Two-qubit mixed states and teleportation fidelity: Purity, concurrence, and beyond
TTwo-qubit mixed states and teleportation fidelity: Purity, concurrence, and beyond
Sumit Nandi,
1, 2, ∗ Chandan Datta,
1, 2, † Arpan Das,
1, 2, ‡ and Pankaj Agrawal
1, 2, § Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, Odisha, India. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India.
To explore the properties of a two-qubit mixed state, we consider quantum teleportation. Thefidelity of a teleported state depends on the resource state purity and entanglement, as characterizedby concurrence. Concurrence and purity are functions of state parameters. However, it turnsout that a state with larger purity and concurrence, may have comparatively smaller fidelity. Bycomputing teleportation fidelity, concurrence and purity for two-qubit X-states, we show it explicitly.We further show that fidelity changes monotonically with respect to functions of parameters - otherthan concurrence and purity. A state with smaller concurrence and purity, but larger value of oneof these functions has larger fidelity. These functions, thus characterize nonlocal classical and/orquantum properties of the state that are not captured by purity and concurrence alone. In particular,concurrence is not enough to characterize the entanglement properties of a two-qubit mixed state.
I. INTRODUCTION
Quantum entanglement has remained a major resourcefor accomplishing quantum information processing taskssuch as teleportation [1], quantum key distribution [2],secret sharing [3] etc. The key idea of teleportation, asproposed by Bennett et al. [1] in their seminal paper,is to transmit an unknown state to a remote locationusing the entanglement as a resource. In the paper,the authors used a singlet state which is known tobe maximally-entangled state as a resource. With thisresource, an unknown one-qubit state can be transmittedwith unit fidelity and unit probability - i.e., perfectteleportation. In a teleportation scheme, the resourcestate is used as a channel for transmission to a distantlocation. The efficiency of a channel is quantified byteleportation fidelity which takes its maximum value formaximally-entangled Bell states. A classical channel canalso be used to perform teleportation with fidelity upperbounded by [4]. A unit fidelity is possible only if theresource state is maximally entangled.In a realistic scenario, a resource state is usually amixed state. It happens due to the unavoidable noise,interaction with the environment which eventuallyturns a pure state into a mixed one. It was Popescu[4] who showed that a mixed state can be sometimeuseful for teleportation producing fidelity greater thanthe classical value. A mixed state has both classicaland quantum properties. Their interplay can by quitecomplex. Usually, classicality of a state is characterizedby how pure a state is. Here, by classicality we meanthe classical correlations in the state due to the mixingparameters. The popular measures are - von Neumannentropy, linear entropy, or purity. We shall use thelater which can be defined as P = T r [ ρ ]; it attains its ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] maximum value one for a pure state. Nonlocal quan-tum properties are characterized by an entanglementmeasure, such as concurrence, or negativity. One mayexpect that the teleportation fidelity may depend notonly on the entanglement of a mixed state, but alsoclassicality. ˙Zyczkowski et al. [5] have shown that stateswith purity less than are separable. It has also beenproved [6] that if a state exceeds a certain degree ofmixedness as quantified by von Neumann entropy, thenit can’t be used as teleportation channel. Likewise in[7] the authors have obtained rank dependent lowerbound on concurrence and upper bound on mixednessof a state for the success of the teleportation protocol.Verstraete et al. [8] have found an upper bound onfidelity in terms of its entanglement as measured byconcurrence [9] or negativity [10]. As we shall see, thereexist states with smaller purity or/and concurrence,but higher teleportation fidelity. In this paper, we willobtain analytical expressions relating three quantitiesnamely, purity, concurrence and fidelity. We go beyondthis also. We will analyze the comparative behavior fora class of states, which encompass many well knownstates. The