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Asymptotic teleportation scheme as a universal programmable quantum processor
Satoshi Ishizaka, Tohya Hiroshima
TL;DR
The paper establishes the equations underlying its teleportation analysis through appendix derivations. It proves the relevant eigenvalue relation and derives the entanglement-fidelity expression.
Problem
The paper addresses establishing the eigenvalue relation and entanglement-fidelity expression used in its analysis.
Method
It proves the eigenvalue relation inductively in N using overlap calculations and eigenvalue equations, then combines operator decompositions to derive fidelity expressions.
Results
The appendix completes the proof of the desired eigenvalue equation and obtains Eq. (4) with Eq. (5) for the entanglement fidelity.
Takeaways & Limitations
The supplied derivations establish the eigenvalue and entanglement-fidelity equations used by the paper’s analysis.
Abstract
from arXiv · showhide
We consider a scheme of quantum teleportation where a receiver has multiple (N) output ports and obtains the teleported state by merely selecting one of the N ports according to the outcome of the sender's measurement. We demonstrate that such teleportation is possible by showing an explicit protocol where N pairs of maximally entangled qubits are employed. The optimal measurement performed by a sender is the square-root measurement, and a perfect teleportation fidelity is asymptotically achieved for a large N limit. Such asymptotic teleportation can be utilized as a universal programmable processor.
APPENDIX A: PROOF OF EQ. (3)
The appendix proves Eq. (3) by induction on N, combining overlap calculations and system-size-labeled eigenvalue relations to derive the desired eigenvalue equation.
- Overlap calculations yield the intermediate relations in Eqs. (A1) and (A2), which are combined after classifying the relevant terms.
- The proof proceeds inductively: Eq. (3) is immediate for N = 1, and the N−1 case is assumed.
- After lengthy calculations, the combined results produce the desired eigenvalue equation, completing the proof.
APPENDIX B: EQ. (4) WITH EQ. (5)
The appendix derives Eq. (4) with Eq. (5) by decomposing ρ[N] into eigenspace operators and combining the resulting expressions.
- The entanglement fidelity F is calculated by decomposing ρ[N] into operators associated with eigenvalues λ∓.
- The derivation obtains intermediate expressions from Eqs. (B1) and (B2).
- Adding the expressions from Eqs. (B3) and (B4) yields Eq. (4) together with Eq. (5).