Super quantum discord for general two qubit X states

0.35 MB·

by  Naihuan Jing

SUPER QUANTUM DISCORD FOR GENERAL TWO QUBIT X STATES
NAIHUAN JING, BING YU∗
Abstract. The exact solutions of the super quantum discord are derived for general
7 two qubit X states in terms of a one-variable function. Several exact solutions of the
1 superquantumdiscordaregivenforthegeneralX-stateovernontrivialregionsofaseven
0 dimensional manifold. It is shown that the super quantum discord of the X state may
2 increase or decreases under the phase damping channel.

b
e
F 1. INTRODUCTION

3
Quantum discord measures the difference between the total correlation and classical cor-
2
relation based on a family of complete mutually orthogonal projectors such as the von

] Neumann measurements [1,2]. It has been investigated in various works, and reveals new
h
p properties in quantum correlations [3–11]. As quantum states are fragile to quantum mea-
- surements; when they are undergone projective measurements, their coherence are likely to
t
n be loosed. In 1988, Aharonov, Albert, and Vaidman have proposed to use weak measure-
a
ments[12], whichcauseonlysmallchangestothestate, anditisexpectedthatthequantum
u
q state may loose partial coherence under the weak measurements. Recently, the quantum
[ discord under weak measurement, called the “super quantum discord” (SQD) by Singh and

2 Pati, brings new hope for deeper insights on the quantum correlation [13]. It is known
v that the weak measurement captures more quantum correlation of a bipartite system than
7
the strong (projective) measurement under certain situation. Since then, SQD has been
7
1 studied in various perspectives [14–17]. It is known that the solution is equivalent to the
6 optimization of a multi-variable function with seven parameters. However, exact solutions
0
of SQD are few for general two qubit X states, except for the case of diagonal states.
.
1 We observe that most of the previous methods claim that the super quantum discords
0
are given by the entropic functions at the endpoints, which is unfortunately an incorrect
7
1 statement (see counterexamples given in Example 3). Thus it is necessary to settle the
: super quantum discord in the general case of X-type states.
v
i The aim of this paper is to propose a brand new method to compute the super quantum
X
discord by reducing the optimization to that of one-variable function. This completely
r
solves the problem in principle. We also give analytical formulas of the SQD for several
a
nontrivial regions of the parameters. To examine the dynamic behavior of SQD under
damping channel, we also analyzed the super quantum discord through the phase damping
channel. ItisshownthatthesuperquantumdiscordoftheXstatemaydecreasesorincrease
under the damping channel. However, there also exists an example of X-state where the
MSC(2010): Primary: 81P40;Secondary: 81Qxx. ∗Correspondingauthor: [email protected].
1
2 NAIHUANJING,BINGYU∗
super quantum discord is stable or even decreasing through the whole process under the
damping channel.
The article is organized as follows: in section II, we shall review the weak measurement
formalism and the definition for super quantum discord. In section III, we shall give an
analytic solution for the SQD of general two qubit X-type states and also show that SQD
is given by the minimum of a one-variable entropy-like function. In section IV, we shall
analyze the dynamics of super quantum discord under phase damping channel. In section
V,we shall conclude our work. Two appendixes present details proofs of lemma 3.1 and
theorem 3.2.
2. THE DEFINITION FOR SUPER QUANTUM DISCORD
Let Π ,Π be a pair of orthogonal projectors such that Π Π = δ Π ,Π +Π = I. In
0 1 i j ij i 0 1
ordertostudymoregeneralsituation, oneconsiderstheweakmeasurementoperatorswhich
are a pair of complete mutual parameterized orthogonal operators that are not necessarily
idempotents. For any real x (cid:62) 0, let
(cid:114) (cid:114)
1∓tanhx 1±tanhx
(2.1) P(±x) = Π + Π .
0 1
2 2
ThenP(±x)2 = I±tanhx(Π −Π ),P(x)P(−x) = P(x)P(−x) = 0andP(x)2+P(−x)2 = I.
