In three papers, Colbeck and Renner [Nat. Commun. 2, 411 (2011); Phys. Rev. Lett. 108, 150402 (2012); e-print arXiv:1208.4123] argued that “no alternative theory compatible with quantum theory and satisfying the freedom of choice assumption can give improved predictions.” We give a more precise version of the formulation and proof of this remarkable claim. Our proof broadly follows theirs, which relies on physically well motivated axioms, but to fill in some crucial details, certain technical assumptions have had to be added, whose physical status seems somewhat obscure.

1.
Beltrametti
,
E. G.
and
Bugajski
,
S.
, “
A classical extension of quantum mechanics
,”
J. Phys. A
28
,
3329
3343
(
1995
).
2.
Braunstein
,
S. L.
and
Caves
,
C. M.
, “
Wringing out better Bell inequalities
,”
Ann. Phys.
202
,
22
56
(
1990
).
3.
Bub
,
J.
,
Interpreting the Quantum World
(
Cambridge University Press
,
1997
).
4.
Colbeck
,
R.
and
Renner
,
R.
, “
No extension of quantum theory can have improved predictive power
,”
Nat. Commun.
2
,
411
(
2011
).
5.
Colbeck
,
R.
and
Renner
,
R.
, “
Is a system’s wave function in one-to-one correspondence with its elements of reality?
,”
Phys. Rev. Lett.
108
,
150402
(
2012
).
6.
Colbeck
,
R.
and
Renner
,
R.
, “
The completeness of quantum theory for predicting measurement outcomes
,” arXiv:1208.4123.
7.
Ghirardi
,
G. C.
and
Romano
,
R.
, “
About possible extensions of quantum theory
,”
Found. Phys.
43
,
881
894
(
2013
).
8.
Laudisa
,
F.
, “
Against the no-go philosophy of quantum mechanics
,”
Eur. J. Philos. Sci.
4
,
1
17
(
2014
).
9.
Leegwater
,
G.
, “
An impossibility theorem for parameter independent hidden variable theories. Manuscript
” (submitted).
10.
Leifer
,
M.
, “
Is the quantum state real? An extended review of ψ-ontology theorems
,”
Quanta
3
,
67
155
(
2014
).
11.
Seevinck
,
M. P.
and
Uffink
,
J.
, “
Not throwing out the baby with the bathwater: Bell’s condition of local causality mathematically ‘sharp and clean
,’”
Explan., Predict., Confirmation
2
,
425
450
(
2011
).
12.
van Dam
,
W.
and
Hayden
,
P.
, “
Universal entanglement transformations without communication
,”
Phys. Rev. A
67
,
060302(R)
(
2003
).
13.
von Neumann
,
J.
,
Mathematische Grundlagen der Quantenmechanik
(
Springer
,
Berlin
,
1932
).
14.

As the notation indicates, μψ depends on ψ only and hence is independent of Z and z. From the point of view of T, a quantum state is a probability measure on Λ, so one might even write ψ for μψ.

15.

Colbeck and Renner look at the setting c as the value of some random variable C, but this is controversial;11 for us, C is simply the set in which c takes values.

16.

Colbeck and Renner treat ψ as a random variable and hence interpret Pψ(Z=z|λ) as a probability conditioned on knowing (that) ψ. We do not do so, yet our mathematical unfolding of (2.2) is similar.

17.

In other words, there is a subset Λ′ ⊂ Λ such that μψ(Λ′) = 0 and Pψ(Z=z|λ)=α(λ) holds for any λ ∈ Λ∖Λ′. If Λ is finite, this simply means that the equality holds for any λ for which μψ({λ}) > 0.

18.

In words, this assumptions states that the probabilities for Alice’s measurement outcomes, given λ, are not only independent of Bob’s choice of his observable Y, but are even independent of his existence altogether, as they are given by the expression that T yields for Alice’s experiment alone (and likewise for Bob). This slightly generalizes the usual Parameter Independence in the context of Bell’s Theorem.3 Note that in our form PI only makes sense because (2.1) and (3.1) imply that for Pψ(Z=z|λ) to be nonzero (in the sense of Sec. II) we must have ziσ(Zi) for each i.

19.

This generality, which is not a common feature of hidden variable theories (and as such is already a significant assumption), is necessary for the Colbeck–Renner argument to work.

20.

This assumption may be replaced by its main consequence, i.e., Lemma 4.2.

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