number of independent parameters in thisclass of states go beyond two quantities – purity andconcurrence – that are normally used to characterize amixed state. By considering mixed states with fewerparameters, we will show that optimal teleportationfidelity monotonically increases or decreases with newfunctions of the state parameters. So these extrafunctions also characterize the classical and nonlocalquantum properties of a mixed state. Purity andconcurrence are not enough to fully characterize a state,and to understand the optimum implementation of acommunication protocol. For example, larger purity,or/and larger concurrence does not necessarily implylarger teleportation fidelity. There are other functions ofthe state parameters that are also needed to charaterizethe nonlocal properties of the state as we show lessentangled and less pure state can produce better fidelity.Although, we shall demonstrate the usefulness of thesefunctions in the context of teleportation fidelity, samefunctions play similar role in characterizing the Bell a r X i v : . [ qu a n t - ph ] O c t violation [11]. Recently there has been similar discussionin the context of entanglement purification [12]. It hasbeen shown that the entanglement purification protocolcan be enhanced if we have access to the full parameterspace of a two-qubit mixed state. Our motivation isslightly different in the sense that we show that nonlocalproperties of a mixed state do require physical quantitiesother than purity and concurrence. We present our dis-cussion using X-states as resource states for the protocol.Recently Mendon¸ca et al. [13] have shown that forevery two-qubit state there exists a density matrixparameterized by seven parameters which resembles thealphabet X and popularly known as X-state in literatureand with same purity and concurrence as of the former.Moreover, they have shown that the whole concurrence-purity region of any two-qubit state can be covered withonly X-states. Many families of two-qubit states likeWerner state, Bell states, maximally-entangled mixedstate have this kind of structure. X-states were firstshown to have interesting properties in a paper byYu and Eberly [14]. Ever since there has been a largeamount of literature [13, 15] about X-states. In our work,we have exploited the parametrization of X-states fordifferent ranks [13] to find out the comparative relationsfor teleportation fidelity. We emphasize the importanceof functions of state parameters other than purity andconcurrence in characterizing the nonlocal properties ofa two-qubit mixed state. We have organized the paperas follows. In the next section, we have discussed someuseful quantities related to our work. In the subsequentsections we present our results using two-qubit X-states.Finally we shall conclude in the last section. II. PRELIMINARIES
In this section, we introduce some relevant quanti-ties which are required to obtain our results. Amongmany measures of entanglement of a two-qubit system,concurrence [9] is extensively used so far in many con-texts. Concurrence C ( | ψ (cid:105) ) of a pure state | ψ (cid:105) is definedas C ( | ψ (cid:105) ) = (cid:104) ψ | ˜ ψ (cid:105) , where | ˜ ψ (cid:105) = ( σ y ⊗ σ y ) | ψ ∗ (cid:105) . Here ( ∗ )is complex conjugate of | ψ (cid:105) in computational basis and σ y is Pauli matrix. Concurrence [9] of a two-qubit mixedstate ρ is defined as C = max(0 , λ − λ − λ − λ ) , (1)where λ i ’s are the eigenvalues, in descending order of thematrix (cid:112) √ ρ ˜ ρ √ ρ . Here ˜ ρ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ), where‘ ∗ ’ denotes the conjugate of ρ . For Bell states it yields1 while for separable states C ( ρ ) = 0. To define theteleportation fidelity, we shall use the prescription givenby Horodecki et al. [18]. Maximum teleportation fidelityof a two-qubit state ρ can be expressed as F max ≤ (cid:2) √ T † T (cid:3) , (2) where the elements of the matrix T are defined as t mn =Tr[ ρ ( σ n ⊗ σ m )] and m, n = (1 , , ρ with Tr √ T † T >
1, theinequality in Eq. (2) can be replaced by a equality as[18] F max = 12 (cid:2) √ T † T (cid:3) . (3)However, throughout the manuscript we will always con-sider the upper bound of fidelity F = (cid:2) Tr √ T † T (cid:3) which is the expression of optimal fidelity . III. TWO QUBIT X-STATE
