann matrices λi [14] would just four of the Gell-Mann matrices (λ1−3,λ8). There-
occur instead of the Pauli matrices. Two of them are fore, in the analog to Eq. (8), although the full U2 no
diagonal and also would stand in the projectors: longer commutes through to disappear in a product ex-
pression,thereisneverthelessconsiderablesimplification.
For this purpose, the structure of Π in Eq. (3) is
i
Π1,2 =(2I 3λ3+√3λ8)/6, Π3 =(I √3λ8)/3. (14) crucial, that they are matrices with only one non-zero
± −
entry, namely unity in the i-th diagonal position, i =
A calculation analogous to that in Sec. I would give dA 1,2,...dA. The U in Eq. (3) is viewed as U1U2 as
matrices A as in Eq. (8) and Eq. (9). Multiplying their in [15, 16], with two unitary factors, the latter block-
kj-th element into the jk-th block of ρAB, and adding diagonalintermsoflowerdimensionalU. Thus,theonly
the blocks would give the dB dB conditional density parameters needed are the z of [15, 16] involved at each
×
matrix from which eigenvalues and classical correlation step in such a construction of U. These sets of complex
are computed. numbersz,ortheiralternativerenderingintermsofpairs
However, an immediate question arises, namely, what of polar angles, are the parameters varied to extremize
is the number of independent parameters needed for the the classical correlation. For A a qutrit, U is given in
1
extremization? We sawinthe previoussectionthatfora Eq.(27) and Eq.(28) of [16]. U has an upper diagonal
2
4
block of a qubit and a pure phase as its lower diagonal and,thereby,thequantumdiscord. Thiscutsthenumber
block. Asaresult,forA inEq.(3),withΠ havingonly of parameters by half.
3 3
one non-zero entry of unity in the lowest diagonal posi- Extension to dimensions larger than a qutrit proceeds
tion,U2 dropsouttoleavebehindU1Π3U1† whichfurther similarly. For A a qudit with dA = 4, the density ma-
becomes trix ρAB is viewedas a 4 4 block matrix, eachblock of
×
d d . InconstructingA ,the 4 4matrixU maybe
B B i 1
× ×
decomposed into blocks either as 4=2+2 or 4=3+1. Ac-
s2c2 s2s c ei(ǫ1−ǫ2) s c c eiǫ1 cording to the procedure in [15], the former will involve
1 2 1 2 2 − 1 1 2
A3=s21−s2sc12ce1−c2ie(ǫ−1−iǫǫ12) −s1cs121ss222e−iǫ2 −s1cc121s2eiǫ2 , andui2amg×obn2earmlzbaltfoorcirxkasofotfo4t2cao×lmo2fplm6exactonrmuicmpeslbe,exeraszczho.rwU1i2t2hthrietesanlchopamasrpatlwmexo-
(15)
eters in all in A . The alternative reduction of 4=3+1,
where we we have abbreviated c = cosθ,s = sinθ and i
followedby3=2+1,willmeanagain3+2+1or6complex
similarly for subscripts 1 and 2.
z. The case of general d proceeds similarly, involving
Thus, A involves only four parameters, those of the A
3 the sequence of triangular numbers for the number of
twocomplexz oftheSU(3)basemanifoldinEq.(27)and
complex z needed, that is, d (d 1) real parameters
Eq.(28) of [16]. A1 and A2 are a little more complicated in all. Instead of the canonicAal rAep−resentation in terms
expressions than Eq. (15), involving both the two com-
of Gell-Mann matrices or their higher-dimensionalcoun-
plex z of the SU(3) U but also the two parameters in
1 terparts, the view of [15, 16] through a sequence of z
Eq. (8) of their SU(2) complex z in U . They are evalu-
2 providesareductioninthenumberofparametersneeded
atedby sandwichingbetweenU andU† the 3 3block-
1 1 × forextremizationandis,therefore,thepreferredrouteto
diagonalmatrix that has in its upper diagonalblock A
± calculatingquantumdiscordorinpossibleotherapplica-
fromEq.(8)andunityinthelastdiagonalposition. They
tions. This is where we differ from previous attempts at
depend, therefore, on 6 parameters. Thus, we have
discord calculations for qutrit-qutrit that also handled
through Gell-Mann matrices [17–19].
