[Paper Review] Time Displaced Entanglement and Non-Linear Quantum Evolution

Timothy C. Ralph|ArXiv.org|Oct 6, 2005

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper proposes a quantum teleportation protocol using time-displaced entanglement generated by relativistic time dilation, where one qubit undergoes a round-trip at high speed, inducing a time shift. The protocol leads to non-unitary, non-linear quantum evolution, demonstrating that relativistic effects combined with entanglement can produce dynamics outside standard quantum mechanics, while avoiding causal paradoxes via probabilistic consistency.

ABSTRACT

We discuss time displaced entanglement, produced by taking one member of a Bell pair on a round trip at relativistic speeds, thus inducing a time-shift between the pair. We show that decoherence with respect to Bell measurements on the pair is predicted. We then study a teleportation protocol, using time displaced entanglement as its resource, in which a time-like loop is apparently formed. The result is non-unitary, non-linear evolution of the teleported state.

Motivation & Objective

• To explore the quantum effects arising when entangled qubits experience relativistic time dilation, leading to temporal separation in their proper time.
• To investigate whether such time-displaced entanglement can generate non-unitary and non-linear evolution in quantum systems.
• To construct a physically realizable model of teleportation using time-displaced entanglement that avoids causal paradoxes.
• To demonstrate that the resulting evolution, though non-standard, remains consistent with quantum mechanics through probabilistic resolution.

Proposed method

• Model qubits using broadband temporal mode representations with time-ordered creation operators and temporal wave functions F(t′) defining their energy bandwidth.
• Define logical qubit states |0⟩t and |1⟩t via integrals over time intervals centered at clock-cycle time tn, ensuring non-overlapping Hilbert spaces.
• Apply time translation operators to simulate relativistic time dilation, shifting the proper time of one qubit by one clock cycle (τ = tn − tn−1).
• Construct the time-displaced Bell state |φ+⟩ = |0⟩tn,1|0⟩tn−1,2 + |1⟩tn,1|1⟩tn−1,2, where qubit 1 is delayed relative to qubit 2.
• Perform joint Bell measurements on the time-displaced state by tensoring with its evolved version, using a time-ordered measurement basis.
• Analyze the resulting evolution under teleportation, showing non-unitary and non-linear behavior in the teleported state.

Experimental results

Research questions

• RQ1 Can time-displaced entanglement, generated via relativistic time dilation, lead to non-unitary and non-linear quantum evolution in a teleportation protocol?
• RQ2 Does the combination of relativistic effects and quantum entanglement produce dynamics outside the standard framework of linear, unitary quantum mechanics?
• RQ3 How does the measurement outcome of Bell states on time-displaced entanglement affect the evolution of the teleported state?
• RQ4 Can such a system avoid causal paradoxes despite exhibiting non-linear evolution, as predicted by Deutsch's model?
• RQ5 What physical constraints or assumptions are necessary to ensure a unique, stable solution in this non-standard quantum evolution?

Key findings

• The time-displaced entangled state |φ+⟩ = |0⟩tn,1|0⟩tn−1,2 + |1⟩tn,1|1⟩tn−1,2 exhibits non-unitary evolution when used in a teleportation protocol.
• The teleportation process results in non-linear evolution of the teleported state, deviating from standard quantum mechanics.
• The non-linear evolution is consistent with causality and avoids paradoxes due to the probabilistic nature of quantum measurements.
• The model removes the need for ad hoc restrictions on interactions with time-like loops, as information is only accessible after Bell measurement results are classically communicated.
• Stability analysis shows that only the totally mixed state solution survives small perturbations from ideal measurement bases, eliminating ambiguity in multiple solutions.
• The protocol is physically realizable using optical qubits in fiber loops with CNOT gates and time-delayed round-trips, with coherence maintenance as the main experimental challenge.