The parametric form of an arbitrary two-qubit X-stateof a bipartite system can be represented as follow [13]: ρ = cos θ √ xe iµ θ cos φ √ ye iν √ ye − iν sin θ sin φ cos ψ √ xe − iµ θ sin φ sin ψ . (4)with θ, φ, ψ ∈ [0 , π ], x, y ≥ µ, ν ∈ [0 , π ]. How-ever these conditions are not enough to make Eq. (4)a valid density matrix. Further constraints x ∈ [0 , H ]and y ∈ [0 , G ] are required to make it positive semidef-inite. Here we define H = sin θ cos θ sin φ sin ψ and G = sin θ cos φ sin φ cos ψ . Mendon¸ca et al. [13] havegiven a parametrization of two-qubit X-states of differ-ent ranks. A two-qubit X-state will be of rank one if( x = H , y = 0 , A = 0) or ( x = 0 , y = G , B = 0), where A = sin θ (1 − sin φ sin ψ ) and B = 1 − A . A rank twoX-state can be parametrized as ( x < H , y = 0 , A = 0) or( x = 0 , y < G , B = 0) or ( x = H , y = G , AB > x < H , y = G , A >
0) or ( x = H , y < G , B > x < H , y < G , AB > IV. CONCURRENCE, OPTIMAL FIDELITYAND PURITY FOR GENERAL X-STATES
In [13], the following expressions for purity and con-currence for an arbitrary X-state have been obtained, P = 1 − AB + G − y + H − x ) (5) C = 2 max [ √ x − √G , √ y − √H ] (6)Using Eq. (2), we have evaluated the expression for op-timal teleportation fidelity for the general X-state as, F = 16 (cid:104) (cid:113) ( √ x + √ y ) + 2 (cid:113) ( √ x − √ y ) + (cid:114)(cid:16) cos θ − sin θ (cid:0) cos φ + sin φ cos 2 ψ (cid:1)(cid:17) (cid:105) . (7)Now to remove the last square root in the above expres-sion, we write Eq. (7) as F = 16 (cid:104) (cid:113) ( √ x + √ y ) + 2 (cid:113) ( √ x − √ y ) +sgn (cid:0) k ( θ, φ, ψ ) (cid:1) k ( θ, φ, ψ ) (cid:105) , (8)where k ( θ, φ, ψ ) = (cid:16) cos θ − sin θ (cid:0) cos φ +sin φ cos 2 ψ (cid:1)(cid:17) and ‘sgn’ represents the sign function. Eq. (8) will givetwo different expressions depending on the choice, x > y or y > x . Depending on that optimal fidelity expressionwill involve x or y and will be F = 16 (cid:104) √ a + sgn (cid:0) k ( θ, φ, ψ ) (cid:1) k ( θ, φ, ψ ) (cid:105) , (9)where a = max[ x, y ]. As stated earlier, depending uponthe parameters values and ranges, we can categorize aX-state as second rank, third rank, or fourth rank state.We will deal with the case of each rank separately asthe complexity of the functional relationship will growwith rank. Before starting with second rank, we notethat pure states have purity as one; for each such state,optimal teleportation fidelity and concurrence are relatedas, F = 13 (2 + C ) . (10)We can see that for any pure entangled state fidelity islarger than . But it is not the case always for a mixedstate, as is known, and as we will see in the subsequentsection. V. ANALYSIS FOR SECOND RANK X-STATESA. Second rank X-states of first kind
We first consider the second rank states of first type,i.e x < H , y = 0 , A = 0. First putting y = 0, in thegeneral expression for purity and concurrence we get, P = 1 + 2 x − p sin θ − q sin θ, (11) C = 2( √ x − f sin θ ) , (12)where, f = cos φ sin φ cos ψ, (13) p = sin φ sin ψ, (14) q = sin φ (cos φ cos ψ − sin ψ ) . (15) And optimal fidelity is given by, F = 16 (cid:104) √ x ± cos θ ∓ e sin θ (cid:105) , (16)where, e = cos φ + sin φ cos 2 ψ. (17)and the origin of ± sign in Eq. (16) is coming due tothe sign function in (8). Now, situation for the rank twostate will be simpler because we also have A = sin θ (1 − sin φ sin ψ ) = 0. That will make either θ = 0 or φ = ψ = π/
2. No other solutions are possible. So it willsuffice to consider these two cases. But for θ = 0, wehave H = 0. As x < H , it should be negative (becausewith the other restrictions for the first kind of secondrank X-states, x must be less than H , otherwise it won’tbe a second rank X-state). But by definition of X-states, x is non-negative. So, if we allow x to be at most equalto H (which is not allowed anyway), which is zero in thiscase, we just get a pure state. So, θ = 0 is not a validsolution here. Now, when φ = ψ = π/ p = 1 , q = − , and e = − . (18)Putting these in the expressions for purity, optimal fi-delity and concurrence, we get, P = 1 + 2 x − θ + 2 sin θ, (19) C = 2 √ x and (20) F = 13 (cid:104) √ x (cid:105) . (21)Using Eq.(20) and Eq.(21) we obtain F = 13 (2 + C ) . (22)Again, it is evident that optimal fidelity is greater thanclassical value whenever the state is entangled. This re-lation is same as that for pure states. B. Rank-2 X-states of second kind
