Teleportation via maximally and non-maximally entangled mixed states

Teleportation via maximally and non-maximally entangled mixed states

Satyabrata Adhikari tapisatya@gmail.com Dept. of Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215, India    Archan S. Majumdar archan@bose.res.in Dept. of Astro Phys. and Cosmology, S. N. Bose National Centre of Basic Sciences, Salt lake, Kolkata-700098, India    Sovik Roy sovik1891@gmail.com Dept. of Mathematics, Techno India, EM 4 / 1 - Salt Lake City, Kolkata-700091, India    Biplab Ghosh quantumroshni@gmail.com Dept. of Phys., Vivekananda College for Women, Barisha, Kolkata 700008, India    Nilkantha Nayak Centre for Applied Physics, Central Univ. of Jharkhand, 835205, India
July 12, 2019
Abstract

We study the efficiency of two-qubit mixed entangled states as resources for quantum teleportation. We first consider two maximally entangled mixed states, viz., the Werner statewerner (), and a class of states introduced by Munro et al. munro (). We show that the Werner state when used as teleportation channel, gives rise to better average teleportation fidelity compared to the latter class of states for any finite value of mixedness. We then introduce a non-maximally entangled mixed state obtained as a convex combination of a two-qubit entangled mixed state and a two-qubit separable mixed state. It is shown that such a teleportation channel can outperform another non-maximally entangled channel, viz., the Werner derivative for a certain range of mixedness. Further, there exists a range of parameter values where the former state satisfies a Bell-CHSH type inequality and still performs better as a teleportation channel compared to the Werner derivative even though the latter violates the inequality.

pacs:
03.67.-a, 03.67.Bg, 03.65.Ud

I Introduction

Quantum teleportation is one of the most relevant applications of quantum information processing. Teleportation requires the separation of a protocol into classical and quantum parts using which it is possible to reconstruct an unknown input state with perfect fidelity at another location while destroying the original copy. The original idea of teleportation introduced by Bennett et al. bennett () is implemented through a channel involving a pair of particles in a Bell State shared by the sender and the receiver. Later, Popescu popescu () showed that pairs in a mixed state could be used for imperfect teleportation. Further, it has been shown that if the two distant parties adopt a ”measure-and-prepare” strategy for teleporting an unknown quantum state, then the average fidelity of teleportation is at most which is the maximum fidelity achievable by means of local operations and classical communications popescu (); massar (); gisin (). A quantum channel could be useful for communication purposes only if its teleportation fidelity exceeds .

In practice it is difficult to prepare pure states, but rather the states obtained are generally mixed in their characteristics. Naturally, a question arises as to whether better average teleportation fidelities compared to that in classical protocols could be obtained if mixed states were used in quantum communication purposes. Therefore, the basic objective is to look for such mixed states which when used as quantum teleportation channels, give fidelity of teleportation higher than the classical fidelity . It has been found that Werner states werner () used as quantum teleportation channels give higher teleportation fidelityhorodecki1 (). Recently, the mixed state obtained from the Buzek-Hillery cloning machinebuzek-hillery () as a teleportation channel has been studiedadhikari ().

Similar to the case of pure entangled states, entangled mixed states can also be divided into two categories: (i) maximally entangled mixed states (MEMS) and (ii) non-maximally entangled mixed states (NMEMS). Those states that achieve the greatest possible entanglement for a given mixedness are known as MEMS, otherwise they are NMEMS. The notion of MEMS was first introduced by Ishizaka and Hiroshima ishizaka (). They proposed a class of bipartite mixed states and showed that entanglement of those states cannot be increased further by any unitary operations (e.g., the Werner state). Later, Munro et.al. (MJWK)munro () studied a class of states which has the maximum amount of entanglement for a given degree of purity and derived an analytical form for that class of MEMS. Apart from maximally entangled mixed states, there are also NMEMS which can be studied for some particular interest. Hiroshima and Ishizaka hiroshima () studied a NMEMS called Werner derivative which can be obtained by applying a unitary transformation on the Werner state.

