Colbeck-Renner theorem

In plain terms

Quantum mechanics gives probabilities, and a natural hope is that some deeper theory holds extra information that would sharpen them into certainties. Colbeck and Renner showed that the hope fails, given one assumption about the experiment: that the measurement settings can be chosen freely, meaning their choice is not tied to whatever the deeper theory holds.

The argument uses a chain. Take two particles prepared in the state where each measurement on one is perfectly anticorrelated with the matching measurement on the other, and lay out many settings around a circle, each one a small angle from the next. Along the chain, neighbouring settings almost always disagree in a fixed way, and the angles are chosen so that after going all the way round you arrive back where you started. Now suppose the deeper theory has information that makes one outcome more likely than the other. Follow that bias around the chain: each link carries it along almost unchanged, and the closing link demands that it reverse. A bias cannot survive a lap that ends by contradicting it, so there is no bias, and the deeper theory's prediction is the quantum one, an even split.

The word free is carrying two ideas at once, and separating them is what makes the result usable. One is that the distant setting does not affect what happens here. The other is that the settings are uncorrelated with the deeper theory's extra information. A theory can give up the first and keep the second, and then the argument does not apply to it.

In CSD

A reader who has met this theorem asks the obvious question, since CSD posits underlying states that fix outcomes, so the corpus answers it with a theorem rather than a paragraph. The chained walk is proved in the library layer and instantiated at the singlet, and the two halves of the free-choice assumption appear as separate hypotheses: that each component of the extension is no-signalling, and that one distribution over the extra information serves every choice of settings.

The programme denies the first and keeps the second. Denying it is not a stance but a result the corpus already had: no way of splitting a probability space into a part for each wing reproduces the singlet correlations, so an underlying model of the singlet cannot have each wing's response depend only on its own setting. The second half, that the settings are uncorrelated with the underlying state, the corpus keeps as an explicit premise and names in the module that uses it. So CSD is not superdeterministic and not retrocausal, and anyone who reads it that way has collapsed the two ideas back into one.

Two cautions travel with this. Escaping a no-go theorem removes an objection and supports nothing, and this escape is inherited from the corpus's Bell result rather than independent of it, so if that result were ever weakened this one weakens with it. The Lean statement is for the maximally entangled pair and for the prediction of one outcome, in mean over the extra information; the extension to all states, which the original paper obtains by an embedding argument, is not claimed.

Mathematically

The library half is ProbabilityTheory.ChainedBell. For a two-outcome outcome law, two marginals of one joint distribution satisfy |marginalA - marginalB| <= P(disagree) and |marginalA + marginalB - 1| <= P(agree). No-signalling makes each wing's marginal a function of that wing's setting alone, which is what lets those one-pair bounds be composed along the alternating walk A 0, B 0, A 1, ..., A n, B n (abs_marginalA_sub_marginalB_chain_le); closing the walk with one reversed link gives abs_marginalA_sub_half_le, and integral_abs_marginalA_sub_half_le transfers the same bound to every component of a mixture, the functional being affine in the law.

The instance is the LF3 singlet kernel P_st with 2n + 2 settings at the angles j(pi - pi/(2n+1)) of a great circle: every link costs (1 - cos(pi/(2n+1)))/2 in disagreement and the closing link is exactly correlated, since (2n+1) steps come to 2n pi (dotR_crA_zero_crB_last). With 1 - cos x <= x^2/2 the bound is pi^2/(8(2n+1)), so no_improved_predictive_power gives, for every positive delta, settings at which every parameter-independent extension predicts Alice's outcome within delta of one half in mean over the extension variable. exists_signalling_of_sharp is the contrapositive: an extension that does sharpen the outcome has a component that signals.

The name

Roger Colbeck and Renato Renner published the theorem in 2011, working at ETH Zurich, and it grew out of their earlier result that free randomness can be amplified, which uses the same chained correlations. Colbeck is now at York and Renner still leads the quantum information theory group at ETH.

Giancarlo Ghirardi and Raffaele Romano in 2013, and Jelle Leegwater in 2016, separated the free-choice condition into its two components and identified which one the argument needs, which is why a theory can deny parameter independence and keep measurement independence. The chained inequalities the proof runs on are Samuel Braunstein and Carlton Caves's, from 1990.

Related entries

• Bell and CHSHNo pre-written answer sheet for all four measurement settings can match what a singlet pair of particles does, and the corpus proves this as a theorem about every possible answer sheet.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.
• Is CSD superdeterministic?No. In CSD's Bell model the distribution of hidden states is provably the same whatever settings the experimenters choose, and the escape from Bell's theorem runs through outcomes that depend on the measurement context and not through settings arranged in advance.
• No-signallingNothing done to the far wing of an entangled pair changes the statistics seen at the near wing, and CSD proves this exactly, for every state, entangled ones included.
• Bell state is not a joinThe arena CSD assigns to two systems together is strictly larger than the two arenas simply paired, and the Bell state is a point of the excess, so entanglement is located exactly.
• Is CSD a hidden-variable theory?Yes, in the technical sense, as something definite underlies the quantum state in CSD, and the programme openly pays the two prices the no-go theorems demand, dependence on measurement context and correlations beyond what locality permits.