Approaching Tsirelson's bound from
theory and experiment
Adán Cabello
Universidad de Sevilla
Quantum Foundations 2016,
Patna, October 18, 2016
Bell inequality scenario
Bell inequality
Tsirelson bound
Non-local (non-signalling) boxes
Why?
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
even when other measurements are performed in between.
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
even when other measurements are performed in between.
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
even when other measurements are performed in between.
Answer: Because we need nice theories

We need theories with sharp measurements: Repeatable
even when other measurements are performed in between

In these theories, the E principle holds (Chiribella and Yuan,
arXiv:1404.3348)
Exclusive events

Event: Post-measurement state (a,…,c|x,…,z), where a,…,c are
the outcomes of compatible sharp measurements x,…,z on a
given input state

Two events are exclusive if they can be perfectly distinguished
by a sharp measurement
The exclusivity (E) principle

E principle: If every two events in a set are exclusive, then all
events in the set are mutually exclusive
The exclusivity (E) principle

E principle: If every two events in a set are exclusive, then all
events in the set are mutually exclusive
(The E principle does not follow from Kolmogorov)

Kolmogorov: The sum of the probabilities of any set of
mutually exclusive events cannot be higher than 1
The exclusivity (E) principle

E principle: If every two events in a set are exclusive, then all
events in the set are mutually exclusive
(The E principle does not follow from Kolmogorov)

Kolmogorov: The sum of the probabilities of any set of
mutually exclusive events cannot be higher than 1

E principle + Kolmogorov: The sum of the probabilities of any
set of pairwise exclusive events cannot be higher than 1
The exclusivity (E) principle

E principle: If every two events in a set are exclusive, then all
events in the set are mutually exclusive
(The E principle does not follow from Kolmogorov)

Kolmogorov: The sum of the probabilities of any set of
mutually exclusive events cannot be higher than 1

E principle + Kolmogorov: The sum of the probabilities of any
set of pairwise exclusive events cannot be higher than 1
Related to:
Orthocoherence in quantum logic (Mackey, 1957, 1963)
Specker’s principle (1960)
Local orthogonality in Bell scenarios (Fritz et al., 2013)
CHSH re-expressed
Consider a Bell-CHSH experiment (Vienna)
And an independent Bell-CHSH experiment (Stockholm)
Consider Vienna-Stockholm events
Vienna
Consider Vienna-Stockholm events
Vienna
Stockholm
Two exclusive events
Two exclusive events
Vienna’s Bob’s results are different
Assumption: V and S experiments are independent

P(a, b, a’, b’ | x, y, x’, y’) = P(a, b | x, y) P(a’, b’ | x’, y’)
Vienna
Stockholm
Besides the measurements needed for CHSH
An observer could perform Alice-Alice’ measurements
So the Vienna-Stockholm events can be relabelled as
Assumption: A0 and A1 are maximally incompatible

Compatible = no violation

Assumption: In this scenario, maximal violation occurs
for maximal incompatibility for one party
Assumption: A0 and A1 are maximally incompatible

Compatible = no violation

Assumption: In this scenario, maximal violation occurs
for maximal incompatibility for one party

This assumption implies
For example, in QT
The standard notion of measurement incompatibility for projective
measurements is non commutativity: Maximal incompatibility = anti
commutativity
Assumption: A0 and A1 are maximally incompatible

Compatible = no violation

Assumption: In this scenario, maximal violation occurs
for maximal incompatibility for one party

This assumption implies
So events like these have well-defined probabilities
These two events were pairwise exclusive
Bob’ results
are different
These two events are also pairwise exclusive
Alice-Alice’ results
are different
These two events are also pairwise exclusive
Alice-Alice’ results
are different
This is a set of 9 pairwise exclusive events
We can apply the E principle
“The sum of the probabilities of any set of pairwise
exclusive events cannot be higher than 1”
and obtain an “E inequality”
Similarly, we can obtain other 15 “E inequalities”
Summing these 16 “E inequalities” we obtain
Tsirelson bound!
What else does the E principle explain?
Simplest Bell inequality
Simplest non-contextuality inequality
Why?
Consider one KCBS experiment (Vienna)
Adjacent vertices = pairwise exclusive events
And an independent KCBS experiment (Stockholm)
Notice that
Viewed from a super observer’s perspective
Viewed from a super observer’s perspective
Viewed from a super observer’s perspective
1010|1212
1010|5134 1010|2345
1010|4551 1010|3423
Assumption: V and S experiments are independent

P(a, b, a’, b’ | x, y, x’, y’) = P(a, b | x, y) P(a’, b’ | x’, y’)
Vienna
Stockholm
E inequality #1
1010|1212
1010|5134 1010|2345
1010|4551 1010|3423
E inequality #2
E inequality #3
E inequality #4
E inequality #5
The 5 E inequalities
Summing them all
“Quantum” bounds because “we need nice theories”

We need theories with sharp measurements: Repeatable
even when other measurements are performed in between.

In all these theories, the E principle holds (Chiribella and
Yuan, arXiv:1404.3348).

If nothing else (local realism, non-contextual realism,
entanglement/nonlocality limited to be n-partite, limited
incompatibility, measurements limited to be m-chotomic,…)
prevents to reach these bounds, then nature should saturate
them.

Conjecture: All “quantum” bounds are the E principle’s
bounds; no physical principle constraints them.
Open problems

We don’t know whether all quantum bounds can be explained
by the E principle, e.g., I3322

We don’t know whether all quantum bounds are reachable
Are the “quantum” bounds really reachable?
when adopting an informationtheoretic definition of
“observer”, involving a limit on
the complexity of the strings
she can store and handle
What do experiments say?
What do experiments say?
The Singapore experiment

Tightest experimental test of Tsirelson’s bound ever

Violates Grinbaum’s bound by 4.3 standard deviations
The Singapore experiment
Moral
 Tsirelson bound seems to be a reachable limit of nature
 There seems to be no principle (e.g., local realism, noncontextual realism, bounded complexity for the strings an
observer can store) beyond the “we need nice theories
principle”
What’s next?
 We have to test sensible “close to quantum theories” (e.g.
“almost quantum”, “fundamentally binary”,…)
 Can we derive QT from the “we need nice theories
principle”?
Fundamentally binary?
Fundamentally binary?
Results
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary quantum
measurements
Results
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary quantum
measurements… and there is already experimental evidence
of these correlations
Results
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary quantum
measurements… and there is already experimental evidence
of these correlations
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary measurements
(including PR boxes!)
Results
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary quantum
measurements… and there is already experimental evidence
of these correlations
 QT predicts correlations which cannot be explained by any
nonsignaling correlations produced by binary measurements
(including PR boxes!)… but testing these correlations is very
difficult. An open problem is identifying and performing such
experiment
M. Kleinmann and AC, Phys. Rev. Lett. 117, 150401 (2016)