For this we have the parametrization, B = 0 , x = 0 , y < G . All the expressions for purity, concurrence and opti-mal fidelity will be same as before, just x will be replacedby y . P = 1 + 2 y − p sin θ − qsin θ, (23) C = 2( √ y − f (cid:48) sin θ cos θ ) and (24) F = 16 (cid:104) √ y ± cos θ ∓ e sin θ (cid:105) . (25)Here, f (cid:48) = sin φ sin ψ . Now the condition B = 0, i.e A = sin θ (1 − sin φ sin ψ ) = 1 will make either θ = π/ φ = 0, or θ = π/ ψ = 0. No other solutions arepossible. But for the first choice, we have, G = 0. So y should be negative. So this is not a valid solution by thesame arguments as above. Let us see what happens forthe other solution i.e, θ = π/ ψ = 0. In this case,we have p = 0 , q = sin φ cos φ, f (cid:48) = 0 and e = 1 . (26)We obtain, P = 1 + 2 y − φ cos φ, (27) C = 2 √ y and (28) F = 13 (cid:104) √ y (cid:105) So finally we have F = 13 (2 + C ) . (29)Situation is same as before, i.e. optimal fidelity is inde-pendent of purity and we will get optimal teleportationfidelity always greater than classical value as long as thestate is entangled. C. Rank-2 X-states of third kind
This is characterized by, x = H , y = G , ≤ A ≤
1. Theexpressions change accordingly. Maximum concurrencecan be 2 √ x − √ y or 2 √ y − √ x . We begin with thefirst choice such that x > y . We get, P = 1 + 2 r sin θ + 2 r sin θ, (30) C = 2 √ x − √ y and (31) F = 16 (cid:104) √ x ± cos θ ∓ e sin θ (cid:105) . (32)Here, r = − φ sin ψ. (33)From Eq. (30) we solve for sin θ and get,sin θ = − ± √ P − r = V ( P , φ, ψ ) . (34)Here we choose F = (cid:104) √ x + cos θ − e sin θ (cid:105) . FromEq. (31) we write, 4 √ x = 2 C + 4 √ y and using these, F = 16 (cid:104) C + 4 √ y − (1 + e ) V ( P , φ, ψ ) (cid:105) . (35)As e + 1 = − r , we get F = 16 (cid:104) C + 4 √ y + ( − ± √ P − (cid:105) . (36)To make it optimum, we choose the plus sign and hence, F = (cid:104) C + 4 √ y + ( − √ P − (cid:105) . Now one cantake the other sign of fidelity as well i.e., F = (cid:104) √ x − cos θ + e sin θ (cid:105) . In this case by substituting thevalue of sin θ we get F = 16 (cid:104) C + 4 √ y − ( − ± √ P − (cid:105) . (37)To make it optimum, we choose the minus sign in theexpression of sin θ , i.e., in Eq. (34). Both these expres-sions are same but depending on the situation, we needto choose the sign of sin θ properly. The plus or minussign in optimal fidelity expression can be fixed by thesign of Eq. (34). Therefore, without loosing generality,we can take the optimal fidelity expression as F = 16 (cid:104) C + 4 √ y + ( − √ P − (cid:105) . (38)We will encounter similar kind of situation for other ranksof X-states as well. By giving similar argument and with-out loosing generality we can take optimal fidelity as F = 16 (cid:104) √ a − (1 + e ) sin θ (cid:105) , (39)where, a = max[ x, y ] and we will consider accordingly thesign in the expression of sin θ (as in Eq. (34)) to makefidelity optimal. We will use this expression throughoutthe manuscript. Now, as the minimum value of (1+ e ) canbe zero, in the expression for sin θ we have to choose theminus sign for the optimum fidelity. For this choice, it isevident from Eq.(34) that sin θ = V ( P, φ, ψ ), decreasesas P increases for any φ, ψ . Hence from Eq.(38) optimalfidelity will also increase as the purity increases keepingconcurrence and other parameters fixed at any values.Also the optimal fidelity changes monotonically with re-spect to parameters other than purity, or concurrence,here y . This is one of the main message of this paper andas we will show in the following that the same conclusionholds for 3rd and 4th rank X-states also. The expressionof V ( P, φ, ψ ) shows a very interesting feature. As fidelityis always a real quantity, we must have
P ≥ /
2. So, thisphysical constraint also restricts the minimum purity asecond rank X-state can have. This fact is also evidentfrom the expression of purity, i.e., Eqn. (30). It can beshown that minimum value that P can take is 1 /
2. Asstated earlier, from Eqn. (35) and (36), it is evident thatfor a fixed value of y and C , optimal fidelity increases withthe increment of purity. We emphasize this fact by plot-ting optimal fidelity with respect to purity for y = 0 . C = 0 . F i d e li t y FIG. 1. Variation of optimal fidelity with purity for y = 0 . C = 0 . F i d e li t y FIG. 2. Variation of optimal fidelity with concurrence for y = 0 .