(A ) = c2(s2+c2c )2+s2c2s2(1 c )2
1 11 2 2 1 2 2 − 1
−2csc2s2(1−c1)(s22+c22c1)cos(ǫ1−ǫ2+φ), IV. APPLICATIONS TO QUTRIT-QUTRIT
(A ) = cs[(s2+c2c )(c2+s2c )e−iφ+s2c2(1 c )2 SYSTEMS
1 12 2 2 1 2 2 1 2 2 − 1
ei(2ǫ1−2ǫ2+φ)] fei(ǫ1−ǫ2),
× − To illustrate the general procedure of Sec. III, we
(A1)13 = css1s2[(s22+c22c1)ei(ǫ2−φ)−c22(1−c1)ei(2ǫ1−ǫ2+φ)] consider some density matrices of qutrit-qutrit systems
+[c2(s2+c2c ) s2s2(1 c )]s c eiǫ1, that havebeen recently studied in differentcontexts and
2 2 1 − 2 − 1 1 2
(A ) = c2s2c2(1 c )2+s2(c2+s2c )2 for whom the geometric discord, a correlation simpler
1 22 2 2 − 1 2 2 1 to evaluate, have been computed [9] as well as entropic
2css c (c2+s2c )(1 c )cos(ǫ ǫ +φ),
− 2 2 2 2 1 − 1 1− 2 discord [20] and other correlations [17–19].
(A ) = css c [(c2+s2c )ei(ǫ1+φ) s2(1 c )
1 23 1 2 2 2 1 − 2 − 1
e−i(ǫ1−2ǫ2+φ)] geiǫ2,
× − A. Bell states
(A ) = c2s2c2+s2s2s2+2css2s c cos(ǫ ǫ +φ), (16)
1 33 1 2 1 2 1 2 2 1− 2
Labelling as usual the three states as 0, 1, and 2, we
withA asimilarexpressioninwhichc2 ands2 areinter-
2 begin with the maximally entangled Bell state, a pure
changed and the sign of cs is changed. We have defined
state described by (00 + 11 + 22 )/√3 and density
forconvenience: f =[c2(s22+c22c1)+s2(c22+s22c1)]c2s2(1− matrix of the form | i | i | i
c )] and g =[c2c2(1 c ) s2(c2+s2c )]s s . All three
1 2 − 1 − 2 2 1 1 2
matrices A are Hermitian and unitary with unit trace
i
and we have A +A +A = I. These properties, to- ρ 0 0 0 ρ 0 0 0 ρ
1 2 3 11 15 19
getherwiththefeaturenotedafterEq.(5)thatA2i =Ai, 0 0 0 0 0 0 0 0 0
will prove crucial in applications in Sec. IV.
0 0 0 0 0 0 0 0 0
The calculation of quantum discord for a qutrit sys-
0 0 0 0 0 0 0 0 0
tem therefore requires at most the 4 or 6 parameters,
ρ= ρ 0 0 0 ρ 0 0 0 ρ , (17)
angles θ and phases (ǫ,φ) in Eq. (15) and Eq. (16), not 51 55 59
the original 8 of the qutrit’s su(3) algebra. Further, we 0 0 0 0 0 0 0 0 0
will see in Section IV that for many density matrices, 0 0 0 0 0 0 0 0 0
not even all of this smaller set are needed (just as for
0 0 0 0 0 0 0 0 0
qubits, often one parameter extremization suffices [6]);
ρ 0 0 0 ρ 0 0 0 ρ
in particular, most of the phase parameters drop out in 91 95 99
computing eigenvalues leaving primarily the θ parame- allnon-zeroentriesequalto1/3fortheBellstate. Byin-
ters in the extremization to get the classical correlation spection, eight of the eigenvalues are zero and one unity
5
sothatthelastterminEq.(2)iszero. TracingoverAor also gives a reduced d d density matrix that is A to
i
×
B in Eq. (17) gives 3 3 density matrices I/3 with cor- a multiplicative factor. With our observation in Sec. III
×
responding entropies in Eq. (2) of log3. Conventionally that the eigenvalues of A are all zero except one which
i
inquantuminformation,logarithmshavebeenevaluated equals unity, the eigenvalues of the conditional density
in base 2 but it seems to fit better for our discussion of matrix thereby reduce to d 1 of magnitude (1 p)/d
− −
qutrits to use base 3 (more generally,base d) so that the and one with [p+(1 p)/d. The results of Eq. (11) and
−
total mutual information in Eq. (2) equals 2. This also Eq. (12) are then immediate, making unnecessary any
fits what follows in the next paragraph for the classical supremum calculation because all the parameters in A
i
correlation. have dropped out. This seems to have escaped notice
The procedure of Sec. III for evaluating the reduced in [9, 18, 19], the “quite difficult” supremization carried
density matrix and its entropy becomes trivial. With all out although the final results reduce to zero values for
blocks in Eq. (17) having one non-zero entry of 1/3, the all angles involved, in itself a pointer to the parameters
prescription of multiplying by Eq. (15) or Eq. (16) and introduced by A , or other SU(3) counterparts in their
i
adding blocks gives back those same A matrices, multi- calculations, dropping out of the claculation.
