Teleportation and Superdense coding with Genuine quadripartite entangled states

Abstract

We investigate the usefulness of different classes of genuine quadripartite entangled states as quantum resources for teleportation and superdense coding. We examine the possibility of teleporting unknown one, two and three qubit states. We show that one can use the teleportation protocol to send any general one and two qubit states. A restricted class of three qubit states can also be faithfully teleported. We also explore superdense coding protocol in single-receiver and multi-receiver scenarios. We show that there exist genuine quadripartite entangled states that can be used to transmit four cbits by sending two qubits. We also discuss some interesting features of multi-receiver scenario under LOCC paradigm.

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arXiv:0705.1917v1 [quant-ph] 14 May 2007
Teleportation and Superdense Coding with Genuine
Quadripartite Entangled States
B. Pradhan, Pankaj Agrawal and A. K. Pati∗
Institute of Physics
Sachivalaya Marg, Bhubaneswar, Orissa, India 751 005
February 1, 2008
Abstract
We investigate the usefulness of different classes of genuine quadripartite entangled
states as quantum resources for teleportation and superdense coding. We examine the
possibility of teleporting unknown one, two and three qubit states. We show that one
can use the teleportation protocol to send any general one and two qubit states. A
restricted class of three qubit states can also be faithfully teleported. We also explore
superdense coding protocol in single-receiver and multi-receiver scenarios. We show
that there exist genuine quadripartite entangled states that can be used to transmit
four cbits by sending two qubits. We also discuss some interesting features of multi-
receiver scenario under LOCC paradigm.
∗email: bpradhan@iopb.res.in,agrawal@iopb.res.in,akpati@iopb.res.in
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1Introduction
One of the intriguing feature of quantum mechanics is the quantum entanglement. This
feature has been exploited to do several amazing tasks which are otherwise impossible. In
particular, the entanglement can be used as a quantum resource to carry out a number
of computational and information processing tasks. Such tasks include teleportation of an
unknown quantum state [1], superdense coding [2], entanglement swapping [3], remote state
preparation [4, 5], secret sharing [6], quantum cryptography [7] and many others. Analysis
of such quantum phenomena may allow us a better understanding of the structure of the
quantum mechanics framework.
A quantum system may consist of two or more subsystems which may correspondingly
have bipartite or multipartite entanglement. Characterization and uses of bipartite entangle-
ment are better understood than those of multipartite entanglement. A number of protocols
which were first introduced in the context of a bipartite system can be extended to a multi-
partite system. However, in the case of multipartite systems, the entanglement environment
is quite complex and its nature is still not fully understood. Such entangled states can be
classified according to different schemes. These classes exhibit different types of entangle-
ment properties. All classes may not be suitable for some of the information processing
tasks. The tripartite states have been classified according to stochastic local operation and
classical communication (SLOCC) into six categories. Two of these categories have gen-
uine tripartite entanglement, viz. GHZ-states and W-states [8]. The utilities of these states
have been explored in a number of papers [9] − [20]. The quadripartite states have also
been classified according to SLOCC [21]. There are nine categories. Some of them have
genuine quadripartite entanglement. But usefulness of the multipartite states beyond tri-
partite states is still to be explored in some details. Such studies may even allow a better
understanding of multiparticle entanglement and classification of quantum states according
to their entanglement properties. It is worth mentioning that, a task-based classification
scheme have been proposed by Bruß et al [22, 23] to classify mixed states and multipartite
states according to their densecodabilty.
In this paper, we study various protocols for quantum teleportation and superdense
coding in the context of quadripartite entangled states. In this scenario, there can be two,
three, or four parties (Alice, Bob, Charlie and Dennis). These parties share four particles in
an entangled state. We shall take these states to be genuinely quadripartite entangled states
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in the sense that these states cannot be written as a direct product of bipartite entangled
states, or as a direct product of a tripartite and a single-particle state.