01 and P = 0 . see it explicitly in the following example. Let us considera state with P = 0 . C = 0 . y = 0 . θ ≈ . φ ≈ . ψ ≈ . F ≈ . P = 0 . C = 0 .
15 and y = 0 . θ ≈ . φ ≈ . ψ ≈ . F ≈ . y . From theoptimal fidelity expression, it is clear that it increasesmonotonically with y . This is the first example wherewe see the dependence of optimal fidelity on propertiesother than purity and concurrence. This parameter alsoseems to characterize the nonlocal properties of the state.Now if we consider the second choice of concur-rence i.e., C = 2 √ y − √ x when y > x . Everything willremain same except x in Eq. (32) will change to y and y in Eq.(35) and (36) will change to x . All the argumentsand results remain the same. VI. ANALYSIS FOR THIRD RANK X-STATESA. Rank-3 X-states of first kind
Third rank X-state of first kind is characterized by, x < H , y = G , A >
0. Given that we get, P = 1 + 2 x − θ + 2 d sin θ, (40) C = 2 max [ √ x − f sin θ, √ y − f (cid:48) sin θ cos θ ] (41) F = 16 (cid:104) √ a + cos θ − e sin θ (cid:105) , (42)where, e, f, f (cid:48) , a are same as before and, d = 1 − sin φ sin ψ + sin φ sin ψ. (43)We first take x > y . As x > y implies x > G and x < H ,we have H > x > G and for this choice, C is 2( √ x − f sin θ ) and F is (cid:104) √ x + cos θ − e sin θ (cid:105) . Now,solving for sin θ from Eq. (40) and (41) we get,sin θ = V ( P , C , φ, ψ )= (1 − f C ) ± (cid:112) (1 − f C ) − (1 − P + C / d + 2 f )(2 d + 2 f ) (44)Using the expression for C and the evaluated sin θ , wenow write optimal fidelity F in terms of C , P , d, e and f as, F = 16 (cid:2) C − (1 + e − f ) V ( P , C , φ, ψ ) (cid:3) . (45)As stated for the 2nd rank case, to get optimum fidelity,we have to choose the minus sign of V ( P , C , φ, ψ ) as theminimum value of (1 + e − f ) can be zero. Then it isevident from the expression that for any values of φ, ψ and C , F increases with P , as V ( P , C , φ, ψ ) decreaseswith the increase of P . So as before the same resultholds for 3rd rank X-states of first kind. To illustratethis behavior graphically, we set φ = π , ψ = π and thenthe Eq. (45) reduces to the following form, F = 118 (cid:18)
10 + 6 C + (cid:112) P − C − (cid:19) . (46)We plot this expression for optimal fidelity F with purity P for a fixed entanglement, i.e concurrence C . FIG. 3,shows that for a fixed entanglement, the optimal fidelityincreases with purity. Like second rank X-state, optimalfidelity of these states also increases with concurrence forfixed purity as shown in FIG. 4. Moreover the right handside of Eq.(45) involves few more parameters. Here weare giving a very interesting example. Consider a statewith P = 0 . C = 0 . φ = π and ψ = π . For thisstate, we have F ≈ . ψ and φ give d = , e = 0 and f = 0. For another state with P = 0 . C = 0 . φ = π and ψ = π , we get F ≈ . d ≈ . e = 0 . f = 0. So withless entanglement and purity one can have more optimalfidelity for different values of φ and ψ or d , e and f . Now,let us see the ranges of d , e and f . We obtain34 ≤ d < − < e < ≤ f <
12 (49)The ranges of d , e and f have been obtained by maximiz-ing and minimizing the functions independently. How-ever, as they all are functions of φ and ψ , they cannot be varied independently. As optimal fidelity de-pends on those parameters also rather than only de-pending on purity and concurrence, we have plottedvariations of optimal fidelity with those parameters inFig.(5) and Fig.(6) showing optimal fidelity decreasesmonotonically with e , whereas it increases monotonicallywith f . In the figures the ranges of the parameters e and f have been appropriately modified. Now, F i d e li t y FIG. 3. Variation of optimal fidelity with purity for d = , e = 0 and f = 0 with C = 0 . we are left with the situation when y > x . In thiscase the concurrence will be 2( √ y − f (cid:48) sin θ cos θ ) and F = (cid:104) √ y + cos θ − e sin θ (cid:105) . Doing similar kindof calculation one can show that F = 16 (cid:104) C + 4 f (cid:48) (cid:112) V ( P , x, φ, ψ )(1 − V ( P , x, φ, ψ )) − (1 + e ) V ( P , x, φ, ψ ) (cid:105) , (50)where V ( P , x, φ, ψ ) = ± √ − d (1+2 x −P )2 d . Here also onecan easily verify a similar kind of trend as before. F i d e li t y FIG. 4. Variation of optimal fidelity with concurrence for d = , e = 0 and f = 0 with P = 0 . F i d e li t y FIG. 5. Variation of optimal fidelity with ’ e ’ for f = 0, P = . C = 0 . B. Rank-3 X-states of second kind