2 1 0
Moreover, (ii) lim P(x) = Π and lim P(−x) = Π . We will call P(x),P(−x) a
x→∞ 1 x→∞ 0
pair of weak measurement operators associated with Π [18].
i
The super quantum discord of a bipartite quantum state ρ with weak measurements
AB
on the subsystem B is the difference between the quantum mutual correlation I(ρ ) and
AB
the classical correlation J(ρ ) [13]. Recall that the quantum mutual information is given
AB
by [19]
I(ρ ) = S(ρ )+S(ρ )−S(ρ ),
AB A B AB
where S(ρ ),S(ρ ),S(ρ ) are the von Neumann entropies of the reduced state ρ =
A B AB A
Tr (ρ ), ρ = Tr (ρ ), and the total state ρ respectively. The classical correlation
B AB B A AB AB
represents the information gained about the subsystem A after performing the measure-
ments PB(x) = P(x) on subsystem B [2] and it is defined as by the supremum
(2.2) J(ρ ) = S(ρ )− min S(A|{PB(x)}),
AB A
{PB(x)}
where
S(A|PB(x)) = p(x)S(ρ )+p(−x)S(ρ ),
A|PB(x) A|PB(−x)
p(±x) = tr[(I ⊗PB(±x))ρ (I ⊗PB(±x))],
A AB A
1
ρ = tr [(I ⊗PB(±x))ρ (I ⊗PB(±x))].
A|PB(±x) p(±x) B A AB A
SUPER QUANTUM DISCORD FOR GENERAL TWO QUBIT X STATES 3
Finally the super quantum discord SD(ρ ) is defined as the difference between I(ρ )
AB AB
and J(ρ ),
AB
SD(ρ ) = I(ρ )−J(ρ )
AB AB AB
(2.3) = S(ρ )−S(ρ )+ min S(A|{PB(x)}).
B AB
{PB(x)}
When limx → ∞, super quantum discord becomes the usual quantum discord under the
vonNeumannmeasurements. Thereforeitscomputationcanbeextremelychallenginggiven
that the discord is a nontrivial optimization problem over a parameterized manifold with
boundary.
3. SUPER QUANTUM DISCORD for TWO QUBIT X STATES
We consider the general two qubit X state written in the matrix form in terms of the
usual basis:
 
ρ 0 0 ρ
11 14
(3.1) ρAB =  00 ρρ3222 ρρ3233 00 .
ρ 0 0 ρ
41 44
As a density matrix, the coefficients ρ are complex numbers and satisfy the following
ij
4
conditions: (cid:80)ρ = 1, ρ ρ ≥ |ρ |2 , ρ ρ ≥ |ρ |2, ρ ∈ R, ρ = ρ∗ , and ρ = ρ∗ .
ii 22 33 23 11 44 14 ii 23 32 14 41
i=1
Introduce real parameters r = ρ − ρ + ρ − ρ , s = ρ − ρ − ρ + ρ , c =
11 44 22 33 11 44 22 33 3
ρ +ρ −ρ −ρ , and complex variables c = 2(ρ +ρ ) and c = 2(ρ −ρ ). Suppose
11 44 22 33 1 23 14 2 23 14
the real and imaginary parts of c are a and b (i = 1,2):
i i i
√
c = a + −1b .
i i i
Then the Bloch form of ρ is
AB
ρ
AB
1 (cid:88)
(3.2) = [I +c σ ⊗σ + a σ ⊗σ +sI ⊗σ +rσ ⊗I +b σ ⊗σ −b σ ⊗σ ],
3 3 3 i i i 3 3 2 1 2 1 2 1
4
i=1,2
where σ are the Pauli spin matrices:
i
√
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
0 1 0 − −1 1 0
σ = ,σ = √ ,σ =
1 1 0 2 −1 0 3 0 −1
It is easy to compute the eigenvalues of ρ :
AB
1 (cid:112)
(3.3) λ = (1+c ± (r+s)2+(a −a )2+(b −b )2),
1,2 3 1 2 1 2
4
1 (cid:112)
(3.4) λ = (1−c ± (r−s)2+(a +a )2+(b +b )2).
