PhysRevA.88.014302
PHYSICAL REVIEW A88,014302(2013)
Quantum discord for two-qubit X states:Analytical formula with very small worst-case error
Yichen Huang*
Department of Physics,University of California,Berkeley,Berkeley,California94720,USA
(Received7April2013;published18July2013)
Quantum discord is a measure of quantum correlation beyond https://www.sodocs.net/doc/847542797.html,puting quantum discord for
simple quantum states is a basic problem.An analytical formula of quantum discord for two-qubit X states is
?rst claimed in[Ali,Rau,and Alber,Phys.Rev.A81,042105(2010)],but later found to be not always correct.I
observe numerically that the formula is valid with worst-case absolute error0.0021.For symmetric two-qubit X
states,I give a counterexample to the analytical formula derived in[F.F.Fanchini et al.,Phys.Rev.A81,052107
(2010)],but observe that the formula is valid with worst-case absolute error0.0006.The formula has been used
in many research papers.The results in all these works are approximately correct,even if they may not be exactly
correct.
DOI:10.1103/PhysRevA.88.014302PACS number(s):03.67.?a,03.65.Ud,03.65.Ta,03.65.Aa
Quite a few fundamental concepts in quantum mechanics do not have classical analogs:uncertainty relations[1–4], quantum nonlocality[5–8],etc.Quantum entanglement is de?ned based on the notion of local operations and classical communication(LOCC):a bipartite quantum state is separable (not entangled)if it can be created by LOCC[7,8].The set of separable states is convex and has nonzero measure(volume), and a lot of effort is devoted to entanglement detection[8–13]. However,it is argued that nontrivial quantum correlation also exists in certain separable states.A number of measures have been reported to quantify quantum correlation beyond entanglement[14].Quantum discord,proposed explicitly in[15]and implicitly in[16],is the most popular such measure and a hot research topic in the past a few years. The set of classical(zero discord)states is nowhere dense and has zero measure(volume)[17].Unfortunately,computing quantum discord seems extremely dif?cult as the de?nition (1)requires the optimization over all measurements.Few analytical results are known even for two-qubit states,and the computational cost of any numerical approach is expected to grow exponentially with the dimension of the Hilbert space.
Let us focus on two-qubit X states,which we frequently encounter in condensed matter systems,quantum dynamics, etc.[18–22].For instance,the two-site reduced density matrix of the symmetry unbroken ground state of a lattice Hamiltonian with Z2symmetry is of the X structure(2).An analytical formula of quantum discord for Bell-diagonal states(a subset of two-qubit X states)is known[23].For general two-qubit X states,the?rst attempt is made in[24],and the analytical formula(5)is claimed.However,(5)is not always correct:a counterexample is given in[25](see also[26]).The reason is that not all extrema are identi?ed in[24],and not all constraints are taken into consideration[25].Hence,the analytical formula of quantum discord for general two-qubit X states is still unknown.Two regions in which(5)is valid are identi?ed in[26].Moreover,there are statistical evidences that(5)and related formulae are pretty good approximations for most states[25,27–29],although these evidences do not rule out the *yichenhuang@https://www.sodocs.net/doc/847542797.html, possibility that an unlucky guy obtains qualitatively incorrect results for some states by using these formulae. Symmetric two-qubit X states are of special interest in condensed matter systems,quantum dynamics,etc.[18–22]. For instance,the two-site reduced density matrix of the symmetry unbroken ground state of a translationally invariant lattice Hamiltonian is symmetric.An analytical formula,which happens to be equivalent to(5),for this subset of two-qubit X states is derived independently in[18].
Computing quantum discord numerically for two-qubit X states is straightforward.Surprisingly,I observe that even if(5) is not always correct exactly,it is always correct approximately with very small worst-case absolute error(6).Technically,I search over the entire space of two-qubit X states with steps small enough to ensure numerical precision.For symmetric two-qubit X states,(in contrast to[18])(5)is still not always correct[(9)gives a counterexample],again because not all extrema are identi?ed in[18].In this case,as expected the worst-case absolute error is smaller(8).Equation(5)has been used in many(about80)research papers:e.g.,[18,24,30–34] (I do not list them all here).The results in all these works are approximately correct,even if they may not be exactly correct.
I.TECHNICAL PERSPECTIVE
Mutual information in classical information theory has two inequivalent quantum analogs.Quantum mutual information I(ρAB)=S(ρA)+S(ρB)?S(ρAB)quanti?es the total cor-relation of the bipartite quantum stateρAB,where S(ρA)=?trρA lnρA is the von Neumann entropy of the reduced density matrixρA=tr BρAB.Let{ i}be a measurement on
the subsystem B.Then,p i=tr( iρAB)is the probability of
the i th measurement outcome,andρi
A
=tr B( iρAB)/p i and
ρ
AB
=
i
p iρi
A
? i are post-measurement states.Classical correlation is de?ned as J B(ρAB)=max{
i}
J{
i}
(ρAB),where J{
i}
(ρAB)=S(ρA)?
i
p i S(ρi
A
)[16].The maximization is taken either over all von Neumann measurements or over all generalized measurements described by positive-operator valued measures(POVM).For simplicity,we restrict ourselves to von Neumann measurements in this work.Quantum discord,
a measure of quantum correlation beyond entanglement,is the difference between total correlation and classical correlation [15]:
D B (ρAB )=min { i }
D { i }(ρAB )=I (ρAB )?J B (ρAB )
=S (ρB )?S (ρAB )+min { i }
i
p i S ρi A
=
min { i }
S B (ρ
AB )?S B (ρAB )(1)
where S B (ρAB )=S (ρAB )?S (ρB )is the quantum conditional entropy,and D { i }(ρAB )=I (ρAB )?J { i }(ρAB ).Quantum discord is invariant under local unitary transformations.
