black-box tomography valerio scarani centre for quantum technologies & dept of physics national...
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Black-box TomographyBlack-box Tomography
Valerio ScaraniCentre for Quantum Technologies & Dept of Physics
National University of Singapore
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THE POWER OF BELLTHE POWER OF BELL
On the usefulness of Bell’s On the usefulness of Bell’s inequalitiesinequalities
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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???
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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…
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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?
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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
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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)
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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).
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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).
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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)
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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
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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 “” “”
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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!
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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
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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!
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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