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arXiv:2306.08067

Condition on n-Qubit State For Getting Perfect Quantum Teleportation

Shamiya Javed, Phool Singh Yadav, Prakash, Ranjana, Hari Prakash

Published Jun 13, 2023• 0 citations
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Abstract

It is shown that standard quantum teleportation (SQT) with multi-qubit resource result in fidelity $(2+C)/3$ where $C$ is concurrence of the resource in bipartite entanglement between qubit going to receiver and rest of the qubits. For perfect SQT, obviously, $C=1$. For a general 3-qubit resource, we find conditions for getting perfect SQT for state expressed in any basis states. Zha et al. [Mod. Phys. Lett. B 22, 2523-2528 (2008)], who studied perfect SQT using 3-qubit resource, reported conditions for perfect SQT for only those resource states which are given in the 3-qubit canonical form of Acin representation. We show that there is an alternative easily derivable representation which gives more generalized results. To illustrate the difference between the two schemes, we build an example of 3-qubit entangled state, giving perfect SQT and not included in Zha et al. results.

DOI
10.48550/arxiv.2306.08067

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It is shown that standard quantum teleportation (SQT) with multi-qubit resource result in fidelity (2+C)/3(2+C)/3(2+C)/3 where CCC is concurrence of the resource in bipartite entanglement between qubit going to receiver and rest of the qubits. For perfect SQT, obviously, C=1C=1C=1. For a general 3-qubit resource, we find conditions for getting perfect SQT for state expressed in any basis states. Zha et al. [Mod. Phys. Lett. B 22, 2523-2528 (2008)], who studied perfect SQT using 3-qubit resource, reported conditions for perfect SQT for only those resource states which are given in the 3-qubit canonical form of Acin representation. We show that there is an alternative easily derivable representation which gives more generalized results. To illustrate the difference between the two schemes, we build an example of 3-qubit entangled state, giving perfect SQT and not included in Zha et al. results.
We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. Lett. 78 (1997) 5022] does not exist for entangled 2N-qubit states with N > 1 but magic partial bases, similar to those discussed recently by Prakash and Maurya [Optics Commun. 284 (2011) 5024] do exist. We give explicitly all magic partial bases for N = 2.
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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Quantum Teleportation is a very useful scheme for transferring quantum information. Given that the quantum information is encoded in a state of a system of distinguishable particles, and given that the shared bi-partite entangled state is also that of a system of distinguishable particles, the optimal teleportation fidelity\textit{optimal teleportation fidelity}optimal teleportation fidelity of the shared state is known to be (Fmaxd+1)/(d+1)(F_{max}d+1)/(d+1)(Fmax​d+1)/(d+1) with FmaxF_{max}Fmax​ being the `maximal singlet fraction' of the shared state. However, Parity Superselection Rule (PSSR) in Fermionic Quantum Theory (FQT) puts constraint on the allowed set of physical states and operations, and thereby, leads to a different notion of Quantum entanglement - locally accessible and locally inaccessible (topological correlation). In the present work, we derive an expression for the optimal teleportation fidelity\textit{optimal teleportation fidelity}optimal teleportation fidelity of locally accessible entanglement preservation, given that the quantum information to be teleported is encoded in fermionic modes of dimension 2N×2N2^N \times 2^N2N×2N using 2N×2N2^N \times 2^N2N×2N-dim shared fermionic resource between the sender and receiver. To get the optimal teleportation fidelity in FQT, we introduce PSSR restricted twirling operations and establish fermionic state-channel isomorphism. Remarkably, we notice that the structure of the canonical form of twirl invariant fermionic shared state differs from that of the isotropic state\textit{isotropic state}isotropic state -- the corresponding canonical invariant form for teleportation in Standard Quantum Theory (SQT). In this context, we also introduce restricted Clifford twirling operation that constitute the Unitary 2-design in case of FQT for experimentally validating such optimal average fidelity. Finally, we discuss the preservation of locally inaccessible entanglement for a class of fermionic teleportation channel.