In the next section, we enumerate the quadripartite states of qubits that we shall be
considering. We shall explore the possibility of teleporting an unknown one-qubit, two-
qubit, or three-qubit state. It is known (and we review it) that it is possible to teleport an
unknown one-qubit state using a variety of quadripartite states. One can do this using a
number of different protocols. These protocols can involve only two-particle, or three-particle
or four-particle von Neumann measurements. The situation is a bit more complicated in the
case of the teleportation of a general two-qubit state. Although some special two-qubit states
can be teleported by a number of different quadripartite states, but a general state often
cannot be. We show that a specific state can be used to teleport a general two-qubit state.
This state is different from the one that was discussed in the literature [28]. A limited set
of three-qubit states can also be teleported by using some of the quadripartite states which
we discuss in the next section. However, to teleport an arbitrary three-qubit state, one may
need an appropriate six-qubit entangled state.
Apart from the teleportation protocol, we discuss superdense coding using quadripartite
states as a quantum resource. We discuss two scenarios: single-receiver and multi-receiver.
In both the cases, there is just one sender. In the case of single-receiver scenario, there
exist several possibilities: i) transmit two-cbits by sending one qubit, ii) transmit three, or
four-cbits by sending two qubits, iii) transmit four-cbits by sending three qubits. Here the
case of sending four cbits by sending two qubits is clearly more interesting. It turns out
that there exist quadripartite states that can be used as a quantum resource to accomplish
the task of transmitting four cbits by sending two qubits. More than four cbits cannot be
sent using quadripartite entangled state because the dimensionality of the Hilbert space of
four qubits is sixteen. We also discuss multi-receiver scenarios in the framework of LOCC
distinguishability of a set of orthogonal states.
The plan of the paper is as follows. In section II, we enumerate the quadripartite en-
tangled states that we consider in this paper. In section III, we discuss the use of these
states as a quantum resource for the teleportation. In section IV, we discuss the protocol of
superdense coding. Finally, in section V, we present our conclusions.
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2Quadripartite Entangled States
The bipartite entangled states are the simplest of the entangled states. Such states can be
classified and their entanglement quantified. All bipartite entangled states belong to one
equivalence class under SLOCC. The representative state of this class can be taken to be
any of the Bell states. These Bell states are maximally entangled states and can be used
fruitfully for the teleportation and the superdense coding as shown in the original papers
that introduced these protocols [1, 2].
One may think that if we go beyond bipartite entangled states to multipartite entangled
states, one may be able to accomplish tasks which are not otherwise feasible. However, the
nature of entanglement in the case of multipartite entangled state is multifaceted and far
from being understood. The genuine tripartite entangled states have also been classified on
the basis of SLOCC [8]. There are two classes: i) the class with the representative state,
|GHZ?, ii) the class with the representative state, |W?. States belonging to these classes
can be used to successfully carry out the protocol of teleportation and superdense coding
[9, 10, 11].
Beyond tripartite entangled states, one may consider quadripartite entangled states. On
the basis of SLOCC, such states have also been classified. We consider a set of states from
this classification and consider the possibility of implementing the protocol of teleportation
and superdense coding. These states are given by
|Q1? ≡ |GHZ? =1
|Q2? ≡ |W? =1
2(|0000? + |1111?)(1)
2(|0001? + |0010? + |0100? + |1000?)
1
√2(|0?|ϕ+?|0? + |1?|ϕ−?|1?)
(2)
|Q3? ≡ |Ω? =
(3)
|Q4? =
1
2(|0000? + |0101? + |1000? + |1110?)
1
2(|0000? + |1011? + |1101? + |1110?)
(4)
|Q5? =
(5)
Here |ϕ±? are Bell states defined in (15). According to the classification of Verstrate et
al [21] quadripartite states |GHZ? and |Ω? belong to Gabcdclass, the states |W?,|Q4? and
|Q5? belong to Lab3,L05⊕3and L07⊕1respectively. Among these five entangled states |GHZ?
and |W? states are symmetric with respect to permutation of particles; thus any quantum
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information task performed using these states are independent of distribution of particles
among the parties. For later three states it varies depending on the distribution of particles
among the parties.