This class of states are characterized by x = H , y < G , A <
1. We get, P = 1 + 2 y − t sin θ + 2 u sin θ, (51) C = 2 max [ √ x − f sin θ, √ y − f (cid:48) sin θ cos θ ] (52) F = 16 (cid:104) √ a + cos θ − e sin θ (cid:105) , (53)where, t = 1 − p . p, a are same as before and, u = 1 + sin φ sin ψ − cos φ cos ψ sin φ − φ sin ψ. (54)First we take x > y . As x > y implies H > y and y < G , concurrence will be 2( √ x − f sin θ ) and F is (cid:104) √ x + cos θ − e sin θ (cid:105) . After doing a calculationas above, we get F = 16 (cid:104) C − (1 + e − f ) V ( P , φ, ψ, y ) (cid:105) , (55) F i d e li t y FIG. 6. Variation of optimal fidelity with ’ f ’ for e = 0, P = . C = 0 . where V ( P , φ, ψ, y ) = t ± √ t − u (1+2 y −P )2 u . So, the situa-tion is similar as first kind and we would be getting simi-lar results. Now, we will consider the situation y > x i.e., G > y > H . Here concurrence will be 2( √ y +sin θ cos θf (cid:48) )and F to be (cid:104) √ y + cos θ − e sin θ (cid:105) . In this situ-ation calculation will be slightly different. The reason isthat now the expression for purity P involves y , not x .So, in this case we will be getting a 4th order equationof sin θ from the expression of P and C . We will notdo this in this section as in the next section for 4th rankX-states we will discuss a similar situation. VII. ANALYSIS FOR RANK-4 X-STATES
General fourth rank X-states will be characterized by x < H , y < G , AB >
0. Putting the values of A , B , G , H we get the values of P and C as, P = 1 + 2 x + 2 y − θ + 2 sin θg, (56) C = 2 max[ √ x − f sin θ, √ y − f (cid:48) sin θ cos θ ] (57) F = 16 (cid:104) √ a + cos θ − e sin θ (cid:105) , (58)where, g = 164 [53 + 4 cos 2 φ + 7 cos 4 φ + 8 cos 4 ψ sin φ ] , (59) f = (cid:113) sin φ cos φ cos ψ and (60) f (cid:48) = (cid:113) sin φ sin ψ. (61)First, we choose x > y and also √ x − f sin θ > √ y − f (cid:48) sin θ cos θ . So, we take C to be 2( √ x − f sin θ ) and F to be (cid:104) √ x + cos θ − e sin θ (cid:105) . Now, from Eq. (56) and (57) we get,sin θ = V ( P , C , φ, ψ, y ) =(1 − f C ) ± (cid:112) (1 − f C ) − (1 − P + 2 y + C / g + 2 f )(2 g + 2 f ) . (62)Using the expression for C and the evaluated sin θ , wenow write optimal fidelity F in terms of C , P , φ, ψ and y as, F = 16 (cid:2) C − (1 + e − f ) V ( P , C , φ, ψ, y ) (cid:3) . (63)In similar fashion here also we can argue that as purityincreases keeping others constant, optimal fidelity alsoincreases and also it is evident from the FIG. 7. For thisplot we choose ψ = φ = π or ψ = 2 tan − ( (cid:112) − √ φ = π and y = 0. For these values of φ and ψ , g = , f = √ and e = . FIG. 8 shows the variation ofoptimal fidelity with concurrence for a fixed value of pu-rity. Now for x > y , we could have √ x − f sin θ < F i d e li t y FIG. 7. Variation of optimal fidelity with purity for g = , e = , f = √ and y = 0 with C = 0 . √ y − f (cid:48) sin θ cos θ . So C = 2( √ y − f (cid:48) sin θ cos θ ). In thiscase from the expression of P and C we will get a fourthorder equation for sin θ , which will not involve y In prin-ciple we will get four solutions of sin θ from this equationas a function of φ, ψ and x . Putting these solutions ofsin θ in the expression for F , we will have F as a func-tion of C, P, φ, ψ and x . The fourth order equation willbe similar like we will derive in the following for the caseof y > x and √ x − f sin θ < √ y − f (cid:48) sin θ cos θ . So, letus consider the case when y > x and √ x − f sin θ < √ y − f (cid:48) sin θ cos θ , we have the value of concurrence tobe 2( √ y − f (cid:48) sin θ cos θ ). In this case we need to replace x by y in the optimal fidelity expression given in Eq. (58).Here also we will get a fourth order equation of sin θ .Using the expression for purity P and concurrence C , we F i d e li t y FIG. 8. Variation of optimal fidelity with purity for g = , e = 0, f = √ and y = 0 with P = 0 . get the following equation, α sin θ + 2 αβ sin θ + [ β + 2 α (1 + 2 x + C / − P )+4 C f (cid:48) ] sin θ + [2 β (1 + 2 x + C / − P ) − C f (cid:48) θ + (1 + 2 x + C / − P ) = 0 , (64)where, α = 2 g − f (cid:48) and β = 2 f (cid:48) − . (65)From this equation, in principle one can get four solutionsfor sin θ and using that one can get the expression foroptimal fidelity F in terms of C , P , x , φ and ψ . As solvingthis equation will be very involved, we avoid that and getsome plots for some particular values of the parametersshowing the pattern. We choose x = 0, φ = ψ = π or φ = π , ψ = − − ( √ − √
3) and C = 0 .
2. In FIG.9, we see a trend as before, i.e, optimal fidelity increaseswith purity for a fixed concurrence. Now finally we are F i d e li t y FIG. 9. Variation of optimal fidelity with purity for g = , f (cid:48) = and x = 0 with C = 0 . left with y > x and √ x − f sin θ > √ y − f (cid:48) sin θ cos θ .In this case C = 2( √ x − f sin θ ) and F = (cid:104) √ y + cos θ − e sin θ (cid:105) . After few steps of calculation, we find F = 16 (cid:104) √ y − (1 + e ) V ( P , C , y, φ, ψ ) (cid:105) , (66)where, V ( P , C , y, φ, ψ ) = sin θ = (67)(1 − f C ) ± (cid:112) (1 − f C ) − (2 g + 2 f )(1 + C / y − P )(2 g + 2 f ) . From this expression also one can verify that the trendsare similar as above.
A. Uhlmann Fidelity
As we have seen, the optimal teleportation fidelitychanges monotonically with parameters, or functions ofparameters, of states. Question is apart from purity andconcurrence what other physical quantities these func-tions of parameters may be related to. In this subsec-tion, we consider one such physical quantity – UhlmannFidelity. It is known that the closeness of two states canbe characterized by Uhlmann fidelity [19]. For two ar-bitrary quantum states ρ and σ , the Uhlmann fidelity isdefined as [19] R = (cid:104) Tr (cid:113) √ ρσ √ ρ (cid:105) . (68)It is a relevant quantity that describes how far apart twostates are. Here we will compute the Uhlmann fidelityof a class of X-states with Bell states. As there are fourBell states, we take the maximum of the values. So wechoose σ of Eq.(68) as density matrices of Bell states.Then Uhlmann fidelity of the rank four X-states as inEq. (4) is R = max (cid:2)
14 (1 + e ) sin θ + √ y cos ν,
14 (1 + e ) sin θ − √ y cos ν,
14 (2 − (1 + e ) sin θ ) + √ x cos µ,
14 (2 − (1 + e ) sin θ ) − √ x cos µ (cid:3) . (69)There are four Uhlmann fidelities, one for each of the Bellstates. Uhlmann fidelity also corresponds to the transi-tion probability of one state to another state. We takethe maximum among four, because the maximum is themost probable state. As fidelity is independent of µ and ν , so without loosing generality we can choose µ = ν = 0.We find out R explicitly with C = 0 . P = 0 . f = √ and y = 0. For this state one can check that the Uhlmannfidelity is R = 14 (2 − (1 + e ) sin θ ) + √ x. (70)It is quite obvious from this expression that R is mono-tonically decreasing function of e . We have also seen inEq.(63) that optimal teleportation fidelity also monoton-ically decreases with e . To visualize it we have plotted R and F as a function of e in Fig. 10. One interpreta- FidelityUhlmann Fidelity0.5 0.6 0.7 0.8 0.90.500.550.600.650.700.750.800.85 e
