Violation of a Monty-Hall constraint on determinism using a single qutrit

This paper proposes a Monty Hall-inspired protocol using a single qutrit to derive an inequality that distinguishes deterministic hidden-variable theories from quantum mechanics, demonstrating that the latter's prediction of a $1/6$ probability violates the $1/3$ bound required by any deterministic theory respecting the Monty Hall condition, thereby proving that determinism is incompatible with quantum coherence in sequential measurements.

The Gist

Imagine you are playing a game show version of the famous "Monty Hall" puzzle. You know the rules: there are three doors (let's call them Door A, Door B, and Door C). Behind one is a prize (the "true state"), and behind the other two are goats. You pick a door, say Door A. The host, who knows where the prize is, opens one of the other two doors to reveal a goat. He never opens the door with the prize. Now, you have a choice: stick with Door A or switch.

In the classic game, switching gives you a 2/3 chance of winning. But this paper isn't about winning a car; it's about a deep question in physics: Does the universe have a "secret script" that decides the outcome before we even look?

Here is the simple breakdown of what Jorge Meza-Domínguez's paper argues:

1. The Big Question: Is the Universe a Scripted Play?

For decades, physicists have debated whether the universe is deterministic (like a movie where the ending is already written, we just haven't seen it yet) or probabilistic (like a dice roll where the outcome is truly random until it happens).

Some theories say the universe is deterministic but "non-local" (meaning things can influence each other instantly across space, like in the de Broglie–Bohm theory). Others say it's random. The paper asks: Can we prove the universe is not a pre-written script using just one tiny particle?

2. The Experiment: A Quantum "Door" Game

The author proposes a test using a single qutrit.

• What is a qutrit? Think of a coin. A normal coin has two sides (Heads/Tails). A qutrit is like a "three-sided coin" that can be in a state of A, B, or C.
• The Setup: The experiment starts with the qutrit in a "superposition," which is like spinning the coin so fast it's effectively all three sides at once.

The Two-Step Game:

1. The "Coherent Discard" (The Host's Move): The experimenter performs a special operation that acts like the Monty Hall host. They "discard" one of the options (say, Door B) in a very specific, quantum way. Crucially, this operation is designed so that if the prize was behind Door A, it is never thrown away. It mimics the rule: "The host never eliminates the true state."
2. The Final Check: Immediately after discarding, the experimenter checks: "Is the prize behind Door A?"

3. The Prediction: Two Different Worlds

The paper calculates what should happen in two different worlds:

• World A: The Deterministic "Scripted" Universe
  If the universe is deterministic, the qutrit already had a definite answer (A, B, or C) before the game started. The "discard" step just removes the wrong doors, but it can't touch the right one because of the Monty Hall rule.

  • Result: If the prize was originally behind Door A (which happens 1/3 of the time), it stays there. If it was behind B or C, it stays there.
  • The Math: The chance of finding the prize at Door A after the discard is exactly 1/3 (33%).

• World B: The Quantum Universe (Our Reality)
  In quantum mechanics, the qutrit doesn't have a definite answer until we look. Because it was in a "superposition" (a blur of A, B, and C), the "discard" step changes the nature of the blur. It doesn't just remove a door; it reshapes the wave of possibilities.

  • Result: The math of quantum mechanics predicts that the chance of finding the prize at Door A drops to 1/6 (16.6%).
  • The Violation: Quantum mechanics predicts a result that is half of what any deterministic "scripted" theory allows.

4. Why This Matters

This is a big deal because previous tests (like Bell's Theorem) required two particles that were "entangled" (linked across vast distances) and assumed the universe was "local" (nothing travels faster than light).

This new test is simpler and stronger in a different way:

• One Particle Only: You don't need two entangled particles; just one qutrit is enough.
• No "Local" Assumption Needed: It doesn't matter if the universe is "non-local" (spooky action at a distance). Even if you believe in non-local deterministic theories (like the de Broglie–Bohm interpretation), this test says: "If your theory is deterministic, you must predict 1/3. If you predict 1/6, your theory is wrong."

5. The Proposed Experiment

The author suggests doing this with photons (particles of light).

• Imagine light traveling through three different paths (like three lanes on a highway).
• You use mirrors and beam splitters to create the "superposition" and the "discard" move.
• Then you count how often the light ends up in the "A" lane.

If the count is around 16.6%, it proves that the universe does not have a pre-existing "secret script" for this measurement. If it were 33%, it would mean the universe is deterministic.

Summary

The paper uses a clever analogy to the Monty Hall game to show that quantum mechanics violates a fundamental rule of deterministic theories.

• Deterministic Theory says: "The answer was already there; we just hid the wrong doors. The odds are 1/3."
• Quantum Mechanics says: "The answer wasn't decided until we looked, and the act of hiding a door changed the odds to 1/6."

The paper claims that if we run this experiment with current technology (using light or trapped ions), we will see the 1/6 result, proving that no deterministic hidden-variable theory (even a non-local one) can explain how this single particle behaves. It closes a loophole that has existed for decades, showing that nature is fundamentally probabilistic, not just "hidden" deterministic.

Technical Summary

1. Problem Statement

The paper addresses a fundamental open question in quantum foundations: whether deterministic hidden-variable theories (whether local or nonlocal, such as de Broglie–Bohm mechanics) can fully reproduce the predictions of quantum mechanics.

