⚛️ quantum physics

Retrievability of information in quantum and realistic hidden variable theories

The authors propose a generalization of macrorealism based on the retrievability of information rather than non-invasive measurability, demonstrate its theoretical connection to quantum error-disturbance uncertainty relations, and experimentally verify its violation in a photonic qubit system.

Original authors: Roope Uola, Erkka Haapasalo, Juha-Pekka Pellonpää, Tom Kuusela

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Roope Uola, Erkka Haapasalo, Juha-Pekka Pellonpää, Tom Kuusela

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

The Big Question: Can We Peek Without Breaking?

Imagine you have a magical, invisible box (a quantum system). You want to know what's inside.

The Old Rule (Macrorealism): In the classical world, if you peek inside the box, you can see the object without moving it, and the object stays exactly as it was. This is called Non-Invasive Measurability. If you peek, then peek again later, the second peek should give you the same result as if you had never peeked the first time.
The Quantum Problem: In the quantum world, this rule breaks. If you look at a quantum particle, you "kick" it. The act of looking changes its state. This is why famous tests (Leggett-Garg inequalities) often fail: the universe seems to say, "You can't look without touching."

However, there's a catch. Critics say, "Maybe the experimenters are just clumsy! Maybe they are accidentally kicking the particle too hard." This is called the "Clumsiness Loophole." It's hard to prove you didn't kick the particle because any measurement kicks it a little bit.

The New Idea: "Retrievability of Information"

The authors of this paper propose a smarter way to test reality. Instead of demanding that we never disturb the system (which is impossible in quantum mechanics), they ask a different question:

"Even if you disturb the system, can you fix it later so that the information is still there?"

They call this Retrievability of Information (RoI).

The Analogy: The Shattered Vase vs. The Mosaic

The Old Way (Non-Invasive): Imagine a pristine vase. You want to take a photo of it. The rule says: "You must take the photo without the camera flash or the lens touching the vase." If the flash cracks the vase, you failed.
The New Way (Retrievability): Imagine the vase is made of a special, self-healing material, or perhaps it's a mosaic. You smash the vase (measure it). But, you have a master craftsman (the "retrieving protocol") who knows exactly how the pieces fell. They can glue the pieces back together perfectly, or rearrange the mosaic tiles, so that the picture on the wall looks exactly the same as it did before you smashed it.

If you can smash the vase and then perfectly restore the picture, the information was retrievable. If you smash it and the picture is permanently ruined, the information was lost.

What Did They Do?

The team built a quantum experiment using photons (particles of light) to test this new rule.

The Setup: They prepared a photon in a specific state (like a spinning coin).
The "Smash" (First Measurement): They measured the photon. This inevitably disturbed it, changing its state.
The "Fix" (Retrieval): Instead of just measuring again, they applied a specific, calculated correction (a "wiring" operation). This is like the master craftsman gluing the vase back together.
The Test: They checked if the final result matched what they would have seen if they had just measured the photon directly without the first "smash."

The Twist: The Quantum Limit

Here is the fascinating part. The authors discovered that in the quantum world, there is a fundamental limit to how well you can "glue the vase back together."

This limit is governed by the Heisenberg Uncertainty Principle (specifically, the Busch-Lahti-Werner relations). Think of it like this:

You can try to fix the vase perfectly, but the universe has a "noise floor."
If you try to retrieve information about two different properties of the particle at once (like its position and its speed, or in this case, two different directions of spin), you will always lose a tiny bit of information. You can't perfectly reconstruct the original picture.

The Result: Quantum Wins

The team ran the experiment and found:

They could retrieve information: They successfully showed that even after disturbing the photon, they could use a clever protocol to get the data back. This proved that "clumsiness" isn't the only reason for quantum weirdness; the disturbance is real, but the information can be recovered.
They hit the limit: They found that they could retrieve the information only up to the limit allowed by quantum physics.
The Verdict: If you try to build a "hidden variable" theory (a classical explanation where the particle has a secret, pre-determined state) that allows for perfect retrieval of information, it fails. The quantum world is strictly more powerful than any classical model that tries to explain it using "retrievable" hidden states.

Summary in One Sentence

The authors replaced the impossible rule of "never disturb the system" with the realistic rule of "can you fix the disturbance?" and proved that while you can fix the disturbance, the universe still keeps a secret that no classical "hidden variable" theory can ever fully explain.

The Takeaway: The universe isn't just "clumsy"; it's fundamentally fuzzy. You can peek, you can even fix the mess you made, but you can never perfectly reconstruct the past without hitting a hard wall of quantum uncertainty.

1. Problem Statement

The paper addresses the challenge of testing macrorealism—the classical view that macroscopic systems possess definite properties at all times and can be measured without disturbance.

The Clumsiness Loophole: Traditional tests of macrorealism rely on the Leggett-Garg inequalities, which assume Non-Invasive Measurability (NIM). NIM posits that measurements do not disturb the system or its future dynamics. However, in quantum mechanics, measurements are inherently invasive. Critics argue that violations of Leggett-Garg inequalities might simply result from "clumsy" experimental setups where the measurement apparatus unintentionally disturbs the system, rather than a fundamental failure of macrorealism.
The Gap: Existing methods to close this loophole (e.g., ideal negative measurements, adroit measurements) are often complex or require specific assumptions about the measurement apparatus. There is a need for a framework that allows for disturbance but still tests the classicality of temporal correlations in a rigorous, theory-agnostic way.