There are a number of ways to see that the above states have genuine quadripartite
entanglement. One way is to find out the states of the system after tracing out one, two or
three particles. If the state is mixed in each case, then this would be an indication of genuine
quadripartite entanglement. This is what we observe. In addition we note the following. The
|GHZ? has a mixed 3-tangle [24] of zero when one of the party is traced out. The mixed
3-tangle for |W? is zero and concurrence of 1/2 when two qubits are traced out. For |Q4?
mixed 3-tangle of 1/2 is obtained if qubit 2,3 and 4 are traced out, zero when qubit 1 is
traced out. By tracing out qubit 1 and (3 or 4) one can get a concurrence equal to 1/2, while
the other concurrence vanish. The state |Q5? has a concurrences equal to zero if two qubits
are traced out and mixed 3-tangle of 1/2 if particle 2,3 and 4 are traced out. The |Ω? state is
the cluster state introduced by Briegel and Raussendorf [31, 32] . This state is considered to
have maximum connectedness and high persistence of entanglement and has been discussed
extensively in the context of one-way quantum computation. The concurrence of this state
is zero with any of the two qubits traced out.
3Teleportation
In this section, we consider the teleportation of the unknown states of one, two, and three
qubits. In the case of the teleportation the arbitrary state of one qubit, a number of situations
may exist. There may be just two parties, or more. There is a possibility of making four-
qubit, three-qubit, two-qubit, and one-qubit von Neumann measurements or a combinations
of them.As making a measurement involving a larger number of qubits may be more
difficult, it would be interesting to know if the protocol would work with the measurement
on fewer particles. In the case of the transmission of unknown two-qubit states, there can
be situations of two or three parties with quadripartite entangled states, or Alice could have
option of making different types of measurements. In the case of transmitting an unknown
three-qubit state, with quadripartite states as a quantum resource, there can be only two
parties: Alice and Bob.
We look at some of the above situations below. Depending on the number of parties, the
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classical communication cost of the protocol would be different. If there are more than two
parties, then the number of transmitted cbits would increase as the information about the
measurement results would be distributed. One issue in classical communication would be
the number of cbits that must be transmitted to Bob who wishes to convert the state of his
qubit to the state of the unknown state of the qubit that Alice has. When there are only
two parties, Alice and Bob, then Alice could encode in the cbits either the results of her
measurements or the unitary operations that Bob should apply to his qubit. When Alice
makes a series of measurements, then the latter option is simpler. Of course, Alice and Bob
would need to have a prior understanding of the option that Alice would use.
3.1Teleportation of a single-qubit state
In this scenario, Alice wishes to teleport an unknown qubit state |ψ?a= α|0?+β|1? to Bob.
They share a quantum channel given by one of the quadripartite states of the last section.
Of the four entangled qubits, Alice has qubits 1, 2, and 3 and Bob has the qubit 4. As in the
conventional teleportation protocol, Alice’s strategy is to make von Neumann measurements
involving particles a, 1, 2, and 3 and communicate the results to Bob. Bob then performs
necessary unitary operation on his qubit according to the received message to convert the
state of his qubit to that of the unknown qubit. As noted above, Alice has several choices
of bases to perform the measurement. She may choose a basis of four particles or successive
two-particle Bell basis or three-particle and one-particle basis for measurement. Let us now
discuss various states of the last section as a quantum resource.
3.1.1Teleportation using |GHZ? state
Alice has the qubits a, 1, 2, and 3. Bob has the qubit 4. Alice wishes to teleport the unknown
state of the particle a,
|ψ?a= α|0?a+ β|1?a.
(6)
Here α and β are complex numbers. The combined state of the five-qubit systems can be
written as:
|ψ?a|GHZ?1234=
1
√2(α|0?a+ β|1?a) ⊗ (|0000?1234+ |1111?1234).