• Context: Bell's theorem rules out local hidden variables, but nonlocal deterministic theories remain compatible with current experimental data.
• Gap: Existing tests (Bell inequalities, Kochen–Specker, Leggett–Garg) rely on assumptions like locality, non-contextuality, or macrorealism. There is a need for a test that isolates determinism itself without requiring entanglement or spacelike separation.
• Goal: To derive an equality that any deterministic theory respecting a specific "Monty Hall" condition must satisfy, and to demonstrate that standard quantum mechanics violates this bound.

2. Methodology

The authors propose a protocol using a single three-level quantum system (qutrit) and a sequence of two measurements. The protocol is an analogy to the classical Monty Hall puzzle, reformulated in quantum language.

The Protocol Steps:

1. State Preparation:

   • A qutrit is prepared in a coherent superposition state: $|\psi_0\rangle = \frac{1}{\sqrt{3}}(|A\rangle + |B\rangle + |C\rangle)$.
   • State $|A\rangle$ is designated as the "bet" (analogous to the player's initial door choice).

2. Step 1: Coherent Discard (Generalized Measurement):

   • A Positive Operator-Valued Measure (POVM) is performed with elements $\{F_1, F_2\}$.
   • $F_1 = \frac{1}{2}(|A\rangle\langle A| + |C\rangle\langle C|)$.
   • Condition: The procedure is designed to mimic the Monty Hall host: it "discards" information about state $|B\rangle$ without eliminating the true state if it is $|A\rangle$ or $|C\rangle$.
   • If outcome $F_1$ occurs, the system collapses to the conditional state $|\psi_1\rangle = \frac{1}{\sqrt{2}}(|A\rangle + |C\rangle)$.
   • If outcome $F_2$ occurs, the state is orthogonal to $|A\rangle$ (probability of finding $|A\rangle$ becomes 0).

3. Step 2: Projective Measurement:

   • A projective measurement of the projector $P_A = |A\rangle\langle A|$ is performed immediately after the discard.
   • The quantity of interest, $Q$, is the total probability of obtaining outcome $|A\rangle$ in the second step (averaged over both branches of the first step).

3. Key Contributions

• Monty-Hall Equality for Determinism: The paper derives a strict bound for any deterministic hidden-variable theory that satisfies the "Monty Hall condition" (the discard procedure never eliminates the pre-existing true value of the system).
  • Theorem: In any such deterministic theory, the probability $Q_{det} = 1/3$.
• Quantum Violation: The authors calculate the quantum mechanical prediction for the same protocol.
  • Result: $Q_{QM} = 1/6$.
• Independence from Locality: The test requires no entanglement and no locality assumption. It applies to a single system, thereby closing the loophole for nonlocal deterministic theories (like de Broglie–Bohm) that usually evade Bell tests.
• Distinction from Other No-Go Theorems:
  • Unlike Kochen–Specker, it does not require non-contextuality; the incompatibility is dynamical (sequential) rather than algebraic.
  • Unlike Leggett–Garg, it does not assume macrorealism or non-invasive measurability; invasive measurements are central to the protocol.

4. Results and Analysis

• The Discrepancy: Quantum mechanics predicts a probability of 1/6, while deterministic theories respecting the Monty Hall condition predict 1/3. This represents a violation by a factor of two.
• Why Determinism Fails:
  • In a deterministic model, the system possesses a pre-existing value $\lambda \in \{A, B, C\}$.
  • The "discard" is a physical process that removes options but never removes the true value $\lambda$.
  • If $\lambda = A$, the discard leaves $A$ (probability 1). If $\lambda = B$ or $C$, the discard removes the other, leaving the true state.
  • Since the initial distribution is uniform ($1/3$ for each), the probability of the system being in state $A$ after the discard remains $1/3$.
  • Quantum Mechanism: The violation arises because quantum mechanics relies on coherent superpositions. The "discard" is a coherent operation that alters the amplitudes, not just the probabilities of pre-existing values. The interference effects reduce the probability of finding $|A\rangle$ to $1/6$.
• Bohmian Mechanics Analysis: The authors address whether de Broglie–Bohm theory could mimic the result. They argue that while Bohmian mechanics is deterministic, it does not assign a pre-existing discrete value $v(\lambda) \in \{A, B, C\}$ to the qutrit before the first measurement in the same way the theorem assumes. The "true state" in Bohmian mechanics is the particle's continuous position, and the interaction with the apparatus (ancilla) changes the trajectory in a way that reproduces the quantum result ($1/6$), effectively violating the specific "Monty Hall condition" defined for discrete outcome-determinism.

5. Significance and Experimental Feasibility

• Closing Loopholes: The test closes the "nonlocal determinism" loophole without needing complex entanglement setups.
• Experimental Proposal: The authors propose an implementation using photonic qutrits utilizing path and polarization degrees of freedom:
  1. Use a tritter to create the superposition.
  2. Implement the coherent discard via a beamsplitter (partial projection) and post-selection.
  3. Perform the final projective filter.
• Robustness:
  • Detection Loophole: Can be closed with current high-efficiency detectors (>90% for photons, near-unity for trapped ions).
  • Noise Tolerance: The inequality holds even with significant white noise ($\epsilon < 1$). Even with 50% noise, the quantum prediction ($Q \approx 0.25$) remains distinguishable from the deterministic bound ($1/3 \approx 0.33$).
• Conclusion: The paper provides a simpler, experimentally accessible demonstration that deterministic completions of quantum mechanics which assign definite pre-existing values to all measurements are incompatible with quantum coherence in sequential measurements. It offers a new operational inequality (related to KCBS but with a distinct Monty Hall interpretation) to test the nature of reality.