The motivation for this work lies in performing a comparitive study of mixed states in their capacity to perform as efficient channels for quantum teleportation. It is known that not all entangled mixed states are useful for teleportation horodecki (). So the question arises as to whether all MEMS states could be used as teleportation channels. To this end we first explore the capability of the MJWK class of states munro () as teleportation channels by finding their average teleportation fidelity. We find an upper bound for mixedness beyond which the MJKW class of states is not useful for teleportation. We further show that Werner states always act as better teleportation channels for all finite values of mixedness, even though they are less entangled compared to MJWK states for a given entropy. We then focus on non-maximally entangled mixed states and probe a question: is there any family of NMEMS which outperforms existing NMEMS such as the Werner derivative stateshiroshima () when used for quantum communication purposes ? To address this issue we construct a new entangled mixed state which is the convex combination of an entangled mixed state and a separable mixed state. Our state is NMEMS since it does not fall in the class of Ishizaka and Hiroshima’s ishizaka () MEMS. We show that this class of NMEMS can serve better as quantum channel for teleportation compared to the Werner derivative for a range of values of mixedness.

The relation between nonlocality of states as manifested by the violation of Bell-CHSH inequalitieschsh () and their ability to perform as efficient teleportation channels is interesting. It has been shown that there exist mixed states that do not violate any Bell-CHSH inequality, but still can be used for teleportationpopescu (). Here we raise this question first with regard to MEMS states and show there exists states in this category which satisfy the Bell-CHSH inequality, but could be still useful for teleportation. We then consider NMEMS states and find a range of parameters for which our constructed state satisfies a Bell-CHSH type inequality but still outperforms the Werner derivative in teleportation, even though the latter violates the Bell-CHSH inequality. Finally, our comparitive study of teleportation by maximally and non-maximally entangled mixed states reveals that whereas in the former case, one class of states, i.e., Werner states, definitely outperforms another, i.e., MJWK states for all values of mixedness, the result for the NMEMS states that we consider depends on their degree of mixedness.

The paper is organized as follows. In section-II, we recapitulate some useful definitions and general results related to mixed states, their violation of local inequalities, and the optimal teleportation fidelities when they are used as teleportation channels. We illustrate these general results with the well-known example of the Werner state kim (). In section-III, we study the efficiency of the MJWK states munro () in teleportation. We then consider two different NMEMS in Section-IV. We first study the Werner derivativehiroshima () as a teleportation channel and also obtain the range of parameter values for which it violates the Bell-CHSH inequality. We next introduce another NMEMS and investigate its entanglement properties and efficiency as a teleportation channel. We further show that this new NMEMS satisfies the Bell-CHSH inequality. In Section-V we present a comparitive analysis of the MEMS as well as the NMEMS channels for teleportation, also highlighting their respective status vis-a-vis the Bell-CHSH inequality. Finally, we summarize our results in Section-VI.

Ii The Werner state as a teleportation channel

The Werner state is a convex combination of a pure maximally entangled state and a maximally mixed state. Ishizaka and Hiroshima ishizaka () showed that the entanglement of formation wootters () of the Werner state cannot be increased by any unitary transformation. Therefore, the Werner state can be regarded as a maximally entangled mixed state. In this section we will review the performance of the Werner state as a teleportation channel. Though most of the results presented here are well known kim (), our discussion is intended to set the stage for the analysis of other MEMS and NMEMS states that we perform later. To begin with, let us recall certain useful definitions on the entanglement, teleportation capacity and mixedness of general states.

The maximal singlet fraction is defined for a general state as bose ()

(1)

where the maximum is taken over all maximally entangled states .