(7)
We can rewrite this combined state depending on the type of the measurement Alice
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wishes to make. If Alice wishes to make four-particle von Neumann measurement, then we
can rewrite the above state as,
|ψ?a|GHZ?1234 =
1
√2(α|0000?a123|0?4+ α|0111?a123|1?4+ β|1000?a123|0?4+ β|1111?a123|1?4)
1
2[|4GHZ+
|4GHZ+
=
1?a123(α|0?4+ β|1?4) + |4GHZ−
2?a123(α|1?4+ β|0?)4+ |4GHZ−
1?a123(α|0?4− β|1?)4+
2?a123(α|1?4− β|0?4)],
(8)
where,
|4GHZ±
1? =
1
√2(|0000? ± |1111?)
1
√2(|0111? ± |1000?).
(9)
|4GHZ±
2? =
(10)
According to the results of the measurement, Alice sends two bits of classical information
to Bob, encoding either the results of her measurements, or the unitary operation that Bob
should apply. Bob performs one of the {σ0,σ1,iσ2,σ3} operations to convert the state of his
qubit to that of the unknown qubit a.
Instead of making a four-particle von Neumann measurement, Alice may wish to make a
three-particle followed by one-particle von Neumann measurements. Or, there can be three
parties, Alice, Bob, and Charlie. In this latter case, Alice may have the qubits a, 1, and 2,
while Charlie has the qubit 3 and Bob has the qubit 4. To see how the protocol would work
in this situation, we rewrite the combined state (8) as,
|ψ?a|GHZ?1234 =
1
√2(α|000?a12|0?3|0?4+ α|011?a12|1?3|1?4+ β|100?a12|0?3|0?4+
β|111?a12|1?3|1?4)
1
2√2(|3GHZ+
(|3GHZ+
(|3GHZ+
(| − 3GHZ+
=
1?a12|+?3+ |3GHZ−
1?a12|−?3+ |3GHZ−
2?a12|+?3− |3GHZ−
2?a12|−?3+ |3GHZ−
1?a12|−?3)(α|0?4+ β|1?4) +
1?a12|+?3)(α|0?4− β|1?)4+
2?a12|−?3)(α|1?4+ β|0?)4+
2?a12|+?3)(α|1?4− β|0?4),
(11)
where,
|3GHZ±
1? =
1
√2(|000? ± |111?)(12)
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|3GHZ±
2? =
1
√2(|011? ± |100?).
(13)
and,
|±? =
1
√2(|0? ± |1?).
(14)
Here in the case of two parties, the simplest thing for the Alice would be to encode in
two cbits the unitary operation that Bob should apply on his qubit after she makes the
measurement. So here classical communication cost would still be two cbits. However, in
the scenario of three parties, there will be need of three cbits of classical communication.
This communication could take many forms. Some examples are: Alice sends two cbits and
Charlie one cbit to Bob about the results of measurements; Charlie sends one cbit to Alice,
who then sends two cbits to Bob, encoding the unitary operation that Bob should apply.
In all cases Bob will make one of the {σ0,σ1,iσ2,σ3} operations on his qubit to convert its
state to that of the unknown qubit a.
Let us now consider the case, where Alice makes two successive Bell measurements, i.e.,
von Neumann measurements using the Bell basis:
|ϕ±? =
1
√2(|00? ± |11?)
1
√2(|01? ± |10?)
|ψ±? =
(15)
In the three-party scenario here, Alice will have qubits a and 1, Charlie will have the
qubits 2 and 3, while Bob would have the qubit 4. In either of the two scenarios, the
teleportation would be possible but with different classical communication cost. It can be
seen by rewriting equation (8) as,
|ψ?a|GHZ?1234 =
1
√2(α|00?a1|00?23|0?4+ α|01?a1|11?23|1?4+
β|10?a1|00?23|0?4+ β|11?a1|11?23|1?4)
1
2√2[(|ϕ+?a1|ϕ+?23+ |ϕ−?a1|ϕ−?23)(α|0?4+ β|1?4) +
(|ϕ+?a1|ϕ−?23+ |ϕ−?a1|ϕ+?23)(α|0?4− β|1?)4+
(|ψ+?a1|ϕ+?23− |ψ−?a1|ϕ−?23)(α|1?4+ β|0?)4+
(|ψ−?a1|ϕ+?23− |ψ+?a1|ϕ−?23)(α|1?4− β|0?4)].
