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Perfect Teleportation of a Single Qubit: Unearthing New W-Class of States

Satyabrata Adhikari

Published Mar 27, 2019• 0 citations
OpenAlex

Abstract

Agrawal and Pati \cite{agrawal} had shown that there exist a class of three-qubit W-states, which act as a resource state for the perfect teleportation of a single qubit. They have used three-qubit measurement basis to execute their teleportation protocol. In this work, we modify Agrawal-Pati teleportation protocol. Our modified teleportation protocol is based on two-qubit measurement basis for the perfect teleportation of a single qubit. We deduce the condition for the successful execution of modified teleportation protocol and this leads us to new class of three-qubit W-states, which is required as a resource state to achieve the modified perfect teleportation protocol. We have constructed operators that can be used to verify the condition of teleportation in experiment. This verification is necessary for the detection of whether the given three-qubit state is useful in modified teleportation protocol or not. Then we have shown that the set containing three-qubit W-states used in modified teleportation scheme is larger than the set of shared three-qubit W-states used in Agrawal-Pati teleportation protocol. Further we quantify the amount of entanglement contained in the newly identified shared three-qubit W-class of states needed for the perfect teleportation of a single qubit. Moreover, we show that the three-qubit W-class of shared states used in the two teleportation protocols can be prepared using NMR set up.

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Agrawal and Pati \cite{agrawal} had shown that there exist a class of three-qubit W-states, which act as a resource state for the perfect teleportation of a single qubit. They have used three-qubit measurement basis to execute their teleportation protocol. In this work, we modify Agrawal-Pati teleportation protocol. Our modified teleportation protocol is based on two-qubit measurement basis for the perfect teleportation of a single qubit. We deduce the condition for the successful execution of modified teleportation protocol and this leads us to new class of three-qubit W-states, which is required as a resource state to achieve the modified perfect teleportation protocol. We have constructed operators that can be used to verify the condition of teleportation in experiment. This verification is necessary for the detection of whether the given three-qubit state is useful in modified teleportation protocol or not. Then we have shown that the set containing three-qubit W-states used in modified teleportation scheme is larger than the set of shared three-qubit W-states used in Agrawal-Pati teleportation protocol. Further we quantify the amount of entanglement contained in the newly identified shared three-qubit W-class of states needed for the perfect teleportation of a single qubit. Moreover, we show that the three-qubit W-class of shared states used in the two teleportation protocols can be prepared using NMR set up.
Abstract In this work, we propose a probabilistic teleportation protocol to teleport a single qubit via three-qubit W-states using two-qubit measurement basis. We show that for the proper choice of the state parameter of the resource state, it is possible to make success probability of the protocol very high. We deduce the condition for the successful execution of our teleportation protocol and this gives us new class of three-qubit W-states which act as a resource state. We have constructed operators that can be used to verify the condition of teleportation in experiment. This verification is necessary for the detection of whether the given three-qubit state is useful in our teleportation protocol or not. Further we quantify the amount of entanglement contained in the newly identified shared W-class of states. Moreover, we show that the W-class of shared state used in the teleportation protocol can be prepared using NMR set up.
As we know, the states of triqubit systems have two important classes: GHZ-class and W-class. In this paper, the states of W-class are considered for teleportation and superdense coding, and they are generalized to multi-particle systems. First we describe two transformations on the shared resources for teleportation and superdense coding. With these transformations, we obtain a sufficient and necessary condition for a state of W-class being suitable for perfect teleportation and superdense coding. For the state which was thought to be not suitable for sending three classical bits by sending two qubits by Agrawal and Pati (2006 Phys. Rev. A 74 062320), we show that it may be used to fulfil that task, if entangled unitary operations on two qubits are allowed. We generalize the states of W-class to multi-qubit systems and multi-particle systems with higher dimension. We propose two protocols for teleportation and superdense coding by using W-states of multi-qubit systems that generalize the protocols by using | W ⟩ 123 proposed by Agrawal and Pati. We obtain an optimal way to partition some W-states of multi-qubit systems into two subsystems, such that the entanglement between them achieves maximum value.
As we know, the states of triqubit systems have two important classes: GHZ-class and W-class. In this paper, the states of W-class are considered for teleportation and superdense coding, and are generalized to multi-particle systems. First we describe two transformations of the shared resources for teleportation and superdense coding, which allow many new protocols from some known ones for that. As an application of these transformations, we obtain a sufficient and necessary condition for a state of W-class being suitable for perfect teleportation and superdense coding. As another application, we find that state ∣W>123=1/2(∣100>123+∣010>123+sqrt2∣001>123)|W>_{123}={1/2}(|100>_{123}+|010>_{123}+\\sqrt{2}|001>_{123})∣W>123​=1/2(∣100>123​+∣010>123​+sqrt2∣001>123​) can be used to transmit three classical bits by sending two qubits, which was considered to be impossible by P. Agrawal and A. Pati [Phys. Rev. A to be published]. We generalize the states of W-class to multi-qubit systems and multi-particle systems with higher dimension. We propose two protocols for teleportation and superdense coding by using W-states of multi-qubit systems that generalize the protocols by using ∣W>123|W>_{123}∣W>123​ proposed by P. Agrawal and A. Pati. We obtain an optimal way to partition some W-states of multi-qubit systems into two subsystems, such that the entanglement between them achieves maximum value.
It is known that two local unitaries (LU) equivalent states possess the same amount of entanglement and can be used to perform the same tasks in quantum information theory (QIT). For a protocol for a task in QIT, we call a protocol LU invariant if two LU-equivalent states are either both suitable for the protocol or neither is. So far, no one has discussed whether a protocol for a task in QIT is LU invariant. In [Phys. Rev. A, 74, 062320 (2006)], Agrawal and Pati proposed the perfect teleportation protocol (PTP) and the protocol for superdense coding to transmit 2-bit classical information by sending one qubit (PSDC-2) and 3-bit classical information by sending two qubits (PSDC-3). In this paper, we show that PTP and PSDC-2 are LU invariant. That is, two LU equivalent states are suitable for PTP and PSDC-2 or neither of them is. We show that PSDC-3 is not LU invariant. We also indicate that the teleportation proposed in <cite>Nielsen</cite> is not LU invariant. We give a necessary and sufficient condition for a state to be suitable for PTP, PSDC-2, and PSDC-3, respectively. Via the LU invariance of PTP and PSDC-2, we prove that a state is suitable for PTP and PSDC-2 if and only if it has 1 ebit of shared entanglement, respectively and find all genuine entangled states and separable states which are suitable for PTP and PSDC-2, respectively. So far, no one has indicated that PTP and PSDC-2 do not require genuine entanglement. Agrawal and Pati suggested to study if there are subclasses of W SLOCC class which are suitable for PSDC-3. So far, it still remains an unsolved question. We show that any state of the SLOCC class W is not suitable for PSDC-3.