Wigner-Araki-Yanase theorem

In plain terms

Measurement is a physical interaction, so it has to obey the conservation laws that every other interaction obeys. Wigner asked what that costs, and the answer is that some measurements become impossible.

Suppose the system and the apparatus together conserve a quantity, with each contributing its own share, as they do for energy, momentum or spin along an axis. Now try to measure an observable of the system that doesn't commute with the system's share of that conserved quantity. Recording the outcome means leaving the apparatus in one of several distinguishable states, and doing so exactly, with no error, turns out to be incompatible with conservation, so no such measurement exists. Read the other way round, exact measurements are available only for observables that commute with the conserved quantity.

The result comes with a side condition, and the condition is doing real work rather than tidying up. The argument needs either that the pointer states carry no information about the apparatus's own share of the charge, which is Yanase's condition, or that the measurement can be repeated with the same answer. Drop both and there are counterexamples, so the theorem isn't a blanket prohibition but a precise one.

In CSD

The corpus proves the theorem on the quantum-mechanical side, as an inner-product argument with no ontology in it, and then states honestly where it bears on the programme, which is nowhere, for different reasons at the two places a measurement is modelled. In the von Neumann tier the hypotheses can be met, since the measurement interaction there is an isometry on a product of two spaces and an additive conserved quantity for it is exhibited, and the conclusion then holds without effort because the recorded basis is already diagonal for the conserved quantity. In the record layer the hypotheses are not met at all, as the stroke that forms a record is a skew product on the programme's arena rather than a linear isometry on a joint Hilbert space, and its twin theorem proves that no encoding of the Hilbert data into any space, with any map and any readout, reproduces the pointer image the stroke produces, because the outcome is selected by the ontic fibre coordinate.

That's a scope statement and not an escape. Nothing here says the programme evades a conservation law, only that no apparatus charge is modelled at either tier, which is a recorded modelling boundary: which interaction an apparatus realises, and hence which quantity it conserves, is an input rather than a derived fact. Two witnesses keep the module honest. A controlled-not interaction meets every hypothesis, through the repeatability branch, so the theorem isn't vacuous. A swap interaction conserves a charge, records exactly, and still fails the conclusion, because it fails both branches of the side condition at once, so a reader cannot believe that condition removable.

Mathematically

Everything follows from arakiYanase_identity: for exact records U(a tensor xi) = phi tensor xi1 and U(a' tensor xi) = phi' tensor xi2 of orthogonal system vectors, with U inner-product preserving and commuting with L = L_S tensor 1 + 1 tensor L_A, one has <a, L_S a'> = <phi, L_S phi'><xi1, xi2> + <phi, phi'><xi1, L_A xi2>. Orthogonal pointers kill the first term and the Yanase condition or repeatability kills the second (arakiYanase_offDiag_eq_zero), so summing over an eigenbasis of A makes L_S block diagonal in the eigenspaces of A: Commute A L_S (wigner_araki_yanase), with no_exact_record_of_not_commute the contrapositive.

The tensor structure enters only through a map with the inner-product factorisation and additivity of L on product vectors, so no linearity of U, no matrix exponential and no unitary group is needed. Witnesses: way_hypotheses_satisfiable (controlled-not conserves one charge and records another exactly and repeatably), swap_exact_record_not_commute (swap conserves, records, and fails both side conditions), sigmaX_no_exact_conserving_record (the no-go in use). The record-layer twin is no_joint_hilbert_map in Empirical/CSD/WignerArakiYanase.lean, stated at the level of any joint lift so that base back-reaction cannot invalidate it. Ozawa's quantitative form, bounding the measurement error below by the inverse apparatus variance, is not formalised.

The name

Eugene Wigner raised the problem in 1952 in a paper on the measurement of quantum observables, and Huzihiro Araki and Mutsuo Yanase gave the general theorem in 1960, with Yanase adding the condition on the pointer states in 1961. All three were thinking about whether the measurement postulate is consistent with the rest of physics rather than about any practical limitation.

Masanao Ozawa turned the structural theorem into a quantitative one in 2002, showing that a gate failing to commute with a conserved charge has a fidelity floor set by the control system's charge variance, which is the form the result takes in quantum computing.

Related entries

• Ozawa error-disturbance relationThe famous error-times-disturbance formula attributed to Heisenberg fails in the laboratory, and the relation that actually holds is Ozawa's, with three terms instead of one, and the corpus proves it.
• Pointer basisA measuring apparatus doesn't record what you ask it to, as it records what it can keep, and what it can keep is decided by its interaction with the surroundings and not by the experimenter.
• Record layerA measurement is over when the world itself carries which outcome happened, and the record layer is where that fact lives, on the arena and not in anyone's notebook.
• 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.