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Black-box Tomography

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.

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Black-box Tomography

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  1. Black-box Tomography Valerio Scarani Centre for Quantum Technologies & Dept of Physics National University of Singapore

  2. The power of bell On the usefulness of Bell’s inequalities

  3. 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 are even teaching it to our students! We have understood that quantum physics is not “crypto-deterministic”, that local hidden variables are really not there… Can’t we move on to something else???

  4. 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

  5. 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?

  6. 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, 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

  7. Black-box Tomography Work in collaboration with: Timothy Liew, Charles-E. Bardyn (CQT) Matthew McKague (Waterloo) Serge Massar (Brussels)

  8. 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).

  9. 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).

  10. 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

  11. 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.

  12. How to get rid of the dimension? Theorem: two dichotomic observables A, A’ can be simultaneously block-diagonalized with blocks of size 1x1 or 2x2. b a b P{a} P{b} a “a” “b” P{a} P{b} b a

  13. Multiple scenarios We have derived “a” “b” P{a} P{b} b a But after all, black-box  it’s also possible to have “a,b” “a,b” P{a} P{b} b(a) a(b) 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!

  14. Partial result “a,b” “a,b” P{a} P{b} b(a) a(b) Fidelity: tight S 22 2 qubits Trace distance: not tight F 2 1/4 1/2 1

  15. Summary of results on D(S) D(S) 3/2 Arbitrary d, any state, scenario (a,b), not tight 1/2 Arbitrary d, pure states, achievable. 2 qubits tight S 22 Note: general bound provably worse than 2-qubit calculation!

  16. 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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