=
(16)
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Here classical communication cost would be same as in the last scenario - two cbits if
there are two parties and three cbits if there are three parties.
For this entangled resource, there is one more possibility. In this case Alice first makes
a Bell measurement on the particles a and 1, followed by one-particles measurement on the
particles 3 and 4. Or, there could be three or four parties. The distribution of the particles
in the three-party scenario will be as above. In the four-party scenario, Alice would have
particles a and 1, Charlie would have particle 2 and Dennis would have particle 3, whereas
Bob would have particle 4. One can easily check that in these scenarios, the teleportation
is also possible because one can write |3GHZ±
states (15) and the single-particle states |±? given in (14). The classical information cost
would depend on the number of parties. For two parties, it would be 2 cbits; for three
1? and |3GHZ±
2? in (12) in terms of the Bell
parties, it would be 3 cbits; whereas for four parties, it would be 4 cbits.
3.1.2Teleportation using |Ω? state
As noted earlier, this is one of the most interesting quadripartite entangled state and most
powerful. As before, Alice wishes to teleport the unknown state |ψ? to Bob. However, the
quantum resource available to her is the state |Ω?. This state is shared by particles 1, 2, 3,
and 4. Alice has the particles 1, 2, and 3. Bob has the particle 4. The combined state of
the the five particles a, 1, 2, 3, and 4 will be
|ψ?a|Ω?1234
= (α|0?a+ β|1?a) ⊗ (1
1
√2(α|00?a1|ϕ+?23|0?4+ α|01?a1|ϕ−?23|1?4+ β|10?a1|ϕ+?23|0?4+
β|11?a1|ϕ−?23|1?4).
√2(|0?1|ϕ+?23|0?4+ |1?1|ϕ−?23|1?4)
=
(17)
One can rewrite this combined state as,
|ψ?a|Ω?1234 = |Ω+
1?a123(α|0?a+ β|1?a) + |Ω−
|Ω+
1?a123(α|0?a− β|1?a) +
2?a123(α|1?a− β|0?a).
2?a123(α|1?a+ β|0?a) + |Ω−
(18)
Here the basis vectors are:
|Ω±
1? =
1
√2|00?|ϕ+? ± |11?|ϕ−?,
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|Ω±
2? =
1
√2|01?|ϕ−? ± |10?|ϕ+?.
(19)
After making the measurement in this basis, Alice can convey her results to Bob using
two classical bits. Bob then can apply appropriate unitary transformation to his qubit, as in
the case of the GHZ-state, and complete the protocol. Interestingly, there exist another set of
four-particle basis vectors that Alice can use to make measurement. This basis, GHZ-basis,
has in addition to the states (9) and (10), following states
|4GHZ±
3? =
1
√2(|0011? ± |1100?),
1
√2(|0100? ± |1011?).
(20)
|4GHZ±
4? =
(21)
In this GHZ-basis, the combined state (17) can be written as,
|ψ?a|Ω?1234 =
1
2√2[(|4GHZ+
(|4GHZ−
(−|4GHZ−
(−|4GHZ+
1?a123+ |4GHZ−
1?a123+ |4GHZ+
2?a123+ |4GHZ+
2?a123+ |4GHZ−
3?a123)(α|0?4− β|1?4) +
3?a123)(α|0?4+ β|1?4) +
4?a123)(α|1?4+ β|0?4) +
4?a123)(α|1?4− β|0?4)].
