Can present be the average of the future?

Abstract: We introduce a two state vector formalism of quantum mechanics by generalizing Bell's hidden variable model to higher dimensions and by attributing a physical significance (a state evolving backward in time) to the hidden variable. A simple deterministic and time symmetric rule for measurement outcomes allows us to obtain the Born rule. It turns out that probabilistic outcomes can be derived from a deterministic assignment and averaging over all possible future states traveling backward in time. The assignment rule provides an alternative statement and demonstration of the Pusey, Barrett, Rudolph theorem.

• The paper introduces a deterministic, time-symmetric two-state vector formalism that derives the Born rule as an average over backward-evolving quantum states.
• It generalizes Bell’s hidden variable model to arbitrary dimensions using inner-product based rules, ensuring consistent measurement outcomes.
• The approach addresses quantum measurement near closed timelike curves, providing fresh insights and supporting the PBR theorem.

Deterministic Time-Symmetric Hidden Variable Formalism for Quantum Measurement

Overview

The paper "Can present be the average of the future?" (2604.11968) introduces a novel two-state vector formalism for quantum mechanics, generalizing Bell's hidden variable model to arbitrary dimensions and imparting physical significance to hidden variables as backward-evolving quantum states. This approach constructs a deterministic, time-symmetric rule for quantum measurement outcomes, demonstrating that Born rule probabilities can emerge as averages over possible future states propagating backward in time. The formalism provides alternative statements and proofs pertaining to the Pusey-Barrett-Rudolph (PBR) theorem and offers new perspectives in the context of quantum mechanics near closed timelike curves (CTCs).

Bell's Hidden Variable Model and Its Generalization

The foundation of the paper rests on Bell's hidden variable model, which utilizes two state vectors for a two-level system, forming a deterministic rule for measurement outcomes based on the sum of projections onto the measurement basis. In this model, averaging over the hidden variable (the second state vector) yields the quantum mechanical probability prescribed by the Born rule. The original Bell model assumes uniform weighting over the hidden variable's distribution, and the generalization to higher-dimensional systems in this work leverages inner-product-based assignment rules that extend to arbitrary Hilbert space dimensionality.

For each pair $(p, q)$, with $p = |\langle \mathbf{m} | a \rangle|^2$ and $q = |\langle \mathbf{n} | a \rangle|^2$, the deterministic assignment rule leads to a probability for measurement outcome equal to $p$, thus directly reproducing the Born rule from a purely deterministic model when averaged over a uniform distribution of possible backward-evolving states.

Time-Symmetric Two-State Vector Formalism

Central to the paper is the attribution of physical significance to the hidden variable as a backward-evolving quantum state. In the generalized formalism, the assignment rule utilizes two quantum states propagating forward and backward in time, denoted $\ket{\Psi_\uparrow}$ and $\ket{\Psi_\downarrow}$. Measurement outcome is determined by the condition $$|\langle \Psi_\uparrow | a \rangle|^2 + |\langle \Psi_\downarrow | a \rangle|^2 > 1$$, with time symmetry encoded explicitly.

This rule is shown to be consistent: mutually orthogonal measurement outcomes cannot be simultaneously realized due to the structure of the condition; summing over projections yields an upper bound that enforces exclusivity. Notably, the approach diverges from traditional two-state formalisms related to weak values by avoiding post-selection, focusing instead on deterministic assignment grounded in forward and backward state propagation.

Elements of Physical Reality and the PBR Theorem

The formalism satisfies the criterion for "elements of physical reality" as proposed by Einstein, Podolsky, and Rosen (EPR), wherein deterministic prediction of measurement outcome with probability unity corresponds to physical reality. The model is extended to mixed states using density matrices and projective measurements, demonstrating compatibility with symmetric informationally complete POVMs (SICPOVMs).

The approach enables a direct demonstration of the PBR theorem: distinct non-orthogonal quantum states are distinguishable via a single measurement in this two-state model, as expansion coefficients in SICPOVM bases differ for any two quantum states, ensuring measurement operators can discriminate between underlying physical realities. The formalism quantifies elements of reality for SICPOVM projectors, highlighting measurable distinctions even for non-orthogonal states.

Closed Timelike Curves and Nonlinear Quantum Mechanics

The paper discusses the implications of the formalism in the context of quantum mechanics near closed timelike curves (CTCs). In contrast with Deutsch's self-consistency model, which posits fixed points for CTC system density matrices under unitary evolution and introduces conceptual nonlinearity, the two-state vector formalism permits transformation of the backward-propagating state upon interaction with a unitary gate, modifying $\rho_{CTC}$ with each circuit traversal. This possibility provides a framework for richer state evolution near CTCs and potentially resolves some conceptual challenges relating to nonlinearity and self-consistency.

Practical and Theoretical Implications

This deterministic, time-symmetric formalism yields Born rule probabilities from ignorance of backward-evolving states, embedding quantum ontology in a fully time-symmetric setting. The theory affirms the existence of deterministic assignment mechanisms underpinning quantum measurement outcomes, potentially reducing reliance on probabilistic axioms. The explicit demonstration of the PBR theorem for a subset of quantum states further strengthens the claim that distinct wavefunctions encode distinct physical realities.

Practically, this model motivates new approaches to quantum measurement and quantum information processing, particularly in contexts where time symmetry and post-selection are relevant (e.g., quantum computing architectures exploiting time reversal or CTCs). Theoretically, the formalism opens avenues for re-examining foundational questions in quantum mechanics, including the origin of probabilistic outcomes and the nature of quantum state realism.

Future Directions

Future research may explore explicit construction and characterization of uniform distributions for backward-evolving states in higher dimensions, refinement of consistency conditions for arbitrary measurements, extensions to multipartite systems and entanglement phenomena, and integration with theories of quantum gravity and CTCs. Analytical and numerical investigations into the statistical properties and empirical verifiability of deterministic assignment rules in quantum experiments will further elucidate the implications of time-symmetric hidden variable models.

Conclusion

The introduced two-state vector formalism, generalizing Bell's hidden variable approach and imparting physical significance to backward-evolving states, reconstructs the Born rule via deterministic assignment and time-symmetric ontology. The model provides direct demonstrations of elements of reality, the PBR theorem, and new perspectives on quantum mechanics in CTC contexts. The work lays a foundation for further exploration of deterministic and time-symmetric hidden variable frameworks in quantum theory and its applications.