⚛️ quantum physics

Can present be the average of the future?

This paper proposes a deterministic, time-symmetric two-state vector formalism that generalizes Bell's hidden variable model to derive the Born rule by averaging over future states evolving backward in time, thereby offering an alternative demonstration of the Pusey-Barrett-Rudolph theorem.

Original authors: Z. Gedik

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Z. Gedik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guess the outcome of a coin toss. In the classical world, if you knew exactly how hard the coin was flipped, the wind speed, and the air pressure, you could predict the result with 100% certainty. The "randomness" we see is just because we are missing some of that information.

In the quantum world (the world of tiny particles like electrons), things seem fundamentally different. We usually say that nature is truly random; even if you knew everything, you could only predict the probability of an outcome, not the outcome itself. This is described by the Born rule, a famous formula that tells us how likely a particle is to be found in a certain place.

This paper by Z. Gedik proposes a mind-bending idea: What if the randomness isn't real? What if the "present" is actually just the average of all possible "futures"?

Here is a simple breakdown of the paper's core ideas using everyday analogies:

1. The Two-Story House (The Two-State Vector)

Usually, we think of a particle's history like a movie playing forward: it starts at point A and moves to point B.

The Old View: We only look at the "forward" movie.
Gedik's New View: Imagine a house with two stories.
- The Ground Floor: This is the particle moving forward in time (from the past to the present).
- The Attic: This is a "ghost" version of the particle moving backward in time (from the future to the present).

The paper suggests that to understand what happens now, you need to look at both floors simultaneously. The "hidden variable" (the secret ingredient that determines the outcome) isn't just a number; it's this backward-moving state from the future.

2. The Deterministic Dice Roll

The author takes an old idea from physicist John Bell. Bell showed that if you have a hidden rule, you can explain quantum weirdness without actual randomness.

The Analogy: Imagine you have a magical die. In the old view, when you roll it, it just lands on a random number.
Gedik's Twist: The die always lands on a specific number based on a strict rule. However, that rule depends on a "future" roll that hasn't happened yet.
The Magic: If you don't know what the future roll is, you have to guess. You average over all possible future rolls. When you do this math, the average perfectly matches the famous Born rule.

The Big Takeaway: The universe might be 100% deterministic (no randomness at all), but because we are stuck in the "present" and don't know the "future" state moving backward, it looks random to us. The present is the average of all possible futures.

3. The Time-Symmetric Mirror

The paper argues that time is symmetric. Just as a mirror reflects an image, the future reflects back on the present.

The Rule: To decide if a particle is in a specific state, you check if the "forward" state and the "backward" state agree. If they both point in a certain direction, the particle is there.
The Result: This creates a perfectly balanced, time-symmetric universe where the past and future are equally real and influence each other.

4. Proving the "Reality" of the Wave Function

There is a famous debate in physics: Is the quantum wave function just a tool for calculating odds, or is it a real physical thing?

The PBR Theorem: A previous theorem (Pusey-Barrett-Rudolph) argued that the wave function must be real.
Gedik's Contribution: Using this new "two-story house" model, the author shows that if two different "future" states exist, you can design a test to tell them apart. This proves that the quantum state isn't just a mathematical trick; it corresponds to a real physical reality, even if that reality involves time traveling backward.

5. Time Travel and Paradoxes

The paper briefly touches on "Closed Timelike Curves" (CTCs), which are theoretical paths in space-time that allow you to travel back to your own past (like in Back to the Future).

The Problem: Usually, time travel leads to paradoxes (e.g., killing your grandfather).
The Solution: In this model, the "backward" state and "forward" state swap roles as they pass through a time loop. This creates a self-consistent loop where the universe avoids paradoxes by constantly adjusting the "average" of the future to match the past.

Summary: The "Present" is a Blur

Think of the present moment not as a sharp point, but as a blurry photo.

The sharp edges of the photo are the "forward" past.
The blurry edges are the "backward" future.
The "randomness" we see in quantum mechanics is just our inability to see the sharp edges of the future.

In a nutshell: This paper suggests that the universe is a giant, deterministic machine where the past and future are locked in a dance. The "probabilities" we see are just the result of averaging over all the possible futures that are traveling backward to meet us right now. If we could see the whole dance (past and future), there would be no randomness at all.

1. Problem Statement

The paper addresses the fundamental distinction between classical and quantum mechanics: the probabilistic nature of quantum outcomes versus the deterministic nature of classical systems.

The Core Issue: Standard quantum mechanics relies on the Born rule ( $P = |\langle \psi | \phi \rangle|^2$ ) to predict probabilities, but it does not explain why these probabilities arise from an underlying deterministic reality.
Historical Context: While von Neumann's proof against hidden variables was later found to contain errors (by Hermann and Bell), Bell's subsequent hidden variable model for two-level systems (qubits) was limited to specific dimensions and did not attribute physical significance to the hidden parameters.
The Goal: The author seeks to reformulate quantum mechanics to show that probabilistic outcomes can be derived from a deterministic, time-symmetric formulation. Specifically, the paper asks if the "present" measurement outcome can be viewed as an average over "future" states evolving backward in time.