3,4 3 1 2 1 2
4
4 NAIHUANJING,BINGYU∗
The marginal state of ρ are then given by
AB
1 1
ρ = diag( (1+r), (1−r)),
A
2 2
1 1
ρ = diag( (1+s), (1−s)).
B
2 2
For |y| (cid:54) 1, define the entropic function
1 1
(3.5) E(y) = 1− (1+y)log (1+y)− (1−y)log (1−y).
2 2 2 2
Here the values at E(±1) are taken as lim E(y) = 0. Then the von Neumann entropies
y→±1
S(ρ ) and S(ρ ) are given by
A B
S(ρ ) = E(r),
A
S(ρ ) = E(s).
B
With these quantities, the quantum mutual information is computed as
4
(cid:88)
I(ρ ) = S(ρ )+S(ρ )+ λ log λ
AB A B i 2 i
i=1
= −4+E(r)+E(s)+E(c )
3
(cid:112) (cid:112)
1+c (r+s)2+|c −c |2 1−c (r−s)2+|c +c |2
3 1 2 3 1 2
(3.6) + E( )+ E( ).
2 1+c 2 1−c
3 3
The weak measurements {PB(x)} are associated with Π , which can be parameterized
i
through the the special unitary group SU(2). Up to a phase factor, any element V of
SU(2) can be written as V = tI+i(cid:80)3 y σ , where t,y ,y ,y are real numbers such that
i=1 i i 1 2 3
t2+y2+y2+y2 = 1. One can directly compute that
1 2 3
(3.7) V†σ V = (t2+y2−y2−y2)σ +2(ty +y y )σ +2(−ty +y y )σ ,
1 1 2 3 1 3 1 2 2 2 1 3 3
and V†σ V, V†σ V are obtained from (3.7) under the cyclic permutations (σ ,σ ,σ ) (cid:55)→
2 3 1 2 3
(σ ,σ ,σ ) and (y ,y ,y ) (cid:55)→ (y ,y ,y ).
2 3 1 1 2 3 2 3 1
Let
z = 2(−ty +y y ), z = 2(ty +y y ), z = t2+y2−y2−y2.
1 2 1 3 2 1 2 3 3 1 2
Then z2+z2+z2 = 1, thus |z| (cid:54) 1. It follows from (2.2) that
1 2
(3.8)
ρ
A|PB(x)
I(1−sztanhx)+(r−c ztanhx)σ −[(z a +z b )σ +(z a −z b )σ ]tanhx
3 3 1 1 2 2 1 2 2 1 1 2
=
2(1−sztanhx)
and ρ is given by replacing x with −x in (3.8). Here p(±x) = 1(1∓sztanhx).
A|PB(−x) 2
SUPER QUANTUM DISCORD FOR GENERAL TWO QUBIT X STATES 5
The eigenvalues of ρ and ρ are given by
A|PB(x) A|PB(−x)
(cid:112)
1−sztanhx± r2−2rzc tanhx+θtanh2x
(3.9) λ± = 3
ρA|PB(x) 2(1−sztanhx)
(cid:112)
1+sztanhx± r2+2rzc tanhx+θtanh2x
(3.10) λ± = 3 ,
ρA|PB(−x) 2(1+sztanhx)
where we have introduced a new variable θ by
(3.11) θ = z2|c |2+z2|c |2+2z z |c ×c |+c2z2.
1 1 2 2 1 2 1 2 3
→−
Recall that c ,c are given complex numbers, and c ×c = (a b −a b )k.
