[Paper Review] A reply to "Problems with modelling closed timelike curves as post-selected teleportation"

Seth Lloyd, Lorenzo Maccone+9 more|arXiv (Cornell University)|Jul 31, 2011

Simulation Techniques and Applications3 citations

TL;DR

This paper refutes a claimed paradox in closed timelike curve (CTC) modeling by showing that Ralph's 'unproven-theorem' paradox relies on an incomplete analysis of Bob's physical operations. By rigorously incorporating unitary purification and post-selected teleportation, the authors demonstrate that the paradox cannot arise, as the P-CTC mechanism enforces self-consistent, non-paradoxical evolution where the theorem's origin is traceable to an author, not a causal loop.

ABSTRACT

In arXiv:1107.4675 Ralph uses our post-selection model of closed timelike curves (P-CTC) to construct an "unproven-theorem" paradox, and claims that this voids our argument that P-CTCs are able to resolve such types of paradoxes. Here we point out that Ralph has not accounted for all the interactions needed for his construction. A more careful analysis confirms that indeed there is no paradox, contrary to his claims.

Motivation & Objective

• To refute Ralph's claim that the post-selection model of CTCs (P-CTC) fails to resolve the unproven-theorem paradox.
• To demonstrate that Ralph's construction omits essential physical interactions, particularly Bob's unitary transformation for reading and writing information into the CTC.
• To show that when these interactions are properly modeled via unitary purification, no genuine paradox emerges.
• To clarify that the P-CTC model enforces self-consistent evolution, preventing information from appearing from nowhere.
• To reinforce the validity of the P-CTC framework as a consistent quantum prescription for CTCs in general relativity scenarios.

Proposed method

• Model Bob's action of reading Alice’s book and sending information back via a phase-flip as a controlled-unitary operation, including environmental degrees of freedom to ensure unitarity.
• Use post-selected teleportation as the core mechanism to enforce consistency in CTC evolution, ensuring only self-consistent outcomes survive.
• Analyze two possible controlled-unitary implementations: a C-NOT in the |±⟩ basis and a copy operation on the |±⟩ basis with an environmental qubit.
• Calculate the amplitude of the full process under post-selection, showing it vanishes for the C-NOT case, indicating impossibility.
• Demonstrate that the copy operation leads to a single fixed point (|+⟩), resulting in a tautological, non-arbitrary theorem, not a paradox.
• Conclude that the theorem's origin is traceable to an author (Alice or Bob), not a causal loop, preserving consistency.

Experimental results

Research questions

• RQ1 Can Ralph’s proposed 'unproven-theorem' paradox be consistently realized within the P-CTC framework?
• RQ2 What physical operations must Bob perform to read a theorem from Alice’s book and send it back via a CTC?
• RQ3 How does post-selected teleportation prevent the emergence of causal paradoxes in CTC models?
• RQ4 Does the P-CTC model allow for arbitrary theorems to emerge from nothing, or is the information source traceable?
• RQ5 What role does unitary purification of Bob’s transformation play in resolving apparent paradoxes?

Key findings

• Ralph’s argument is incomplete because it omits the unitary transformation Bob must perform to read and write information into the CTC.
• When Bob’s action is modeled as a controlled-unitary with environmental degrees of freedom, the amplitude for the process vanishes in the C-NOT case, making the paradox impossible.
• In the copy-operation case, only a single fixed state (|+⟩) can be written, resulting in a tautological theorem that is not arbitrary or unproven.
• The P-CTC mechanism ensures that any information in the loop has a traceable origin, so no theorem appears from nothing.
• The model enforces self-consistency: if Bob knows Alice’s state, he can write a theorem, but he is then its author, not a recipient of unproven knowledge.
• Thus, the unproven-theorem paradox does not arise under the P-CTC model, confirming its consistency and validity.