2. Methodology

The authors propose a generalization of macrorealistic hidden variable models by replacing the NIM postulate with a new assumption: Retrievability of Information (RoI).

Core Concept (RoI): Instead of requiring that a measurement leaves the system undisturbed, RoI assumes that if a measurement is performed, the information about the system's state (specifically, the statistics of a subsequent measurement) can be retrieved by using the outcome of the first measurement to adapt the second measurement.
- Mathematically, for a macrostate $\lambda$ , the statistics of a final measurement $y$ (with no prior measurement) must be recoverable on average by performing a first measurement $x$ and then a conditional measurement $y_a$ based on outcome $a$ .
- Equation: $\sum_\lambda p(\lambda)p(b|0, y, \lambda) = \sum_{a,\lambda} p(\lambda)p(a, b|x, y_a, \lambda)$ .
Quantum Implementation:
- The authors restrict the analysis to qubit systems and utilize the Lüders instrument (the minimal disturbance state update rule) for the first measurement.
- They connect RoI to the concept of Joint Measurability. In quantum theory, a sequence of measurements can retrieve information if and only if the observables are jointly measurable.
- They leverage the Busch-Lahti-Werner (BLW) uncertainty relations, which define the fundamental precision limit for approximate joint measurements of non-commuting observables (e.g., $Z$ and $X$ ).
Experimental Setup:
- Platform: Photonic qubits using polarization states ( $|H\rangle, |V\rangle$ ).
- Source: Spontaneous parametric down-conversion (SPDC) generating photon pairs.
- Protocol:
  1. State Preparation: Preparation of specific superposition states.
  2. First Measurement (Noisy Z): Implemented using a noisy Positive Operator-Valued Measure (POVM) $A_\gamma$ controlled by a parameter $\gamma$ . This simulates a "disturbing" measurement.
  3. Retrieving Measurement: A final projective measurement (Sharp $X$ or noisy $X$ ) is performed. The setup allows for the first measurement outcome to effectively influence the second via the optical path (wiring).
- Optimization: The experiment targets $\gamma = \pi/8$ , which corresponds to the optimal trade-off point for the BLW uncertainty relation (minimizing the sum of errors for joint $Z$ and $X$ measurements).

3. Key Contributions

Generalization of Macrorealism: The paper introduces Retrievability of Information (RoI) as a strictly broader class of hidden variable theories than macrorealism. RoI allows for disturbance, provided the information can be recovered via adaptive protocols.
Simplification of Quantum Tests: Unlike previous macrorealism tests that require optimizing over all possible state updates to close the clumsiness loophole, the RoI framework shows that it is sufficient to consider only the Lüders instrument (the standard minimal disturbance update present in every quantum measurement).
Connection to Uncertainty Relations: The authors establish a direct link between the ability to retrieve information in sequential measurements and the Busch-Lahti-Werner error-disturbance uncertainty relations. They prove that the optimal retrieving protocol is bounded by these fundamental quantum limits.
Extension to Generalized Measurements: The model extends macrorealistic concepts from Hermitian observables (sharp measurements) to generalized measurements (POVMs). It demonstrates that while a sharp observable measured twice yields the same result, generalized measurements do not necessarily allow simple repetition to retrieve information unless they are jointly measurable.

4. Results

The authors implemented the proposed protocol experimentally and reported the following:

Violation of RoI Models: The experimental data showed a violation of the inequalities derived from the RoI hidden variable models.
Optimal Retrieving: When $\gamma = \pi/8$ , the system achieved the optimal approximate joint measurement of $Z$ and $X$ . The experimental probabilities matched the theoretical predictions for the quantum mechanical optimal retrieving protocol.
No-Retrieving Scenario: When attempting to retrieve a "sharp" $X$ observable after a non-trivial first measurement (where $\gamma > 0$ ), the data violated the RoI condition. This confirms that quantum mechanics does not allow the retrieval of sharp information from a disturbed system beyond the limits set by the BLW uncertainty relations.
Data Consistency: The experimental results (with statistical errors around $\pm 2\%$ ) closely aligned with quantum theoretical predictions, confirming that quantum theory is more general than the proposed classical retrieving models.

5. Significance

Closing the Loophole: This work provides a robust method to test macrorealism that is immune to the "clumsiness loophole." By explicitly allowing for disturbance and focusing on retrievability, it removes the need to prove that a measurement was "non-invasive" in the classical sense.
Fundamental Limits: It reinforces the fundamental nature of quantum uncertainty. The inability to retrieve information in the "no-retrieving" case is not due to experimental imperfection but is a consequence of the fundamental precision limits of quantum theory (BLW relations).
Practical Application: The study demonstrates that these tests are feasible with current photonic technology.
Future Directions: The framework opens new avenues for investigating temporal correlations in open quantum systems (where information leaks to the environment) and adaptive feed-forward mechanisms in quantum thermodynamics.

In conclusion, the paper successfully demonstrates that quantum mechanics violates a generalized form of macrorealism based on information retrievability, establishing that the fundamental limits of quantum measurement precision are the ultimate barrier to classical descriptions of temporal correlations.