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General teleportation channel in Fermionic Quantum Theory

Sanam Khan, R. Jehadeesan, Sibasish Ghosh

TL;DR

This work derives the maximum subsystem teleportation fidelity for information encoded in fermionic N×N mode systems under parity superselection, by developing a PSSR-respecting twirling framework, a fermionic state-channel isomorphism, and a restricted Clifford unitary-2 design. The authors obtain a canonical, Werner-type invariant fermionic state under restricted twirls and show that any fermionic channel can be reduced to a form described by twirl-invariant parameters F_f and Υ1, yielding a fidelity bound F_max = 4(1+|Υ1|_opt)/(d^2+4) + (F_f)_max[1 − 4(1+|Υ1|_opt)/(d^2+4)]. They also distinguish locally accessible entanglement from topological correlation, and demonstrate how teleportation channels can preserve topological correlation, potentially through phase-damping-like action between even and odd parity sectors; the framework provides a pathway for experimental validation via restricted Clifford twirling and Majorana-based platforms. Overall, the paper advances the operational understanding of fermionic teleportation under PSSR, linking resource state structure, channel twirling, and optimal fidelity in a parity-constrained setting.

Abstract

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 $\textit{optimal teleportation fidelity}$ of the shared state is known to be $(F_{max}d+1)/(d+1)$ with $F_{max}$ 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. In the present work, we derive an expression for the $\textit{optimal teleportation fidelity}$ of locally accessible entanglement preservation, given that the quantum information to be teleported is encoded in fermionic modes of dimension $2^N \times 2^N$ using $2^N \times 2^N$-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 $\textit{isotropic state}$ -- 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.

General teleportation channel in Fermionic Quantum Theory

TL;DR

This work derives the maximum subsystem teleportation fidelity for information encoded in fermionic N×N mode systems under parity superselection, by developing a PSSR-respecting twirling framework, a fermionic state-channel isomorphism, and a restricted Clifford unitary-2 design. The authors obtain a canonical, Werner-type invariant fermionic state under restricted twirls and show that any fermionic channel can be reduced to a form described by twirl-invariant parameters F_f and Υ1, yielding a fidelity bound F_max = 4(1+|Υ1|_opt)/(d^2+4) + (F_f)_max[1 − 4(1+|Υ1|_opt)/(d^2+4)]. They also distinguish locally accessible entanglement from topological correlation, and demonstrate how teleportation channels can preserve topological correlation, potentially through phase-damping-like action between even and odd parity sectors; the framework provides a pathway for experimental validation via restricted Clifford twirling and Majorana-based platforms. Overall, the paper advances the operational understanding of fermionic teleportation under PSSR, linking resource state structure, channel twirling, and optimal fidelity in a parity-constrained setting.

Abstract

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 of the shared state is known to be with 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. In the present work, we derive an expression for the of locally accessible entanglement preservation, given that the quantum information to be teleported is encoded in fermionic modes of dimension using -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 -- 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.
Paper Structure (20 sections, 13 theorems, 133 equations, 1 figure)

This paper contains 20 sections, 13 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Fermionic Hilbert Space
  4. PSSR and physical states
  5. Operations on fermionic states
  6. Fermionic Observables
  7. Entanglement and topological correlation in fermionic system
  8. Fermionic teleportation for N$\times$N mode systems
  9. Qudit Teleportation in SQT
  10. Optimal Teleportation in FQT
  11. $U \wedge U$ invariant fermionic state and reduction under unitary twirling
  12. Reduction of $N \times N$ mode fermionic states under clifford twirling
  13. State-Channel isomorphism in FQT
  14. Optimal teleportation fidelity
  15. Teleportation of Topological correlation
  16. ...and 5 more sections

Key Result

Lemma 1

Any anti-block diagonal unitary in the basis $\mathcal{B_A^P} \equiv \{B_e,B_o\}$ can always be embedded into the form of a block-diagonal unitary using a single ancilla system.

Figures (1)

  • Figure 1: Teleportation in FQT. In this scheme $\omega_{AB}^f$ is the resource state used to to teleport one half of the initial state $\rho_{\tilde{A}\tilde{A}' }^f(initial)$ and produces the final state $\rho_{B\tilde{A}'}^f(final)$.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 1
  • Corollary 1.1
  • Corollary 1.2
  • ...and 3 more