1 2 1 2 1 2 2 1
Notethatformulas(3.9)-(3.10)alwaysgiverealnumbersasθ (cid:62) c2z2, andtheyalsoimply
3
that
(3.12) r2±2rzc tanhx+θtanh2x (cid:54) (1±sztanhx)2,
3
which shows that θ is bounded above.
Tocalculatethesuperquantumdiscordofρ using(2.2)and(2.3), weneedtocalculate
AB
theclassicalcorrelationandminimizeS(A|{PB(x)})withrespecttotheweakmeasurements
{PB(±x)}. Using the formulas for the eigenvalues we find out that
(3.13) min S(A|{PB(x)}) = 1+minG(θ,z)
{PB(x)} z,θ
with
1 1+sztanhx+R
+
G(θ,z) =− (1+sztanhx+R )log
4 + 2 1+sztanhx
1 1+sztanhx−R
+
− (1+sztanhx−R )log
4 + 2 1+sztanhx
1 1−sztanhx+R
−
− (1−sztanhx+R )log
4 − 2 1−sztanhx
1 1−sztanhx−R
−
− (1−sztanhx−R )log ,
4 − 2 1−sztanhx
(cid:112)
where R = r2±2rc ztanhx+θtanh2x. The minimum is taken over a 2-dimensional
± 3
region such that |z| (cid:54) 1 and θ is implicitly bounded by (3.12).
Observe that G(θ,−z) = G(θ,z), so we only need to consider z ∈ [0,1]. Furthermore, we
can reduce the optimization of the two variable function to that of one variable.
√
Lemma 3.1. Let b2 = |c1|2+|c2|2+ (|c1|2−|c2|2)2+4|c1×c2|2, then the minimum of the quantity
2
S(A|{PB(x)}) is given by
(3.14) min S(A|{PB(x)}) = 1+ min F(z),
{PB(x)} z∈[0,1]
6 NAIHUANJING,BINGYU∗
where
1 1+sztanhx+H
+
F(z) =− (1+sztanhx+H )log
4 + 2 1+sztanhx
1 1+sztanhx−H
+
− (1+sztanhx−H )log
4 + 2 1+sztanhx
1 1−sztanhx+H
−
− (1−sztanhx+H )log
4 − 2 1−sztanhx
1 1−sztanhx−H
−
− (1−sztanhx−H )log
4 − 2 1−sztanhx
(cid:113)
and H = b2(1−z2)tanh2x+(r±c ztanhx)2.
± 3
See Appendix for a proof.
Theorem 3.2. The super quantum discord of the general two qubit X-state ρ is
AB
SD(ρ ) =S(ρ )−S(ρ )+ min S(A|PB(x))
AB B AB
{PB(x)}
(cid:112)
1+c (r+s)2+|c −c |2
3 1 2
=E(s)+E(c )+ E( )
3
2 1+c
3
(cid:112)
1−c (r−s)2+|c +c |2
3 1 2
(3.15) + E( )−3+ min F(z)
2 1−c3 z∈[0,1]
where E(y) is the entropic function defined in (3.5) and F(z) is given by Lemma 3.1.
The above result essentially determines the quantum super discord completely, as it is
expressedastheminimumofone-variablefunctionF(z)on[0,1]. Foragivenx,thefunction
F(z) depends on the complex parameters c ,c and there real parameters r,s,c , therefore
1 2 3
F(z) lives on a 7-dimensional manifold. Theorem 3.2 effectively reduces the parameters to
4 real ones b,c ,r,s together with the measurement parameter x. In the following we give
3
exact results for several nontrivial regions of the parameters of the quantum state ρ .
AB
Corollary 3.3. For the general two qubit X-type quantum state, the super quantum discord
is explicitly computed according to the following cases.
