[Paper Review] Quantum Learnability is Arbitrarily Distillable

Joe H. Jenne, David R. M. Arvidsson-Shukur|arXiv (Cornell University)|Apr 19, 2021

Quantum Computing Algorithms and Architecture2 references4 citations

TL;DR

This paper demonstrates that quantum negativity enables the unbounded, lossless distillation of Fisher information across multiple parameters in quantum learning, allowing arbitrarily many input states to be compressed into a single output state without information loss. The key result is a postselection protocol that preserves quantum Fisher information while concentrating it, leveraging nonclassical quasiprobabilities to achieve arbitrarily high information concentration beyond classical limits.

ABSTRACT

Quantum learning (in metrology and machine learning) involves estimating unknown parameters from measurements of quantum states. The quantum Fisher information matrix can bound the average amount of information learnt about the unknown parameters per experimental trial. In several scenarios, it is advantageous to concentrate information in as few states as possible. Here, we present two "go-go" theorems proving that negativity, a narrower nonclassicality concept than noncommutation, enables unbounded and lossless distillation of Fisher information about multiple parameters in quantum learning.

Motivation & Objective

• To generalize postselected quantum metrology beyond single-parameter estimation to multiparameter learning with multiple unknowns.
• To identify nonclassical resources enabling unbounded information concentration in quantum learning protocols.
• To establish that negativity, not just noncommutation, is a fundamental resource for information distillation.
• To design a lossless protocol that concentrates information from many quantum states into few, minimizing experimental overhead.
• To provide a theoretical framework linking quasiprobability distributions, quantum Fisher information, and experimental cost reduction.

Proposed method

• Derives a general formula for the postselected quantum Fisher information matrix using perturbative expansion in the postselection strength.
• Introduces a Kirkwood-Dirac quasiprobability distribution to characterize classical vs. nonclassical bounds on Fisher information entries.
• Uses a controlled postselection protocol with a tunable parameter t to scale the effective information content in the postselected state.
• Demonstrates that the postselected Fisher information scales as 1/t², enabling unbounded concentration for large t.
• Shows that geometric quantumness (Uhlmann curvature) remains invariant under postselection, confirming robustness of nonclassical structure.
• Establishes that negative quasiprobabilities enable anomalous Fisher information entries outside classical bounds.

Experimental results

Research questions

• RQ1 Can quantum information about multiple unknown parameters be concentrated from many input states into a single output state without loss?
• RQ2 What nonclassical resource enables unbounded distillation of Fisher information in multiparameter quantum learning?
• RQ3 How does the presence of negativity in quasiprobability distributions affect the bounds on quantum Fisher information?
• RQ4 Can postselection protocols maintain information integrity while reducing experimental costs in quantum metrology?
• RQ5 Is the geometric quantumness of a state preserved under information-distilling postselection operations?

Key findings

• The postselected quantum Fisher information matrix scales as 1/t² relative to the original, enabling unbounded information concentration for large t.
• Negative entries in the Kirkwood-Dirac quasiprobability distribution allow Fisher information entries to exceed classical bounds, enabling nonclassical enhancement.
• The protocol is lossless: no information is wasted during distillation, preserving the full quantum Fisher information content.
• The geometric quantumness measure remains constant under postselection, indicating stability of nonclassical structure.
• The method applies generally to multiparameter estimation, overcoming the limitation of prior protocols requiring prior knowledge of all but one parameter.
• The results show that negativity is a sufficient and arbitrarily powerful resource for information distillation in quantum learning.