Black-box TomographyBlack-box Tomography

Valerio ScaraniCentre for Quantum Technologies & Dept of Physics

National University of Singapore

THE POWER OF BELLTHE POWER OF BELL

On the usefulness of Bell’s On the usefulness of Bell’s inequalitiesinequalities

Bell’s inequalities: the old storyBell’s inequalities: the old story

Measurement on spatially separated entangled particles correlations

Can these correlations be due to “local variables” (pre-established agreement)?

Violation of Bell’s inequalities: the answer is NO!

OK lah!!We have understood that quantum physics is not

“crypto-deterministic”, that local hidden variables are

really not there…

We are even teaching it to our

students!

Can’t we move on to something

else???

A bit of historyA bit of history

Entanglement Theory

Bell ineqs

Around the year 2000, all serious physicists were not concerned about Bell’s inequalities.

All? No! A small village…

Bell’s inequalities: the new storyBell’s inequalities: the new story

Bell’s inequalities = entanglement witnesses independent of the details of the system!

• If violation of Bell and no-signaling, then there is entanglement inside…• … and the amount of the violation can be used to quantify it!

Counterexample: 1 ZZYYXX• Entanglement witness for two qubits, i.e. if X=x etc• But not for e.g. two 8-dimensional systems: just define

)3()2()1( ,, xxx ZYX

Quantify what?

TasksTasks

• Device-independent security of QKD– Acín, Brunner, Gisin, Massar, Pironio, Scarani, PRL 2007– Related topic: KD based only on no-signaling (Barrett-Hardy-

Kent, Acin-Gisin-Masanes etc)

• Intrinsic randomness– Acín, Massar, Pironio, in preparation

• Black-box tomography of a source– New approach to “device-testing” (Mayers-Yao, Magniez et al)– Liew, McKague, Massar, Bardyn, Scarani, in preparation

• Dimension witnesses– Brunner, Pironio, Acín, Gisin, Methot, Scarani, PRL2008– Related works: Vertési-Pál, Wehner-Christandl-Doherty, Briët-

Buhrman-Toner

BLACK-BOX TOMOGRAPHYBLACK-BOX TOMOGRAPHY

Work in collaboration with:Work in collaboration with:

Timothy Liew, Charles-E. Bardyn (CQT)Timothy Liew, Charles-E. Bardyn (CQT)

Matthew McKague (Waterloo)Matthew McKague (Waterloo)

Serge Massar (Brussels)Serge Massar (Brussels)

The scenarioThe scenario

• The User wants to build a quantum computer. The Vendor advertises good-quality quantum devices.

• Before buying the 100000+ devices needed to run Shor’s algorithm, U wants to make sure that V’s products are worth buying.

• But of course, V does not reveal the design U must check everything with devices sold by V.

• Meaning of “V adversarial”:

= “V wants to make little effort in the workshop and still sell his products”

“V wants to learn the result of the algorithm” (as in QKD).

Usual vs Black-box tomographyUsual vs Black-box tomography

22? CC xz

??? CC ??

Usual: the experimentalists know what they have done: the dimension of the Hilbert space (hmmm…), how to implement the observables, etc.

Black-box: the Vendor knows, but the User does not know anything of the physical system under study.

Here: estimate the quality of a bipartite source with the CHSH inequality.(first step towards Bell-based device-testing, cf. Mayers-Yao).

Reminder: CHSH inequalityReminder: CHSH inequality

',,', BBAA

2)','()',(),'(),( BAEBAEBAEBAES

dichotomic observables

)()(),( baPbaPBAE

• Two parties• Two measurements per party• Two outcomes per measurement• Maximal violation in quantum physics: S=22

(Clauser, Horne, Shimony, Holt 1969)

Warm-up: assume two qubitsWarm-up: assume two qubits

1|minmax)( UUSD

BA uuUS

The figure of merit:

Trace distance: bound on the prob of distinguishing

2

12/1)(

2

SSD

11sin00cos SS

Solution:Tight bound, reached by

Proof: use spectral decomposition of CHSH operator.

: the ideal state

U: check only S=CHSH up to LU

S: the amount of violation of the CHSH inequality

How to get rid of the dimension?How to get rid of the dimension?

??? CC

Theorem: two dichotomic observables A, A’ can be simultaneously block-diagonalized with blocks of size 1x1 or 2x2.

??? CC

??? CC “” “”

Multiple scenariosMultiple scenarios

??? CC “” “”

We have derived

??? CC “” “”

But after all, black-box it’s also possible to have

i.e. an additional LHV that informs each box on the block selected in the other box (note: User has not yet decided btw A,A’ and B,B’).

Compare this second scenario with the first: • For a given , S can be larger D(S) may be larger.• But the set of reference states is also larger D(S) may be smaller. No obvious relation between the two scenarios!

Partial resultPartial result

??? CC “” “”

F

S22

2

1/4 1/2 1

2 qubits

)(2)1(22

)4/1()1(1SF

ppS

ppF

Fidelity: tight

4/)1( )2,2()1,1( Ipp

)(1)( SFSD Trace distance: not tight

Summary of results on D(S)Summary of results on D(S)

S

D(S)

22

1/2

3/2

2 2.2 2.4 2.6 2.80

0.4

0.8

2 qubitstight

Arbitrary d, pure states,achievable.

Arbitrary d, any state, scenario (), not tight

Note: general bound provably worse than 2-qubit calculation!

ConclusionsConclusions

• Bell inequality violated Entanglement

• No need to know “what’s inside”.

• QKD, randomness, device-testing…

• This talk: tomography of a source– Bound on trace distance from CHSH– Various meaningful definitions

• No-signaling to be enforced, detection loophole to be closed