(a) If stanhx (cid:62) 0, rc tanhx (cid:54) 0 and c2−b2 (cid:62) src , then the super quantum discord is
3 3 3
given by (3.15) with min F(z) = F(1), and
z∈[0,1]
1 1+r+(s+c )tanhx
3
F(1) =− (1+r+(s+c )tanhx)log
4 3 2 1+stanhx
1 1−r+(s−c )tanhx
3
− (1−r+(s−c )tanhx)log
4 3 2 1+stanhx
(3.16)
1 1+r−(s+c )tanhx
3
− (1+r−(s+c )tanhx)log
4 3 2 1−stanhx
1 1−r−(s−c )tanhx
3
− (1−r−(s−c )tanhx)log
4 3 2 1−stanhx
(b) If stanhx (cid:54) 0, rc tanhx (cid:62) 0 and c2−b2 (cid:62) src , then the super quantum discord is
3 3 3
given by the same formula as in (a).
SUPER QUANTUM DISCORD FOR GENERAL TWO QUBIT X STATES 7
(c) If r = s = 0 and c2 ≤ b2, then the super quantum discord is given by (3.15) with
3
min F(z) = F(0), where
z∈[0,1]
F(0) =E(btanhx)−1
1
=− (1+btanhx)log (1+btanhx)
(3.17) 2 2
1
− (1−btanhx)log (1−btanhx).
2 2
(d) If s = rc , b2 = c2, and r2 +c2tanh2x±rc tanhx ≥ 1, then the super quantum
3 3 3 3
discord is given by (3.15) where min F(z) = F(0), where
z∈[0,1]
(cid:113)
F(0) =E( r2+c2tanh2x)−1
3
1 (cid:113) (cid:113)
(3.18) =− (1+ r2+c2tanh2x)log (1+ r2+c2tanh2x)
2 3 2 3
1 (cid:113) (cid:113)
− (1− r2+c2tanh2x)log (1− r2+c2tanh2x).
2 3 2 3
See Appendix for a proof.
Remark. Theabovecorollaryshowsthatthesuperquantumdiscordismostlydetermined
byF(1), buttherearestillothersolutionsnotcoveredbythisresult. Forexample, Example
3 below is not covered by the above result, and cannot be solved by any existing algorithms.
It is imperative to find a new method to resolve the situation. The following formula will
fill up the gaps in the literature, and covers all the situations for the general X-state.
Theorem 3.4. The exceptional optimal points of F(z) are determined by the iterative
formula:
F(cid:48)(z )
n
(3.19) zˆ= lim (z − ),
n→∞ n F(cid:48)(cid:48)(zn)
where
1 (cid:26) ((1+sztanhx)2−H2)(1−sztanhx)2
F(cid:48)(z) = − stanhxln +
4ln2 ((1−sztanhx)2−H2)(1+sztanhx)2
−
(3.20)
(cid:27)
1+sztanhx+H 1−sztanhx+H
+H(cid:48) ln + +H(cid:48) ln −
+ 1+sztanhx−H − 1−sztanhx−H
+ −
(cid:40)
1 (s2tanh2x+H(cid:48)2)(1+sztanhx)−2stanhxH H(cid:48)
F(cid:48)(cid:48)(z) = − + + +
2ln2 (1+sztanhx)2−H2
+
(s2tanh2x+H(cid:48)2)(1−sztanhx)+2stanhxH H(cid:48)
+ − − −
(1−sztanhx)2−H2
−
2s2tanh2x 1 1+sztanhx+H 1 1−sztanhx+H (cid:27)
(3.21) − + H(cid:48)(cid:48) ln + + H(cid:48)(cid:48) ln − .
1−s2z2tanh2x 2 + 1+sztanhx−H 2 − 1−sztanhx−H
+ −
As F(cid:48)(0) = F(3)(0), the iteration usually starts with z = 1.
