Entanglement vs Nonlocality
Massimiliano RotaDurham University

June 16, 2016

YITP, Kyoto University

Work in progress withMukund Rangamani

t=0CBA

What is the nature of quantum correlations between A-B or A-B-C ?

HA ⌦ H BAiii⇢AB = ⇢A ⌦ ⇢B

LOCCB

⇢AB =Xp( )⇢A ⌦ ⇢B

By definition a state is entangled if it is not separable

The definition can naturally be extended to the multipartite case

Measures involving combinations of entropies like the mutual informationor I3 do not work because they mix classical and quantum correlations.

Measures like distillable entanglement and entanglement costare very hard to compute even for few qubits.

For the bipartite case one can use the negativity, but in field theory it notoriously
difficult to compute even in the vacuum of a 1+1 dim CFT.
(Coser, Tonni, Calabrese ‘15)

For multipartite entanglement the situation is even worst, as for mixedstates there is no proposed computable measure, even for qubits.

For gauge theories, issue with tensor product structure.

t=0CBA

Other ways to look at quantum correlations ?

no-signaling conditions
X
X
0
PA (a|i) =P(ab|ij) =P(ab|ij )b

PB (b|j) =Xa(Popescu, Rohrlich ‘94)

8 a, i, j, jb

P(ab|ij) =

X

0

0

0

P(ab|i j)

8 b, i, i , jaP(ab|ij)

PA (a|i)

ij

A

Bab

PB (b|j)BAP(ab|ij) =X

p( )PA (a|i )PB (b|j )jlocality!

iIf the probability distribution does not admit this form, it is nonlocal
As for entanglement, the definition can be extended to a multipartite setting
The definition does not require a tensor product structure!
Even more generally, it is completely theory-independent!

(2, m, o)P(ab|ij) 2 R

i, j 2 {1...m}a, b 2 {1...o}m2 o2
Bell inequality

LX
abij

NSabSij P(ab|ij)  Sk

P(ab|ij) = h | ⇧a|i ⇧b|j | i
QM:

⇥+

*

Q = {PQM (ab|ij)}✓

QFT:⇧a|i ⇧b|j ⌘ ⇧a|i ⌦ ⇧b|j⇤

Q = {P? 0Q⇢Q

Tsirelson problem⇧a|i , ⇧b|j ⌘ 00

QFT(ab|ij)}Separable states are local
This follows immediately from the construction that uses a shared random variable.

Nonlocal states are not separable, i.e. entangled
Indeed Bell inequalities (BI) are used in experiments to detect entanglement,
they are entanglement witnesses.

But some entangled states are local!
There are known families of states which are non-separable (and hence entangled) but
nevertheless only locally correlated.

How “strong” can quantum nonlocality be?
In the simplest scenario (2,2,2), the local polytope is specified by a single BI,
the CHSH inequality.

hBi  2 (L)phBi  2 2 (Q) Tsirelson bound(Tsirelson ‘80)

Are no-signaling and nonlocality sufficient to define QM? No!

hBi  4(Popescu, Rohrlich ‘94)

(N S)

P(ab|ij)

Tsirelson inequality

LQ

NSThe space of quantum behaviours is not a polytope and it can be specified
only by an infinite number of inequalities.

t=0

B
A

(2, 2, o)i, j 2 {1, 2}

a, b 2 {1...o}We can assume that “o” is still finite (some lattice regularisation) but typically it would
be infinite and we want to be able to take the continuum limit. Now in this scenariochoose a BIXabSijP(ab|ij)  Skabij

We want to construct “experiments” that could be used to detect its violation, so weneed to rewrite this inequality in terms of expectation values of appropriately chosenobservables.
For the (2,2,2) scenario the answer is obvious and indeed this is how the CHSH
inequality is usually presented:

hO1 Õ1 i + hO1 Õ2 i + hO2 Õ1 ihO2 Õ2 i  2

For an arbitrary BI in the more general scenario (2,2,o) there is also a trivial construction.
Each probability that appears in the inequality can be obtained as the expectation valueof the corresponding projector:

P(ab|ij) ! h | ⇧a|i ⇧b|j | iThis transformation maps the original BI to itself, but in a different scenario

2(2, 2, o) ! (2, 4o , 2)
In the case where “o” is infinite, this construction implies that in order to be able to detecta violation, infinitely many measurements are necessary. We want instead to rewrite theoriginal inequality within the original scenario (2,2,o), using expectation values ofobservables with a spectrum containing “o” different eigenvalues.

Xi↵i hOi i +Xj↵
˜ j hÕj i +Xijij hOi Õj i  Sk

A BI then imposes a set of constraints on the spectrum of the observables that should
be used to detect its violation.
Is there always a solution to this problem and what are the properties of theseoperators? A formulation in terms of AQFT seems natural in this context, but ultimately
one would hope to translate the construction into a more “practical” formulation whereone can investigate the possible violation of BI by computing correlation functions.

P(ab|ij)

0Q ⌘QLNS

P(ab|ij)

Q0L

QNS

P(ab|ij)

0Q ⌘ NSLQ

Are causality, nonlocality and covariance sufficient to “define” relativistic QFT?