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arXiv:1603.08866 · Electronic Proceedings in Theoretical Computer Science

Tight Reference Frame–Independent Quantum Teleportation

Dominic Verdon, Jamie Vicary

Published Jan 1, 2017• 2 citations
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Abstract

We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group. Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size. In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.

DOI
10.4204/eptcs.236.13

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We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group. Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size. In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.
We present a scheme for teleporting a quantum state between two parties whose local reference frames are misaligned by the action of a finite symmetry group. Unlike other proposals, our scheme requires the same amount of classical communication and entangled resources as conventional teleportation, does not reveal any reference frame information, and is robust against changes in reference frame alignment while the protocol is under way. The mathematical foundation of our scheme is a unitary error basis which is permuted up to a phase by the conjugation action of the group. We completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases.
We present two schemes for quantum teleportation between parties whose local reference frames are misaligned by the action of a compact Lie group GGG. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference-frame alignment during execution, suiting them to situations of rapid reference-frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially---this includes the case of qubit teleportation under arbitrary SU(2)\mathrm{SU}(2)SU(2) reference-frame uncertainty---while communicating no information about either party's reference-frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference-frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of GGG.
Part I: We present two new schemes for teleportation of a quantum state between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially ---including the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no information about either party's reference frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G; we completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases. Part II: Quantum pseudo-telepathy is a phenomenon whereby two non-communicating parties can use the non-signalling correlations from shared quantum entanglement to perform a task that would ordinarily be impossible without communication. Here we consider the graph isomorphism game: an instance of the game is defined by a pair of graphs, and a perfect deterministic classical strategy precisely corresponds to an isomorphism of those graphs. We call a pair of graphs for which a perfect strategy using shared entanglement (a quantum strategy) exists pseudo-telepathic, and call a quantum strategy a quantum isomorphism between the graphs. Using a deep connection between noncommutative mathematics and nonlocal games, we classify all graphs quantum isomorphic to a given graph in terms of the quantum automorphism group of the graph. This implies a construction of quantum isomorphisms from central type subgroups of the ordinary automorphism group; we give an explicit description of the resulting quantum strategy and the new graph. We consider generation of new examples of quantum contextuality from central type groups. For abelian central type groups we show that this reduces to construction of quantum solutions of a binary linear constraint system from the abelianisation of its homogeneous solution group.
Quantum teleportation is a fundamental building block of quantum communications and quantum computations, transferring quantum states between distant physical entities. In the context of quantum secret sharing, the teleportation of quantum information shared by multiple parties without concentrating the information at any place is essential, and this cannot be realized by any previous scheme. We propose and experimentally demonstrate a novel teleportation protocol that enables one to perform this task. It is jointly performed by distributed participants, while none of them can fully access the information. Our scheme can be extended to arbitrary numbers of senders and receivers and to fault-tolerant quantum networks by incorporating error-correction codes.
FINDING: Quantum teleportation is a protocol for transferring an unknown quantum state via shared entanglement and classical communication — not a transfer of matter or energy. | MATH: Core protocol: Bell state ∣Φ+⟩=12(∣00⟩+∣11⟩)|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)∣Φ+⟩=2​1​(∣00⟩+∣11⟩); teleported state ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle∣ψ⟩=α∣0⟩+β∣1⟩ with ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1∣α∣2+∣β∣2=1. After Bell measurement, Bob applies one of four Pauli corrections: I,X,Z,XZI, X, Z, XZI,X,Z,XZ (identity, bit-flip, phase-flip, both). Success probability = 1 (deterministic) given the classical 2-bit channel. No faster-than-light signaling — information rate limited by classical bits. | CONNECTION: The four Bell states form a basis of the 2-qubit Hilbert space C2⊗C2\mathbb{C}^2 \otimes \mathbb{C}^2C2⊗C2, isomorphic to the root system of D2D_2D2​ (or A1×A1A_1 \times A_1A1​×A1​) — a crystallographic symmetry. The coefficients α,β\alpha, \betaα,β are complex numbers on the unit sphere S3S^3S3, whose Hopf fibration relates to the golden rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com