Quantum squeezing cannot beat the standard quantum limit

Quantum squeezing cannot beat the standard quantum limit because, when comparing measurement precision to unentangled ensembles, squeezed states provide no fundamental advantage. This work rigorously demonstrates that an ensemble of N squeezed particles cannot surpass the precision achievable with N independent particles under a given set of conditions.

The Gist

A single non-separable squeezed state provides fundamentally no better precision, per unit time, than a single particle.

Standard Quantum Limit and Approaches to Enhancement

The standard quantum limit (SQL) sets the uncertainty bound per unit time that is impossible to surpass with a given number of identical and independent spins. Two distinct approaches exist to overcome the SQL using entanglement:

1. Making the spin response greater, such as with entangled NOON, CAT or GHZ states, where the response is theoretically scaled by N while measurement noise remains related to a single particle's uncertainty.

2. Reducing the uncertainty in measuring the spin direction, where squeezed states are claimed to have a reduced noise factor of N but an unchanged signal response.

Mathematical Framework for Comparison

The analysis compares precision using two key metrics: the state response to the signal, quantified by the derivative term, and information bounds derived from Fisher information. The paper establishes that for any single non-separable state vector, the minimum uncertainty per unit time is bounded by its state response to the signal.

(1) The bound based on measurement precision:

The analysis shows that if the optimum precision of any non-separable state vector can be characterised by the state response to a parameter, then squeezed states cannot improve this optimum precision without improving the state response.

(2) The bound based on information (Fisher Information):

The Fisher information, denoted as I[θ, t], is used to compare uncertainties. The key requirement is that the central claim in squeezing enhanced metrology is that squeezed states have an improved intrinsic noise (uncertainty) compared to a single spin.

Proof Against Squeezing Enhancement

The core proof relies on the counting argument based on the number of indivisible state vectors.

1. For an ensemble containing N > 1 particles, if it is entangled to any degree, it must contain less indivisible state vectors than an unentangled ensemble.

2. The information provided by a squeezed ensemble is shown to be less than the information provided by the unentangled ensemble because a squeezed ensemble can be separated into a maximum of M independent state vectors where M < N (following the definition of entanglement).

3. This counting argument leads to the conclusion that increasing squeezing/entanglement is continuously increased, then at some point the uncertainty must get worse.

Refutation of Squeezing Claims

The paper addresses common claims by reviewing analytic errors and physical interpretations in experimental literature:

(1) Conflation of noise and uncertainty:

A common analytical mistake is Conflation of noise and uncertainty, where reduced measurement variance is incorrectly equated with improved precision. The paper notes that the measurement signal (c.f. the derivative in Eq. (5)), is replaced with a constant term that does not depend on θ or the measurement basis.

(2) Replacement of the signal gradient with contrast:

The error occurs when the measurement signal (c.f. the derivative in Eq. (5)), is replaced with a constant term that does not depend on θ or the measurement basis. This ignores the necessary dependence of precision on the state response to θ.

Conclusion

The final conclusion is that squeezed ensembles cannot outperform unentangled ensembles in sensing, as the information bound derived from counting separable states is lower for entangled ensembles. The paper asserts that The signal is the noise! and suggests that experimental claims contradicting this proof are due to Analytic errors or comparisons to different limits, rather than fundamental flaws in the mathematical derivation.

Proof Assumptions

The rigorous mathematical proof relies on several assumptions:

1. The postulates of quantum mechanics are correct.

2. The entire dependence on θ is encoded in Hˆ (θ, t), and the starting state ψ0i and the measurement operator are independent of θ.

3. Each copy of the same state is independent and measurements on copies of the same state are identical. This ensures that any enhancement over the SQL comes solely from entanglement, not other correlations.

4. The analysis focuses on information bounds rather than measurement uncertainty, as relating them requires assumptions about prior probability distributions that can lead to violations of the Cramér-Rao bound.

Alternative Proof Sketches

The paper outlines alternative approaches to obtaining the same result:

1. Using a binary outcome measurement expression (Eq. 5) and showing that any modification increasing I[θ, t] necessarily increases the signal term.