i
plied by 1/3. Given that these matrices are Hermitian The classicalcorrelationin Eq.(12) and quantum dis-
and unitary, the eigenvalues are 0, 0, and 1 so that the cordin Eq.(13) areeasy algebraicexpressions evaluated
entropy in Eq. (10) of these reduced 3 3 matrices is from these eigenvalues,
×
zero. Hence, no supremization is necessary and, from
Eq. (11) and Eq. (12), the classical correlation equals
=[(d 1)(1 p)/d]log(1 p)+[p+(1 p)/d]log(dp+1 p),
log3=1, againusing base 3. So, just as with Bellstates
C − − − − −
(19)
of qubit-qubit, we can conclude that all such maximally
and
entangledBellstatesofqutrit-qutrit,oranyqudit-qudit,
have =2,equallydividedintoclassicalcorrelationand
I
quantum discord of = =1.
C Q =[(d 1)(1 p)/d2]log(1 p)+[(d2p+1 p)/d2]
Q − − − −
log(d2p+1 p) [(dp+1 p)/d]log(dp+1 p).(20)
× − − − −
B. Werner and pseudo-pure states
These results valid for all d coincide for d = 3 with the
expressions given as Eqs.(8) and (9) in [18] for qutrits.
The Bell states of the previous sub-section are pure
Since theyhavepresentedplotsofthesecorrelationsasa
states. A class of mixed states obtained by mixing them
function of p, we present instead our values (using base
withacompletelymixedstatethatisessentiallytheunit
3 for logarithms) in Table I.
matrix, I/d, with equal diagonal elements and all off-
Limiting results for these correlations are interesting.
diagonal coherence terms zero (“white noise”), has been
Both vanish for d=1 whereas for d large, and assuming
much studied both for qubit-qubit and higher dimen-
that p is neither too close to 0 or 1 so that pd>>1 p,
sions. Terminology(andnotation)hasvariedinreferring −
we have = plogd rising linearly with p, a result also
tothemasWernerstates[9,17–19]ormakingaslightdis- Q
in [20]. With the choice of base d for any dimension,
tinctionbetweenthemandso-called“pseudo-pure(PP)”
this reduces to the mixing parameter p, independent of
states [20] but, in essence, they are given by the density
dimension. This might be used as another justification
matrix,
for using logarithms to the base d, the observation [20]
of correlations growing without bound with d merely a
ρ =[(1 p)/d2]I+pΨ Ψ, (18) reflection of the growth in log2d when the base is fixed
AB
− | ih | at2. Wealsonotethat[19]generalizedtheresultsof[18]
with0 p 1and Ψ the BellstateasinEq.(17)with to what are called “Werner derivatives” by introducing
≤ ≤ | i
all non-zero entries equal to 1/d. anotherparameterbesidesp. Again,alaborioussuprem-
The evaluation of quantum discord is straightforward. izationwascarriedoutbutisunnecessaryasperourpro-
For a qudit-qudit state in Eq. (18), by inspection, one cedure,thefinalcorrelationssimplealgebraicexpressions
eigenvalue is p + (1 p)/d2 and the remaining d2 1 depending on the two parameters for any d. The maxi-
are (1 p)/d2, and S− in Eq. (2) follows immediat−ely. mization of classical correlations seems to not enter into
AB
−
Tracingovereitherone ofthe qudits clearlygivesacom- such Werner and pseudo-pure states, correlations inde-
pletely mixed state, I/d, so that S = S = logd in pendent of measurements on one of the sub-systems, no
A B
Eq. (2). Next, the calculation of the conditional den- doubt because of the mixing of isotropic configurations
sity matrix in Eq. (5) is equally straightforward. Given in the density matrix.