The linear entropy for a mixed state is defined by munro ()

(2)

The concurrence for a bipartite state is defined as wootters ()

(3)

where ’s are the square root of eigenvalues of in decreasing order. The spin-flipped density matrix is defined as

(4)

The efficiency of a quantum channel used for teleportation is measured in terms of its average teleportation fidelity given by horodecki2 ()

(5)

where is the input pure state and is the output state provided the outcome is obtained by Alice. The quantity which is a measure of how the resulting state is similar to the input one, is averaged over the probabilities of outcomes , and then over all possible input states ( denotes the uniform distribution on the Bloch sphere ). It has been shown horodecki () that if a state is useful for standard teleportation, the optimal teleportation fidelity can be expressed as

(6)

where and ’s are the eigenvalues of the matrix . The elements of the matrix are given by

(7)

where ’s denote the Pauli spin matrices. Now, in terms of the quantity , a general resulthorodecki () holds that any mixed spin- state is useful for (standard) teleportation if and only if

(8)

The relation between the optimal teleportation fidelity and the maximal singlet fraction is given by horodecki1 ()

(9)

From Eqs.(6) and (9) it follows that

(10)

Now using the inequality verst ()

(11)

where denotes the negativity of the state, we have

(12)

We now recall a useful result on the the violation of the Bell-CHSH inequality by mixed states. Any state described by the density operator violates the Bell-CHSH inequality chsh () if and only if the inequality

(13)

holds, where ’s are eigenvalues of the matrix horodecki ().

Let us now review the Werner state as a resource for teleportation kim (). Though the Werner state can be represented in various ways, in the present work we express it in terms of the maximal singlet fraction. The Werner state can be written in the form

(14)

where is the singlet state and is the maximal singlet fraction corresponding to the Werner state. is also related to the linear entropy as

(15)

The concurrence of is given by

(16)

When the Werner state is used as a quantum channel for teleportation, the average optimal teleportation fidelity is given by horodecki1 (); badziag (); mista ()

(17)

Similarly, the relation between the teleportation fidelity and the concurrence of the Werner state is given by

(18)

In terms of the linear entropy , Eq.(17) can be re-written as

(19)

Further, we have

(20)

Now using the inequality (12) in equation (20), we have

(21)

which is the upper bound of the singlet fraction for the Werner state in terms of negativity.

We now review the status of the violation of the Bell-CHSH inequality by the Werner state. Using Eq.(7) the eigenvalues of the matrix are given by , where denotes the elements of the matrix . The Werner state violates the Bell-CHSH inequality iff , where is given by

(22)

Using Eq.(16) it follows that the Werner state satisfies the Bell-CHSH inequality although it is entangled when the maximal singlet fraction lies within the range

(23)

The optimal teleportation fidelity in terms of is given by

(24)

Moreover, from Eqs.(17) and (24) it follows that the Werner state can be used as a quantum teleportation channel (average optimal fidelity exceeding ) even without violating the Bell-CHSH inequality in the above domain.

Iii Teleportation via the Munro-James-White-Kwiat maximally entangled mixed state

Munro et al. munro (); wei () showed that there exist a class of states that have significantly greater degree of entanglement for a given linear entropy than the Werner state. In this section we will investigate whether the class of states introduced by Munro et al. could be used as a teleportation channel. We begin with the analytical form of the MEMS given by

(25)

where

(26)

with C denoting the concurrence of (25).
The form of the linear entropy is given by

(27)

To see the performance of the MEMS state (25) as a teleportation channel, we have to calculate the fidelity of the teleportation channel. We use the result given in Eq.(9) relating the optimal teleportation fidelity and the singlet fraction of a state . The maximal singlet fraction of the state described by the density operator using the definition (1) is found out to be

(28)

Using Eqs.(9) and (26), the optimal teleportation fidelity is given by

(29)

Now inverting the relation (27), i.e., expressing C in terms of , we can rewrite Eq.(29) in terms of the linear entropy as

(30)

It follows that the MJKW munro () maximally entangled mixed state (25) can be used as a faithful teleportation channel when the mixedness of the state is less than the value .