(22)
By encoding the unitary operations in the two cbits, Alice can convey the information
to Bob who can complete the protocol.
Let us now consider the next scenario where Alice makes a three-particle von Neumann
measurement followed by a one-particle measurement. As in the case of |GHZ?, there can
be two situations. There can be two parties or three parties. In this case we can rewrite (17)
as,
|ψ?a|Ω?1234
= (α|0?a+ β|1?a) ⊗ (1
1
4[(|3GHZ+
(α|0?4+ β|1?4) +
(|3GHZ+
(α|0?4− β|1?)4+
(|3GHZ+
√2(|0?1|ϕ+?23|0?4+ |1?1|ϕ−?23|1?4)
=
1?a12|−?3+ |3GHZ−
1?a12|+?3+ |3GHZ+
3?a12|+?3− |3GHZ−
3?a12|−?3)
1?a12|+?3+ |3GHZ−
1?a12|−?3− |3GHZ+
3?a12|−?3+ |3GHZ−
3?a12|+?3)
4?a12|+?3+ |3GHZ−
4?a12|−?3+ |3GHZ+
2?a12|−?3− |3GHZ−
2?a12|+?3)
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(α|1?4+ β|0?)4+
(|3GHZ+
(α|1?4− β|0?4)].
4?a12|−?3+ |3GHZ−
4?a12|+?3− |3GHZ+
2?a12|+?3+ |3GHZ−
2?a12|−?3)
(23)
Here one can use three-particle GHZ-basis. It has in addition to the states given in (12)
and (13), we have the following states,
|3GHZ±
3? =
1
√2(|001? ± |110?),
1
√2(|010? ± |101?).
|3GHZ±
4? =
(24)
As in the case of |GHZ? state, in the two-party situation, Alice needs to send two cbits
to Bob (e.g., encoding the four unitary operations). In the three-party situation, combined
classical information cost will be three cbits. As before, this classical communication could
take many forms. For example, Charlie can send one cbit of information about his measure-
ment to Alice. On the basis of her results, Alice can send two cbits of information to Bob,
encoding the unitary transformation. After receiving the classical communication, Bob can
complete the protocol by applying a suitable unitary operation.
The strategy of making two successive Bell measurements also works if we make mea-
surements on suitably chosen qubits. (This is because, this state is not symmetric under the
permutation of qubits.) If Alice makes a measurement on qubits ‘a2’ and ‘13’, then we can
write the combined state as:
|ψ?a|Ω?1234
=
1
√2(α|0?a+ β|1?a)(|0?1|ϕ+?23|0?4+ |1?1|ϕ−?23|1?4)
1
4[(|ϕ+?a2|ϕ+?13+ |ϕ−?a|ϕ−?13+ |ψ+?a2|ψ−?13+ |ψ−?a2|ψ+?13)(α|0?4− β|1?4) +
(|ϕ+?a2|ϕ−?13+ |ϕ−?a|ϕ+?13+ |ψ+?a2|ψ+?13+ |ψ−?a2|ψ−?13)(α|0?4+ β|1?4) +
(|ϕ+?a2|ψ+?13− |ϕ−?a|ψ−?13+ |ψ+?a2|ϕ−?13− |ψ−?a2|ϕ+?13)(α|1?4+ β|0?4) −
(|ϕ+?a2|ψ−?13− |ϕ−?a|ψ+?13− |ψ+?a2|ϕ+?13+ |ψ−?a2|ϕ+?13)(α|1?4− β|0?4)].(25)
=
Here the classical communication cost will be two cbits for two-party situation and four
cbits in the three-party situation.