the linearity of that expression, each term in Eq. (18)
can be treated separately. The first, proportional to the
unit operator, commutes through the two A and, with C. X-states
i
the square being A with unit trace, is proportional to
i
a d-dimensional unit matrix. And, as noted in Sec. IV There is an extensive qubit literature for what are
A, in our procedure for a Bell state, the second term called X-states. Originally so named [7] because of
6
p SAB C Q I
0 2 0 0 0
ρ 0 0 0 0 0 0 0 ρ
0.1 1.97 0.008 0.022 0.03 11 19
0 ρ 0 0 0 0 0 ρ 0
0.2 1.89 0.034 0.072 0.106 22 28
0 0 ρ 0 0 0 ρ 0 0
1 1.84 0.053 0.106 0.159 33 37
4
13 1.74 0.094 0.169 0.263 ρ(X) = 00 00 00 ρ044 ρ0 ρ046 00 00 00
0.4 1.64 0.135 0.226 0.361 55
1 1.47 0.21 0.324 0.534 0 0 0 ρ64 0 ρ66 0 0 0
2
2 1.11 0.377 0.509 0.886 0 0 ρ73 0 0 0 ρ77 0 0
3
34 0.903 0.485 0.612 1.097 0 ρ82 0 0 0 0 0 ρ88 0
0.9 0.441 0.735 0.824 1.36 ρ91 0 0 0 0 0 0 0 ρ99
(21)
0.95 0.25 0.85 0.90 1.75
has 16 parameters, fewer than the 80 of a completely
1 0 1 1 2
filled 9 9 matrix. Again, this results in a great econ-
×
omy in handling so that these states are natural targets
TABLEI:CorrelationsforthequtritWernerstateinEq.(18) for studying quantum correlations and other character-
as a function of the mixing parameter p. The entropies SA istics of qudit-quDit systems. In the Appendix, we will
and SB equal 1, while SAB, classical correlation, quantum considerasymmetry structureofthese statesjust ashas
discord, and total mutual information are shown. beendescribedfor qubits [8,10] thatalsopermits exten-
sion to multipartite systems.
Fora discordcalculationwith suchX-states,eigenval-
ues involved in the calculation of Eq. (2) are simple, as
it is in the case of qubits, the central 55 element itself
an eigenvalue and the rest as pairs of 2 2 sub-spaces.
×
The same applies to the reduced densities upon tracing
over either A or B, each a 3 3 matrix with once again
the appearance of a 4 4 qubit-qubit density matrix ×
the central element decoupled from the others. For the
×
with non-zero entries only along the diagonal and anti-
conditional density upon measurements over A, our pre-
diagonal resembling the letter X, their attraction lies in
scriptioninSec. IIIofmultiplyingeachofthe9blocksby
thefewerparametersinvolvedinthem. Withtracefixed,
the corresponding transposed element of A and adding
i
3realelementsalongthe diagonaland2complexonesin
the blocks clearly results in an X-state again. Eigen-
the off-diagonal amount to 7 parameters in all. This is
values again follow easily. Also, as can be seen from
less than the 15 of a general qubit-qubit density matrix
Eq. (15) and Eq. (16) for a qutrit, parameters in their
and makes for easier handling. At the same time, many
12 and 23 elements drop out, reducing the number re-
specializedstatesofinterestbelongtothisclassandmany
quired for supremization. Whereas our prescription ap-
physical phenomena may be studied with them without
plies quite generally to any qutrit-quDit density matrix
necessarilyinvolvingmoregeneralmatrices. Forbipartite
giving a D D conditional density, for X-states and ex-
systems with only one a qubit, the slightly larger class ×
tended X-states, there is such a reduction from 4 to 3
of “extended X” states as in Eq. (1) also prove almost
(ǫ dropping out) for A and from 6 to 5 for A . In
2 3 1,2
as convenientbecause in forming the conditional density
particular, many of the phase angles drop out leaving
matrix,thatalsoends upasanX-state. Eigenvaluesare
a small subset of them plus the theta angles which are
then evaluated through simple quadratic equations, cal-
d (d 1)/2 in number. Mutual phases between suc-
A A
culationsbreakinginto2 2sub-spaces. Forqubit-qubit, −
cessivesub-spacesofSU(d )involvedinthereductionof
× A
extended X-states include all possible 4 4 density ma-
unitarymatrices,asper[15],addd 2foratotalofap-
× A
triceswith15parameters[10]ofsuchasystem. Through −
proximatelyd (d +1)/2parameterstobevaried,which
A A
variouslocaltransformations,higherdimensionaldensity
is alsothe number of POVMs(PositiveOperatorValued
matrices of a more general nature can also be brought
Measures [23]). Finally, we note also other studies with
intoX-form[21,22],lendingfurtherimportancetothem.