Note that since in specific cases of teleportation the teleportation fidelity depends upon the input states, it gives better results for some input states and worse for some other input states. But here we use the formula for average teleportation fidelity averaging over all input states. However, for specific cases of input states it is possible to perform a calculation for the best (or the worst) teleportation fidelity (rather than the averaged optimum) as we illustrate now. For example, if we consider the input state to be teleported is of the form

(31)

and if the teleportation channel is given by , the teleported state (using the standard teleportation protocol) after performing suitable unitary transformations corresponding to the four Bell-state measurement outcomes ,, and is given by (for the following two cases):
(i) For

(32)

and, (ii) for

(33)

To determine the efficiency of the teleportation channel, we calculate the distances between the input and output state using Hilbert Schmidt norm, and they are given by
(i) for

(34)

where and , and (ii) for

(35)

where where and . The teleportation fidelity can be easily calculated by using the formula . Clearly, the fidelity depends on the input state and hence one can easily calculate the best (or worst) fidelity by choosing some particular input state. However, the puprose of the present paper is to compare the average perfomance of various teleportation channels, and to this end henceforth in this work we will deal further with average optimal teleportation fidelities only.

Next, we return to the nonlocal properties of the state . Wei et al. wei () have studied the state from the perspective of Bell’s-inequality violation. Here we focus on the parametrization of the state given by Eq.(25) and demarcate the range of concurrence where the Bell-CHSH inequality is violated. In order to use the result (13) we construct the matrix as

(36)

The eigenvalues of the matrix are given by

(37)

In accord with Eq.(26), the eigenvalues (37) take two different forms which are discussed separately below:
Case-I: . The eigenvalues (37) reduce to

(38)

When , the eigenvalues can be arranged as . Therefore,

(39)

One can easily see that when , and hence, in this case the state violates the Bell-CHSH inequality.
Case-II: . The eigenvalues given by Eq.(37) reduce to

(40)

Now we can split the interval into two sub-intervals and , where the ordering of the eigenvalues are different.
(i) when , the ordering of the eigenvalues are . In this case one has

(41)

From Eq.(41) it is clear that when . Hence, the Bell-CHSH inequality is satisfied by .
(ii) when , the ordering of the eigenvalues are . Therefore, the expression for () is given by

(42)

From Eq.(42), it follows that when and hence the state violates the Bell-CHSH inequality. On the contrary, when , and hence the state satisfies the Bell-CHSH inequality although it is entangled. It was noticed earlier jmodopt () that the MJKW state needs a much higher degree of entanglement to violate the Bell-CHSH inequality compared to the Werner states. Our above results revalidate this fact.

We next consider a wider class of maximally entangled mixed states as proposed by Wei et al. wei (). The general form of a two qubit density matrix comprising a mixture of the maximally entangled Bell state and a mixed diagonal state is given by

(43)

where , , , and are non-negative real parameters. The normalization condition gives The entanglement of is quantified by

(44)

Therefore, the state is entangled only if . The correlation matrix for is given by:

(45)

The eigen values of the symmetric matrix are given by , , and . Now, the quantity is given by

(46)

Here one is led to the following two cases. Case (i): , when either and , for or and for ; and Case (ii): when either , for or for . In either case the Bell-CHSH inequality is violated if .

Now, our task is to find the condition when the state could be used as a teleportation channel. Hence, we have to find the condition under which . In this case is given by

(47)

Therefore, we have

(48)

It follows from Eq.(47) that

(49)

Writing the optimal teleportation fidelity in the above form enables a useful comparison with the teleportation capacity of the Werner state. Note that for either , or , one has . Hence, it follows that the average optimal teleportation fidelity of the Werner state can be written as

(50)

From Eqs.(49) and (50) it immediately follows that

(51)

which shows that the Werner state performs better as a teleportation channel than the general MEMS.