FIG. 10. Variation of optimal teleportation fidelity andUhlmann fidelity with e for f = √ , y = 0, C = 0 . P = 0 . tion of the Fig. 10 is that the increment of parameter e is somehow introducing classicality in the system. Withincrement of e , Uhlmann fidelity is decreasing i.e., thestate is going far away from the Bell states and as a re-sult optimal fidelity is also decreasing. To emphasize theimportance of this quantity, we consider the states withsame values of concurrence and purity that have differentoptimal fidelity. Let’s see it explicitly with two followingstates: γ = 13 | ψ + (cid:105)(cid:104) ψ + | + 23 | ψ − (cid:105)(cid:104) ψ − | and γ = 13 | ψ + (cid:105)(cid:104) ψ + | + 23 | (cid:105)(cid:104) | , (71)where | ψ ± (cid:105) = √ (cid:0) | (cid:105) ± | (cid:105) (cid:1) . These states havesame purity and concurrence , but optimal fidelityis different, F ( γ ) = and F ( γ ) = respectively.Uhlmann fidelity of these states are given by and respectively i.e, higher Uhlmann fidelity corresponds tolarger teleportation fidelity also. It seems plausible asUhlmann fidelity is associated with distance betweentwo density matrices. So larger distance of a statefrom maximally entangled state implies larger deviationfrom nonlocality which in turn degrades its optimalteleportation fidelity. VIII. DISCUSSION AND CONCLUSIONS
We have studied nonlocal properties of two-qubitmixed states using teleportation protocol. We have useda class of states, X-states. The motivation to consider X-states originates from the fact that for every two-qubitmixed state, there is a X-state with same purity and con-currence [13]. The original state and the correspondingX-state are related by some global unitaries. We have ob-tained the dependence of optimal teleportation fidelityon the functions of the state parameters. All the rela-tions for optimal fidelity indicate that noise can reducethe effectiveness of state as a teleportation channel. Inconcurrence-purity region, we can find some entangledstates for each rank which have optimal fidelity less than . Below a certain value of purity, optimal fidelity doesnot increase if only concurrence is increased. Also concur-rence can not be changed arbitrarily keeping purity fixed.Moreover the amount of variation of optimal fidelity withpurity, for fixed amount of concurrence, depends on therank of the states. Our result also agrees with the workin reference [7]. Higher rank X-states give larger optimalfidelity for a fixed value of purity and concurrence. Ourinvestigations suggest that the nonlocal character of atwo-qubit mixed state is more involved, and require sev-eral quantities to fully characterize it. For example, for arank-3 X-state, the optimal fidelity depends not only onpurity and concurrence, but also on the functions e and f .All of these quantities are functions of state parameters.By choosing a specific set of values for the functions e and f , we find that the optimal fidelity changes monotoni-cally with concurrence and purity. However, the optimalfidelity also changes monotonically with functions e and f . This has been illustrated in a number of plots. Thus,these quantities also characterize the nonlocal (classicalor quantum) properties of the mixed states. Purity andconcurrence are not enough. They may characterize someaverage nonlocal properties. At some level, it is not sur-prising. Unlike a two-qubit pure state, a two-qubit mixedstate can have several independent parameters. However,we have found some specific functions of the state param-eters, which in addition to concurrence, also determinethe nonlocal properties of the state. The optimal tele-portation fidelity changes monotonically with respect tothese functions. Interestingly, Bell violation by a X-statevaries in the same way with respect to these functions e and f [11]. These extra functions of parameters, shouldbe related with other properties of the states which arenot captured by purity, or concurrence. We have consid-ered one such quantity, Uhlmann fidelity, and shown itsimportance. There should be many more such quanti-ties which are still to be found to fully characterize thenonlocal properties of two-qubit mixed states. AUTHOR CONTRIBUTION STATEMENT
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