0
8 NAIHUANJING,BINGYU∗
Example1. Letρ = 1(I+(cid:80)3 c σ ⊗σ )betheBell-diagonalstate. Thenr = s = 0. This
4 i=1 i i i
isaspecialcaseofCorollary3.3,sotheminimumofF(z)on[0,1]isF(0)orF(1). Ifc2 ≥ b2,
3
thenmin F(z) = F(1) = −1(1+c tanhx)log (1+c tanhx)−1(1−c tanhx)log (1−
z∈[0,1] 2 3 2 3 2 3 2
c tanhx); If c2 ≤ b2, then min F(z) = F(0) = −1(1+btanhx)log (1+btanhx)−
3 3 z∈[0,1] 2 2
1(1−btanhx)log (1−btanhx). Thus the super quantum discord of ρ is
2 2
1
SD(ρ) = (1−c +c +c )log (1−c +c +c )
4 3 1 2 2 3 1 2
1
+ (1−c −c −c )log ((1−c −c −c )
4 3 1 2 2 3 1 2
1
(3.22) + (1+c +c −c )log ((1+c +c −c )
4 3 1 2 2 3 1 2
1
+ (1+c −c +c )log ((1+c −c +c )
4 3 1 2 2 3 1 2
1 1
− (1+Ctanhx)log (1+Ctanhx)− (1−Ctanhx)log (1−Ctanhx).
2 2 2 2
This solution was first given in [14] with C = max{|c |,|b|}.
3
Note that the Werner state ρ = a|ψ−(cid:105)(cid:104)ψ−| + 1−aI, 0 ≤ a ≤ 1, is a special case with
4 √
r = s = 0,c = −a,c = c = −a. Here |ψ−(cid:105) = (|01(cid:105)−|10(cid:105))/ 2.
3 1 2
Example 2. Let ρ be the following density matrix:
 
0.437 0 0 0.100
 0 0.154 0 0 
(3.23)  .
 0 0 0.037 0 
0.100 0 0 0.372
Here r = 0.182,s = −0.052,c = 0.618,c = 0.2,c = −0.2, so b = 0.2. One sees that this
3 1 2
belongs to Corollary 3.3 (a) and (b), so min F(z) = F(1). Fig.1 shows that F(z) as
z∈[0,1]
a function of x ≥ 0 and z ∈ [0,1], we can observe the behaviour of F(z) more intuitively.
The eigenvalues of ρ are λ = 0.509649,λ = 0.299351,λ = 0.154,λ = 0.037. Following
1 2 3 4
(3.15), the super quantum discord of ρ is given by
1 1
SD(ρ) =2− (1+s)log (1+s)− (1−s)log (1−s)
2 2 2 2
(3.24) 4
(cid:88)
+ λ log λ +F(1) = 0.3899+F(1).
i 2 i
i=1
where
1 1.182+0.566tanhx
F(1) =− (1.182+0.566tanhx)log
4 2 1−0.052tanhx
1 0.818−0.64tanhx
− (0.818−0.64tanhx)log
4 2 1−0.052tanhx
1 1.182−0.566tanhx
− (1.182−0.566tanhx)log
4 2 1+0.052tanhx
1 0.818−0.64tanhx
− (0.818+0.64tanhx)log
4 2 1+0.052tanhx
SUPER QUANTUM DISCORD FOR GENERAL TWO QUBIT X STATES 9
Figure 1. The behaviour of F(z) for x ≥ 0 and z ∈ [0,1] with parameters
r = 0.182,s = −0.052,c = 0.618,b = 0.2.
3
One can easily prove that F(1) is an even function of x, and when x ≥ 0, F(1) decreases
withincreasingx. Whichmeansthesuperquantumdiscordofthisstateρisamonotonically
decreasing function of the measurement strength. This is consistent with the Theorem.2
in [13].
Example 3. The following example cannot be solved by any of the currently available
algorithmsuntilthispaper. Usingournewmethod, itsexactsolutionisobtainedasfollows.
Consider the density matrix ρ given by
 
0.0783 0 0 0
 0 0.1250 0.1000 0 
(3.25)  .