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Other situation is a Bell measurement followed by two one-particle measurements. As
before, one could have two-party, three-party and four-party situations. The protocol would
work as before with appropriate classical communication cost. This is because one can
rewrite |3GHZ±
states (15).
n? (n = 1 − 4) in (23) in terms of the single-particle states (14) and the Bell
3.1.3Teleportation using |W? state
Next we consider the W-state given in (2). This state does not allow the faithful teleportation
of an unknown qubit state. However, a modified version of this state that also belongs to
the W-state category under the SLOCC classification can work. This is shown below. The
distribution of the four qubits is as earlier. The combined state of the particle a, 1, 2, 3, and
4 can be written as
|ψ?a|W?1234 = (α|0?a+ β|1?a) ⊗1
1
2(α|0100?a123|0?4+ α|0010?a123|0?4+ α|0001?a123|0?4+ α|0000?a123|1?4+
β|1100?a123|0?4+ β|1010?a123|0?4+ β|1001?a123|0?4+ β|1000?a123|1?4).(26)
2(|1000?1234+ |0100?1234+ |0010?1234+ |0001?1234)
=
It does not seem possible to rewrite the above state to teleport the qubit faithfully. In
the case of three-qubit W-state, it was shown[11] that instead of the W state, one needs to
√2+2n(|100? +√n|010? +√n + 1|001?). This state can be
used for the perfect teleportation of an unknown qubit. Analogously, one could construct
consider the state |Wn?, which is
1
the state for the case of four qubits. We can consider the state
|Wmn? =
1
√2m + 2n + 2(|1000? + meiρ|0100? + neiη|0010? +√m + n + 1eiσ|0001?).
(27)
Here m and n are real numbers. For simplicity, one could set the phases to unity and
choose m = n = 1,
|W11? =
1
√6(|1000? + |0100? + |0010? +
√3|0001?.
(28)
With this quantum resource, the combined state of five particles would be
|ψ?a|W11?1234 =
1
√6(α|0?a+ β|1?a) ⊗ (|1000?1234+ |0100?1234+ |0010?1234+
√3|0001?1234)
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=
1
√6(α|0100?a123|0?4+ α|0010?a123|0?4+ α|0001?a123|0?4+
β|1100?a123|0?4+ β|1010?a123|0?4+ β|1001?a123|0?4+
1
√6[|η+?a123(α|0?4+ β|1?4) + |η−?a123(α|0?4− β|1?4) +
|ζ+?a123(β|0?4+ α|1?4) + |ζ−?a123(α|0?4− β|1?4,
√3α|0000?a123|1?4+
√3β|1000?a123|1?4)
=
(29)
where,
|η±? =
1
√6(|0100? + |0010? + |0001? ±
1
√6(|1100? + |1010? + |1001? ±
√3|1000?),
√3|0000?).
|ζ±? =
(30)
Now Alice can send the two classical bits of information to Bob to inform him about
the results of his measurement, or the unitary operation that he should apply. Bob then
completes the protocol by applying appropriate unitary transformation.
One can make a more general observation about constructing a suitable state. This state
can be p|1000? + q|0100? + r|0010? + s|0001? where |p|2+ |q|2+ |r|2= |s|2which is suitable
for teleportation of qubit. For a more general case of N-qubits the suitable state would be
a1|10...0?+a2|010...0?+...+aN|00....1? with coefficients satisfying |a1|2+|a2|2+...+|aN−1|2=
|aN|2.
As earlier, there also exist scenarios of Alice making three-particle measurement followed
by one-particle measurement; two successive Bell measurements; one Bell measurement fol-
lowed by two one-particle measurements. In all of these scenarios, there could exist multi-
party situations. These scenarios may work out with suitable W-class state. As before, more
parties would mean more classical communication cost.
