Black-box Tomography Valerio Scarani Centre for Quantum Technologies & Dept of Physics National University of Singapore
THE POWER OF BELL On the usefulness of Bell’s inequalities
Bell’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 history Around the year 2000, all serious physicists were not concerned about Bell’s inequalities. All? No! A small village… Bell ineqs Entanglement Theory
Bell’s inequalities: the new story Bell’s inequalities = entanglement witnesses independent of the details of the system! Counterexample: • Entanglement witness for two qubits, i. e. if X=sx etc • But not for e. g. two 8 -dimensional systems: just define • If violation of Bell and no-signaling, then there is entanglement inside… • … and the amount of the violation can be used to quantify it! Quantify what?
Tasks • 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, Mc. Kague, Massar, Bardyn, Scarani, in preparation • Dimension witnesses – Brunner, Pironio, Acín, Gisin, Methot, Scarani, PRL 2008 – Related works: Vertési-Pál, Wehner-Christandl-Doherty, Briët. Buhrman-Toner
BLACK-BOX TOMOGRAPHY Work in collaboration with: Timothy Liew, Charles-E. Bardyn (CQT) Matthew Mc. Kague (Waterloo) Serge Massar (Brussels)
The 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 tomography 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 inequality (Clauser, Horne, Shimony, Holt 1969) dichotomic observables • Two parties • Two measurements per party • Two outcomes per measurement • Maximal violation in quantum physics: S=2 2
Warm-up: assume two qubits The figure of merit: S: the amount of violation of the CHSH inequality F+: the ideal state Trace distance: bound on the prob of distinguishing U: check only S=CHSH up to LU Solution: Tight bound, reached by Proof: use spectral decomposition of CHSH operator.
How to get rid of the dimension? Theorem: two dichotomic observables A, A’ can be simultaneously block-diagonalized with blocks of size 1 x 1 or 2 x 2. b a a a P{a} P{b} “a” “b” P{a} P{b} b b
Multiple scenarios We have derived a “a” P{a} P{b} “b” b But after all, black-box it’s also possible to have a(b) “a, b” P{a} P{b} “a, b” b(a) 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 r, 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 result “a, b” a(b) 2 2 P{a} P{b} “a, b” Fidelity: tight S 2 qubits F 2 1/4 1/2 1 Trace distance: not tight b(a)
Summary of results on D(S) 3/2 1/ 2 Arbitrary d, any state, scenario (a, b), not tight Arbitrary d, pure states, achievable. 2 qubits tight S 2 2 Note: general bound provably worse than 2 -qubit calculation!
Conclusions • • 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
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