2. Methodology

The author employs a Two-State Vector Formalism (TSVF) generalized from Bell's hidden variable model. The methodology involves three main steps:

A. Generalization of Bell's Model

Bell's Original Model: For a two-level system, the measurement outcome is determined by two state vectors, $|m\rangle$ (forward) and $|n\rangle$ (hidden). The condition for finding the system in state $|a\rangle$ is:
$| \langle m | a \rangle |^2 + | \langle n | a \rangle |^2 > 1$
By averaging over all possible hidden states $|n\rangle$ (uniformly distributed on the Bloch sphere), the model reproduces the Born rule.
Generalization to Higher Dimensions: The author replaces the specific two-level vectors with arbitrary states in $d$ -dimensions, denoted as $|\Psi_\uparrow\rangle$ (forward evolving) and $|\Psi_\downarrow\rangle$ (backward evolving). The deterministic rule becomes:
$| \langle \Psi_\uparrow | a \rangle |^2 + | \langle \Psi_\downarrow | a \rangle |^2 > 1$
Here, $|\Psi_\downarrow\rangle$ represents a state propagating backward in time.

B. Derivation of the Born Rule

The author assumes a uniform distribution over the possible backward-evolving states ( $|\Psi_\downarrow\rangle$ ).
By calculating the ratio of pairs $(|\Psi_\uparrow\rangle, |\Psi_\downarrow\rangle)$ that satisfy the inequality above relative to all possible pairs, the probability of finding the system in state $|a\rangle$ is derived as $P = |\langle \Psi_\uparrow | a \rangle|^2$ .
Key Insight: The probabilistic nature of quantum mechanics emerges not from intrinsic randomness, but from the observer's ignorance of the specific backward-evolving state.

C. Application to Physical Reality (PBR Theorem)

The paper utilizes Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) to analyze the "elements of physical reality."
It applies the Pusey-Barrett-Rudolph (PBR) theorem logic: If two distinct quantum states $(|\Psi_0\rangle, |\Psi_1\rangle)$ correspond to different underlying physical realities (hidden variables), there must exist a measurement that distinguishes them with certainty.
The author demonstrates that in this two-state formalism, distinct pairs of forward/backward states can be distinguished by a single measurement, thereby providing a demonstration of the PBR theorem within this framework.

3. Key Contributions

Time-Symmetric Determinism: The paper proposes a fully time-symmetric ontology where the "present" is determined by the average of future states traveling backward in time. This offers a deterministic underpinning for the Born rule.
Generalized Hidden Variable Model: Unlike Bell's original model restricted to qubits, this formalism is applicable to quantum systems of arbitrary dimensions.
Alternative Derivation of the Born Rule: It provides a novel derivation showing that quantum probabilities are the result of averaging over a uniform distribution of backward-evolving states.
Demonstration of the PBR Theorem: The formalism provides a specific set of quantum states and a measurement strategy to distinguish non-orthogonal states, offering an alternative proof/statement of the PBR theorem regarding the reality of the wavefunction.
Implications for Closed Timelike Curves (CTCs): The author suggests that this formalism offers a consistent way to handle quantum mechanics near CTCs (as proposed by Gödel and analyzed by Deutsch). By allowing the exchange of forward and backward states upon passing through a unitary gate, the model avoids the nonlinearity issues often associated with CTCs in standard quantum mechanics.

4. Key Results

Reproduction of Quantum Statistics: The deterministic rule $| \langle \Psi_\uparrow | a \rangle |^2 + | \langle \Psi_\downarrow | a \rangle |^2 > 1$ , when averaged over a uniform distribution of $|\Psi_\downarrow\rangle$ , yields exactly the standard quantum mechanical probabilities.
Mutual Exclusivity: The formalism proves that mutually exclusive states (orthogonal eigenstates) cannot satisfy the condition simultaneously, ensuring consistency with standard quantum measurement outcomes.
Distinguishability: For any two distinct pairs of states (representing different physical realities), a single measurement exists that can distinguish between them, supporting the "ontic" nature of the quantum state in this model.

5. Significance

Ontological Shift: The paper challenges the standard Copenhagen interpretation by suggesting that quantum randomness is epistemic (due to lack of knowledge about the future) rather than ontological (intrinsic to nature).
Bridging Classical and Quantum: It attempts to bridge the gap between classical determinism and quantum probability by embedding quantum mechanics in a time-symmetric framework.
Foundational Implications: By linking the Born rule to time symmetry and hidden variables, it offers a fresh perspective on the measurement problem and the nature of physical reality, potentially influencing how we understand quantum gravity and time travel (CTCs).
Alternative to Weak Values: The author clarifies that while similar to the Two-State Vector Formalism used for weak values, this model applies to strong measurements without post-selection, making it a more general description of standard quantum processes.

In conclusion, Gedik's work posits that the "present" measurement outcome is indeed the statistical average of all possible "future" states evolving backward in time, providing a deterministic, time-symmetric foundation for the probabilistic laws of quantum mechanics.