3.1.4Teleportaion using |Q4? state
The state |Q4? can also be used to teleport an unknown state |ψ?. Unlike the GHZ-state,
here the state changes with permutation of the particles. The states obtained on permutation
would also belong to the same SLOCC class. However, for different states, one would need
different distribution of particles for the measurement. If the particles are distributes such
that particles a, 1, 2, and 3 are with Alice and 4 with Bob then this state cannot be used
for teleportation; but if the distribution is such that Alice has particles a, 1, 3, 4, and Bob
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has particle 2, it would lead to faithful teleportation. We can see how the protocol works as
follows. The combined state of the five particles can be written as
|ψ?a|Q4?1234 =
1
2(α|0?a+ β|1?a)(|0000?1234+ |0101?1234+ |1000?1234+ |1110?1234
1
2[α|0000?a134|0?2+ α|0001?a134|1?2+ α|0100?a134|0?2+ α|0011?a134|1?2+
β|1000?a134|0?2+ β|1001?a134|1?2+ β|1100?a134|0?2+ β|1011?a134|1?2]. (31)
=
Now Alice can use one of the following set of basis vectors to make four-particle von-
Neumann measurements. One set of basis vectors are
|ρ±
|ρ±
1? =
1
2[(|0000? + |0100?) ± (|1001? + |1011?)]
1
2[(|0001? + |0011?) ± (|1001? + |1100?)],
(32)
2? =
(33)
while the other set is,
|τ±
1? =
1
√2(|0000? ± |1001?)
1
√2(|0001? ± |1000?)
1
√2(|0100? ± |1011?)
1
√2(|0011? ± |1100?).
(34)
|τ±
2? =(35)
|τ±
3? =
(36)
|τ±
4? =
(37)
Using the basis (32)-(33), one can rewrite (31) as
|ψ?a|Q4?1234 =
1
2[|ρ+
|ρ+
1?a134(α|0?2+ β|1?2) + |ρ−
2?a134(α|1?2+ β|0?2) + |ρ−
1?a134(α|0?2− β|1?2) +
2?a134(α|1?2− β|0?2)],
(38)
while using the basis (34)-(37), we can rewrite (31) as,
|ψ?a|Q4?1234 =
1
2√2[(|τ+
(|τ+
1? + |τ+
4?)(α|1? + β|0?) + (|τ−
3?)(α|0? + β|1?) + (|τ−
1? + |τ−
3?)(α|0? − β|1?) +
2? + |τ+
2? + |τ−
4?)(α|1? − β|0?)].
(39)
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Irrespective of the basis set Alice uses, she needs to send only two classical bits of infor-
mation to Bob. Then, Bob can apply suitable unitary operator to convert the state of his
qubit to that of (7).
It is interesting to note that if the particles are distributed such that Alice has particles
a, 1, 2, 3 and Bob has 4, then these exists a state in this SLOCC class,
|Q411? =
1
√6(|0000? + |1000? + |1110? +
√3|0101?,
(40)
which can be used for the teleportation if the measurement is performed in the basis,
|η?±
=
1
√6(|0000? + |0100? + |0111? ±
1
√6(|1000? + |1100? + |1111? ±
√3|1010?)
√3|0010?).
(41)
|ζ?±
=
(42)
As earlier, there exist the scenarios where Alice chooses to make a three-particle mea-
surement followed by a one-particle measurement; or she makes two successive Bell measure-
ments; or she makes a Bell measurement followed by two one-particle measurements. These
scenarios could have multiparty situations. One needs to investigate further whether these
scenarios could be realized with this |Q4? state. One can also explore other states of this
class for their suitability for realizing various scenarios.
3.1.5 Teleportation using |Q5? state
This entangled state can also be used as a suitable quantum resource. The distribution of
the four qubits is as before. The combined state of the particle a, 1, 2, 3, and 4 can be
written as:
|ψ?a|Q5?1234 = (α|0?a+ β|1?a) ⊗1
1
2(α|0000?a123|0?4+ α|0101?a123|1?4+ α|0110?a123|1?4+ α|0111?a123|0?4+
β|1000?a123|0?4+ β|1101?a123|1?4+ β|1110?a123|1?4+ β|1111?a123|0?4).(43)
2(|0000?1234+ |1011?1234+ |1101?1234+ |1110?1234)
=
It turns out that one can teleport the state |ψ?, if Alice makes a four-particle von Neu-
mann measurement using at least two different sets of basis vectors. If Alice uses the following
set of basis vectors:
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