qutrits [24–26] where our prescription could be applied
A different direction of generalization [22] to what have
to simplify the numericsofthe extremizationinvolvedin
been termed “true generalized X” (TGX) states gives
them.
states that differ from our extended X states.
V. APPENDIX: SYSTEMATICS OF X-STATES
It seems then natural to look at similar X-states in
dealingwithhigher-dimensionalsystems. Thus,aqutrit- For a 4 4 qubit-qubit system, X-states were intro-
×
qutritstatewithentriesonlyalongthediagonalandanti- duced andnamedfor their visualresemblanceto the let-
diagonal, ter [7] but, as states using just 7 parameters out of the
7
15 of a general density matrix, they are characterized numberof qubits 1 2 3 4 n
by their symmetry under the su(2) u(1) su(2) sub- general 3 15 63 255 22n−1
⊗ ⊗
algebra of the full su(4) algebra that applies to the pair X-states 3 7 15 31 2n+1−1
of qubits, whether or not the density matrix looks like
su(2)s 1 2 4 8 2n−1
X [8]. This perspective also permits the generalization
u(1)s 0 1 3 7 2n−1−1
to X-states of n qubits. The recognition that for a sin-
gle qubit, the 2 2 density matrix is immediately an X, extended X states 3 15 31 63 2n+2−1
×
and that the pair system involves two such 2 2 su(2) numberof qutrits 1 2 3 4 n
×
spaces with a phase between them, represented by the general 8 80 728 6560 32n−1
u(1), points to the same iteration when more qubits are X states 4 16 52 160 2(3n−1)
added. Thus, for 3 qubits, where the full algbera is of
extended X states 4 44 152 476 2(3n−5)
su(23 =8) with 82 1=63 parameters, the sub-algebra
−
[su(2) u(1) su(2)] [u(1)] [su(2) u(1) su(2)],
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
a15-parametersub-algebraofthefullsu(8)algebragives
TABLE II: Enumeration of the number of states in a gen-
their X-states, and so on [10]. The number for n qubits eral, an X-, and an extended X- state of n qubit and qutrit
is 2n+1 1, the iteration described algebraically as the systems.
inductio−n 2n+1 1 = 2(2n 1)+1. This constitutes a
sub-algebra of t−hat dimensi−on of the full su(2n) algebra
of n qubits. A connection to finite projective geometries
[27, 28] is also worth noting. qudit bipartite system has 4d2 1, X-state has 4d 1,
− −
A similar development for qutrits or higher dimen- and extended X-state 8d 1 for d even and 8d 5 for
− −
sionalsystemscanbe seenfromthe structureofEq.(21) d odd as the number of parameters. The results for a
as a three-dimensional central 3 3 block of a single system of n qubits follows by setting d = 2n−1 so that
×
qutrit’s X-state (with two real and one complex or four thereare22n 1,2n+1 1and2n+2 1statesforthegen-
− − −
parameters) embedded in six other dimensions around eral, X-,and extended X-,state, respectively. Similarly,
it. Thus, a qutrit-qutrit X-state has 16 parameters. As a qutrit-qudit bipartite system has 9d2 1 for a general
−
withthepreviousparagraphforqubits,theiterationsub- density matrix, 6d 1 for d even and6d 2 for d odd in
− −
algebra of the full su(9) algebra (of 80 parameters) can X-states, and 18d 1 for d even and 18d 10 for d odd
be describedas three copies now of the startingsu(X)(3) in extended X-stat−es. Again, setting d=−3n−1 gives the
withu(1)sinbetweenandattheendsforthe12+4num- results for n qutrits: 32n 1, 2(3n 1) and 2(3n+1 5),
− − −
berofparameters. Thenextstepsimilarlyrepeatsthrice respectively.
