Ozawa error-disturbance relation

In plain terms

Heisenberg's microscope is the story everyone is told. Look at an electron closely enough to pin down its position and the light you use to see it gives it a kick, spoiling its momentum, so you measure precisely and disturb proportionally. The trade-off is usually written as a formula, that the error of the measurement times the disturbance it causes is at least a fixed amount, and that formula is quoted constantly.

It isn't true as stated. In 2012 two groups measured it failing, Erhart and colleagues with neutron spins in Vienna (Nature Physics 8, 185), and Rozema and colleagues with photon polarisation in Toronto, and this wasn't a subtle correction, as the product of error and disturbance came out below the supposed floor.

What's true is Ozawa's version, and the difference is instructive, as it has three terms rather than one. Besides the error and the disturbance, it involves the spreads of the two quantities in the state you prepared, meaning how widely each would scatter if measured on its own before any disturbance, so the extra terms pair something about the measurement with something about the prepared state. You still can't have an exact measurement that disturbs nothing, but the honest bookkeeping needs the preparation spreads in it, and the naive formula left them out.

Two things are worth keeping apart. The trade-off between error and disturbance is real, and the famous formula for it was wrong.

In CSD

This is a coverage result rather than a CSD one, as the corpus proves it because a reader expects it, and it records in a theorem, not a footnote, that the relation doesn't belong to CSD's own measurement account.

The relation is about a measurement model in the textbook sense, a system, a probe, an interaction between them, and a meter read at the end. CSD's record layer, its account of how a measurement leaves a lasting mark in the underlying state space, has no such model, as its measurement step is a deterministic map on the arena, the underlying state space, rather than a linear transformation of a joint quantum state of system and probe, so the error and the disturbance the relation talks about aren't defined there at all. This is proved rather than asserted, as it follows from the same theorem that shows the record layer has no instance of the Wigner-Araki- Yanase measurement setting.

CSD therefore neither explains nor predicts the error-disturbance trade-off. What the corpus gains is completeness, as it proves Robertson's inequality, the statistical uncertainty relation, and alongside it the measurement-theoretic relation a reader aware of the 2012 experiments expects to find next to it.

Mathematically

For a measurement model with A_in and B_in the observables entering, A_out the meter read-back and B_out the disturbed observable, the relation reads eps(A) eta(B) + eps(A) sigma(B) + sigma(A) eta(B) >= 1/2 |<psi, [A,B] psi>| with sigma the standard deviations. The corpus proves this at the level the proof actually uses, four symmetric operators on one inner-product space with A_out and B_out commuting, and no tensor product appears. That commutation is the whole content, as expanding the commutator of the two "out" operators gives four terms whose left side vanishes, and the three survivors are bounded by one Cauchy-Schwarz inequality each.

The naive form eps eta >= 1/2 |<[A,B]>| isn't universally valid rather than false, as it holds for unbiased measurements, and the corpus makes no Lean claim about it. It does refute the two-term variant, which shows that the third term is load- bearing.

The record layer's exclusion is no_ozawa_model_of_jointLift, a corollary of the theorem that gives Wigner-Araki-Yanase no record-layer instance, as the record stroke is a skew product on the arena and not a linear map on a joint Hilbert space, so the model's error and disturbance are undefined there.

The name

Masanao Ozawa, born in 1950, worked at Nagoya and Chubu on the mathematical foundations of quantum measurement, the theory of instruments, measuring processes, and what a measurement can and can't achieve. The 2003 relation came out of taking the measurement process itself as the object of study rather than assuming the textbook trade-off, and it took nine years for the experiments to catch up with it.

Related entries

• Robertson uncertainty relationRobertson's inequality is the precise form of the uncertainty principle, and in CSD the spreads it bounds are the spreads of quantities genuinely distributed over the underlying state space and not merely abstract expectation values.
• Luders ruleThe textbook's second rule, which says how a state changes when it's measured, isn't a second rule in CSD, as it's what the record layer does.
• Measurement flowA deterministic, volume-preserving flow carries out a measurement, and the volumes of its pointer blocks are the Born weights, for every preparation.
• Fisher informationThere's a floor under every error bar, classical or quantum, and Fisher information is what sets it.
• CollapseIn CSD the quantum state never jumps, and that's a theorem about volume-preserving maps rather than a preference among interpretations.