2. Utilizing Wootters' distance metric interpretation, showing that if state evolution is the same for two states, then the information is also the same.

Based on the provided scientific paper, here are specific improvements that could be made to AI systems, categorized by their potential applications:

) Improve Quantum Sensing and Metrology AI Systems:

AI systems designed for quantum sensing (e.g., gravitational wave detectors like LIGO, atomic clocks) should shift their theoretical framework from relying on squeezed states to focusing on the fundamental information bounds dictated by the state's response to the signal, rather than assuming noise reduction is equivalent to precision improvement.

AI systems could be improved by:

1. Implementing rigorous checks against common analytical errors identified in the paper, such as conflating measurement noise with uncertainty (as seen in Section 7). The AI should be trained to distinguish between reducing measurement variance and improving the actual estimation uncertainty on the signal parameter.

2. Integrating a framework that utilizes Fisher Information or related metrics derived from quantum state evolution, rather than relying on intuitive descriptions of squeezing (as discussed in Section 4). This would allow the AI to rigorously determine if an experimental setup is approaching a fundamental limit based on information theory, rather than just observing reduced noise.

3. Developing algorithms that account for the necessary assumptions regarding prior probability distributions and measurement statistics (as detailed in Section 25), ensuring that uncertainty bounds are not violated by pathological priors.

) Improve Quantum Computing and Information Processing AI Systems:

AI systems used for quantum computation or quantum information processing should be guided by the principle that entanglement, on its own, does not automatically guarantee a surpassment of the Standard Quantum Limit (SQL).

1. The AI should prioritize tasks where the unitary evolution of a single state is analyzed to determine if it remains in an optimal state for all evolution times. This helps in designing quantum algorithms whose performance is guaranteed by the fundamental information bound, regardless of entanglement manipulation.

2. AI systems should be trained to recognize that squeezed states often lead to metrologically useless states when measured via collective observables (like a collective spin measurement along Z), as demonstrated in Section 11. This prevents the AI from optimizing for apparent noise reduction while ignoring the loss of information on critical parameters (like phase).

) Improve Experimental Design and Validation AI Systems:

AI systems tasked with designing next-generation quantum experiments should adopt a skeptical verification approach, treating claims of SQL-beating as hypotheses requiring rigorous proof.

1. The AI must be explicitly programmed to look for counterexamples to the prevailing experimental claims (as suggested by McGuinness's challenge in Section 2). It should actively search literature for analyses that compare achieved precision directly to the SQL, rather than relying on tautological definitions of precision or modified quantum limits (Section 7.A).

2. The AI should be equipped with tools to calculate the information bound based on the gradient of the state vector with respect to the signal, and then compare this calculated bound against experimental results that claim performance beyond it, rather than accepting claims solely based on reduced noise figures (Section 6).

) Improve Quantum State Preparation AI Systems:

AI systems responsible for preparing quantum states should be guided by a clear understanding of the necessary conditions for entanglement to yield an advantage.

1. The AI should be penalized or constrained when generating states that are described by simplified pictorial representations (like the Bloch sphere representation used in Section 22) without explicitly defining the underlying mathematical structure (like Eq. 8).

2. The system must be trained to recognize that entanglement alone is insufficient for surpassing the SQL; it requires specific, non-trivial correlations with a well-defined phase relationship between basis states (as noted in Section 17).

In summary, the core improvement is shifting AI focus from noise reduction as a proxy for precision enhancement to a rigorous information-theoretic analysis of state response and entanglement structure.

Quantum entanglement between particles is expected to allow one to perform tasks that would otherwise be impossible. In quantum sensing and metrology, entanglement is often claimed to enable a measurement precision that cannot be attained with the same number of particles and time, forgoing entanglement. Two distinct approaches exist: creation of entangled states that either i) respond quicker to the signal, or ii) are associated with lower noise and uncertainty. The second class of states are generally called squeezed states. Here we show that if our definition of success is a precision that is impossible to achieve using the same resources but without entanglement then squeezed states cannot succeed. In doing so we show that a single non-separable squeezed state provides fundamentally no better precision, per unit time, than a single particle.