 0 0.1000 0.1250 0 
0 0 0 0.6717
In terms of the Bloch form, r = s = −0.5934,c = 0.5,c = c = 0.2, so b = 0.2. The
3 1 2
eigenvalues of ρ are λ = 0.025,λ = 0.0783,λ = 0.2250,λ = 0.6717. By symmetry, it
1 2 3 4
is enough to consider x > 0. The function F(z) in deciding the super quantum discord is
shown on the left side of Fig. 1 as a function of x ≥ 0 and z ∈ [0,1]. The right side of
Fig. 2 shows contour pictures of F(z) by choosing x = 1,2,3,4. The red dot on each line
is the minimum of F(z). The graphs reveal that the optimal point zˆ(cid:54)= 0,1, which can also
be computed explicitly by (3.19). This example shows that the claim that the maximum is
always given at either z = 0 or z = 1 is incorrect (cf. [17]), see the third and fourth graphs
in Fig. 2 for more information.
For example, set x = 3. Starting with with z = 1, (3.19) gives that z = 0.8305,z =
0 1 2
0.6718,z = 0.5582,z = 0.4964,z = 0.4788,z = 0.477467,z = 0.4774675,z = 0.4774676...,
3 4 5 6 7 8
thuszˆ= 0.47747istheoptimalpointofF(z). Itfollowsfrom(3.15)thatthesuperquantum
discord of ρ is SD(ρ) = 2− 1(1+s)log (1+s)− 1(1−s)log (1−s)+(cid:80)4 λ log λ +
2 2 2 2 i=1 i 2 i
F(0.47747) = 0.1332.
10 NAIHUANJING,BINGYU∗
Similarly if x = 4, z = 1, (3.19) gives that z = 0.9042,z = 0.8561,z = 0.8467,z =
0 1 2 3 4
0.84638901,z = 0.846388659..., thus zˆ= 0.84639 is another critical point of F(z). Finally
5
thesuperquantumdiscordturnsouttobeSD(ρ) = 2−1(1+s)log (1+s)−1(1−s)log (1−
2 2 2 2
s)+(cid:80)4 λ log λ +F(0.84639) = 0.1328.
i=1 i 2 i
Figure 2. The behaviour of F(z) for x ≥ 0 and z ∈ [0,1] with parame-
ters r = −0.5934,s = −0.5934,c = 0.5,b = 0.2. The red dot on each line
3
represents the optimal point. The 3rd and 4th graphs show that the maxi-
mum is not given by z = 0 or z = 1, instead they are respectively given by
zˆ= 0.47747 and zˆ= 0.84639 as shown by the red dots.
4. DYNAMICS of SUPER QUANTUM DISCORD under PHASE DAMPING
CHANNEL
In this section, we discuss the behavior of the general 2-qubit X state ρ through the
AB
phase damping channels [20] with the Kraus operators {K }, where (cid:80) K†K = 1. Under
i i i i
the phase damping ρ evolves into
AB
(cid:88)
(4.1) ρ˜ = KA⊗KB ·ρ ·(KA⊗KB)†.
AB i j AB i j
i,j∈1,2
√
A(B) A(B)
where the Kraus operators are given by K = |0(cid:105)(cid:104)0| + 1−γ|1(cid:105)(cid:104)1|, and K =
√ 1 2
γ|1(cid:105)(cid:104)1|, with the decoherence rate γ ∈ [0,1]. Thus we have
1 (cid:88)
(4.2) ρ˜= [I +c σ ⊗σ +sI ⊗σ +rσ ⊗I + a (1−γ)σ ⊗σ
3 3 3 3 3 i i i
4
i=1,2
+b (1−γ)σ ⊗σ −b (1−γ)σ ⊗σ ],
2 1 2 1 2 1
The parameter γ also determines how severely the noise in the channel affects the super
quantum discord. Clearly, when γ = 0, super quantum discord is preserved. As we have
mentioned that the super quantum discord tends to decrease when the strength x increases