Iv Non-maximally entangled mixed states as teleportation channels

iv.1 The Werner Derivative

Hiroshima and Ishizaka hiroshima () studied a particular class of mixed states - Werner derivative - obtained by applying a nonlocal unitary operator on the Werner state, i.e., . The Werner derivative is described by the density operator

(52)

where with . The state (52) is entangled if and only if hiroshima ()

(53)

which futher gives a restriction on as .

Our aim here is to study how efficiently the Werner derivative works as a teleportation channel. To do this, let us start with the matrix for the state given by

(54)

The eigenvalues of the matrix () are . The Werner Derivative can be used as a teleportation channel if and only if it stisfies Eq.(8), i.e., , where

(55)

It follows that the Werner Derivative can be used as a teleportation channel if and only if

(56)

where . Solving (56) for the parameter , we get

(57)

Therefore, teleportation can be done faithfully via when the parameter a satisfies the inequality (53).

The fidelity of teleportation is given by

(58)

When , the Werner derivative reduces to the Werner state, and the teleportation fidelity also reduces to that of the Werner state given by Eq.(17). From Eq.(58), it is clear that is a decreasing function of , and hence from Eq.(57), one obtains

(59)

Further, we can express the teleportation fidelity given in Eq.(58) in terms of linear entropy as

(60)

Now we investigate whether the state violates the Bell-CHSH inequality using the condition given in Eq.(13). The real valued function for the Werner derivative state is given by

(61)

It follows that

(62)

where

(63)

For and to be real, . From the expression of and Eq.(53), it is clear that as . Hence . Next, from the expression of , it follows that . Now, we consider the following three cases separately:
Case-I: If and , then . In this case the Bell-CHSH inequality is respected by the state although the state is entangled there.
Case-II: If and , then . Thus in this range of the parameter a the Bell-CHSH inequality is violated by the state .
Case-III: Here we consider the situation when . In this case and hence holds for . The equality sign is achieved when . Therefore, in the case when the Werner derivative satisfies the Bell-CHSH inequality although it is entangled.

iv.2 A new non-maximally entangled mixed state

We construct a two-qubit density matrix as a convex combination of a separable density matrix and an inseparable density matrix where and denote the three-qubit GHZ-stateghz () and the W-statewstate () respectively. This construction is somewhat similar in spirit to the Werner state which is a convex combination of a maximally mixed state and a maximally entangled pure state. We exploit here the properties that the GHZ state and the W state are two qubit separable and inseparable states, respectively, when a qubit is lost from the corresponding three qubit states. By constructing this type of a non-maximally entangled mixed state, our aim is to show that it can be used as a better teleportation channel compared to the Werner derivative state.

The two-qubit state described by the density matrix can be explicitly written as

(64)

The matrix representation of the density matrix in the computational basis is given by

(65)

Since the state described by the density matrix (65) is of the form

(66)

its amount of entanglement bruss () is given by

(67)

Therefore, is entangled only if , i.e., when .

Note that in the limiting case of the state reduces to

(68)

where . The state is maximally entangled since it can be put into Ishizaka and Hiroshima’s ishizaka () proposed class of MEMS. The concurrence of this state is . When this state is used as a teleportation channel, the teleportation fidelity becomes . Moreover, it can be checked that the state satisfies the Bell-CHSH inequality although it is an entangled state.

To obtain the teleportation fidelity for the state , we first construct the matrix using Eq.(7), which is given by

(69)

The eigenvalues of () are given by and . When , one has . Therefore, the teleportation fidelity becomes . Hence for , the state cannot be used as an efficient teleportation channel since it does not overtake the classical fidelity. But when , , and hence can be used as an efficient teleportation channel. In this case the average optimal teleportation fidelity is given by

(70)

and it follows that

(71)

We note here an interesting fact that the state cannot be used as an efficient teleportation channel when although the state is entangled there.

When is used as a quantum teleportation channel the mixedness of the state is given by

(72)

Therefore, the teleportation fidelity in terms of is given by