Labeling the basis vectors |1 =|00 ,|2 =|01 ,|3 =|10 ,|4 =|11 ,the density matrix of a two-qubit X state
ρAB =?
??
??a 00α0b β00ˉβ
c 0ˉα00
d ??
???(2)has nonzero elements only on the diagonal and the antidiag-onal,where a,b,c,d 0satisfy a +b +c +d =1,and the
positive semide?niteness of ρAB requires |α|2 ad,|β|2 bc .The antidiagonal elements α,βare generally complex numbers,but can be made real and nonnegative by the local unitary transformation e ?iθ1σz ?e ?iθ2σz with suitable θ1,θ2,where σis the Pauli matrix;assume without loss of generality α,β 0.Hereafter I follow and generalize the approach of [22].Parametrizing a von Neumann measure-ment { i =|i i |}by two angles θ,φ:|0 =cos(θ/2)|0 +e iφsin(θ/2)|1 and |1 =sin(θ/2)|0 ?e iφcos(θ/2)|1 ,(1)is reduced to a minimization over two variables.The eigenvalues
of the post-measurement state ρ
AB are
λ1,2=(1+(a ?b +c ?d )cos θ
±{[a +b ?c ?d +(a ?b ?c +d )cos θ]2
+4(α2+β2+2αβcos 2φ)sin 2θ}1/2)/4,λ3,4=(1?(a ?b +c ?d )cos θ
±{[a +b ?c ?d ?(a ?b ?c +d )cos θ]2+4(α2+β2+2αβcos 2φ)sin 2θ}1/2)/4;(3)
the eigenvalues of ρ
B are 1,2=[1±(a ?b +c ?d )cos θ]/2.We would like to minimize the quantum condi-tional entropy
S B (ρ AB )= 1ln 1+ 2ln 2?
4 i =1
λi ln λi .(4)
Thanks to the concavity of the Shannon entropy,the min-imization over φcan be worked out exactly:cos 2φ=1.
Indeed,one can explicitly verify ?S B (ρ
AB
)/?cos(2φ) 0.Then,a single-variable minimization suf?ces.There are at least two extrema:θ=0,π/2,and the measurements are σz ,σx ,respectively.It is tempting (but not always correct)to write down the analytical formula:
D B (ρAB )=min {D σx (ρAB ),D σz (ρAB )}.
(5)
As is rephrased by [26],(5)is equivalent to the main result
of [24],obtained in a different approach.In the case that a =d ,b =c the algebra is greatly simpli?ed;the validity of (5)can be veri?ed explicitly;the main result of [23]is reproduced.Setting φ=0,the single-variable expression (4)we would like to minimize (over θ)is lengthy and complicated.This is strong evidence that for general two-qubit X states quantum discord cannot be evaluated analytically,even if it is straight-forward to compute numerically.Surprisingly,only a very small absolute error occurs when there are additional extrema besides θ=0,π/2:
D B (ρAB )>min D σx (ρAB ),D σz (ρAB )
?0.0021.(6)Technically,for our purpose the density matrix ρAB can be parametrized by four free parameters:a,b,c,d with the constraint a +b +c +d =1and α+βas α+βappears as a combination in (4);?ipping the ?rst qubit and/or the second qubit if necessary,assume without loss of generality a +b c +d and a b .This reduced space of two-qubit X states is searched over with different steps in different regions for ef?cient use of computational resources;the steps are kept very small,e.g.,10?6in the vicinity of (7),in the region the absolute error of (5)is large to ensure numerical precision.The state with the largest absolute error 0.002047(and θ=0.607573)I ?nd is
ρAB
=?
????0.027180000.141651
00.000224
00000.027327
0.1416510
0.945269
?
????.(7)
A two-qubit X state ρA
B is symmetric if ρA =ρB or b =c in (2).For symmetric two-qubit X states,I observe
D B (ρAB )>min D σx (ρAB ),D σz (ρAB )
?0.0006(8)by similar numerical analysis with one fewer free parameter.The analytical formula derived independently in [18],which happens to be equivalent to (5),is also not always correct.The state with the largest absolute error 0.000573(and θ=0.477918)I ?nd is
ρAB
=?
????0.021726000.128057
00.010288
00000.010288
0.1280570
0.957698
?
????.(9)
ACKNOWLEDGMENTS
The author would like to thank Joel E.Moore for useful suggestions,and Felipe F.Fanchini and Nicolas Quesada for discussions.This work was supported by the United States Army Research Of?ce via the Defense Advanced Research Projects Agency–Optical Lattice Emulator program.
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