the previous with 4 additional u(1)s. For n qutrits, the
number of parameters is 2(3n 1) and the inductive it- Evenmorebroadly,a qudit-quditbipartitesystemhas
erationisgivenby2(3n 1)=−3[2(3n−1 1)]+4. Again, d4 1 parameters in a general ρ, 2d2 1 for d even and
− − 2d2− 2for dodd inX-states,and2d3− 1ford evenand
in the language of symmetry groups and algebras, these
− −
are sub-algebras of the full su(3n) algebra of n qutrits. 2d3 d2 1 for d odd extended X-states. For d = 4,
this−numb−er for X-states of n such qudits is 2(4n) 1
Table II presents values for various classes of states of whichmeansthrough2(4n) 1=4[2(4n−1) 1]+3 t−hat
such pair systems of dimensions 2 and 3. Also shown − −
eachsuccessivequ4itmeansaniterationof4copiesofthe
for qubits are the number of su(2) and u(1) in the sub-
previouswithu(1)sinbetweenthecopies,parallelingthe
algebra of X-states of n qubits. Three times the former
iterationfornqubits. Again,thesehavethesymmetryof
numberplusthelatteris,ofcourse,thenumbershownin
a sub-algebra of the large su(dn) algebra. An asymmet-
the rowforX-states. Thatnumberisalsothenumberof
ric qudit-quDit system has similar enumerations. There
points in the projective geometry PG(n, 2), whereas the
are in general d2D2 1 parameters, X-states having the
numberintherowaboveofthegeneralstateisPG(2n 1, −
− smallernumber2dD 1fordD evenand2dD 2fordD
2). In particular, PG(2, 2) with 7 points is the Fano − −
odd.
Plane,the simplestprojectiveplane, ofextensive mathe-
maticalinterest [29]. It is interesting that the number of It is also interesting to note as a complement to the
X-statesofqubitsrunsthroughallsuccessiveintegerval- aboveparagraphsthatforpurestates,thedensitymatri-
ues n of PG(n, 2) whereas the number for generalstates ces have fewer parameters. With the eigenkets them-
skips evenvalues, depending as it does on 2n 1. Alter- selves available, a normalization and an overall phase
−
natively,this canbe statedin terms ofskipping half-odd droppingout,therearedn 1complexelementsortwice
integer values of n, that is, that PG(2, 2), PG(4, 2), that number of real param−eters defining such a density
etc., are absent in the general count. There is a whiff matrix. This number is very similar to the number for
ofquantum-mechanicalspinangularmomentuminthese corresponding X-states noted above. Finally, so-called
sequences of numbers! quantum-classicaldensitymatricesareofinterestinsome
Other general enumeration of the number of parame- discussionsofcorrelations. WithAdescribedintermsof
tersforarbitrarydimensionfollowsfromstraightforward quantum ket-bra and B a classical density matrix, they
if slightly tedious algebra. As examples,a generalqubit- have the form
8
sity matrices of sub-system B. As noted for pure states,
theseare2(d 1)innumber,thereared 1probabilities,
d and each clas−sicaldensity matrix has D−2 1 parameters
ρ= pi|ψiihψi|ρ(iB), (22) foratotalofd(D2+d 1) 1parameters−,againsmaller
Xi=1 than the full d2D2 1−in−a general ρ. Also, again the
−
withp probabilities(0 p 1),andρ(B) classicalden- number is asymmetric in d and D.
i i
≤ ≤
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