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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: XX YY ZZ 1 • Entanglement witness for two qubits, i.e. if X=sx etc • But not for e.g. two 8-dimensional systems: just define X sx , Y sx , Z sx (1) ( 2) (3) • 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-HardyKent, 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, BriëtBuhrman-Toner BLACK-BOX TOMOGRAPHY Work in collaboration with: Timothy Liew, Charles-E. Bardyn (CQT) Matthew McKague (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. sz ?C C 2 2 sx Black-box: the Vendor knows, but the User does not know anything of the physical system under study. ? ?C C ? ? ? 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) A, A' , B, B' dichotomic observables E ( A, B) P(a b) P(a b) S E ( A, B) E ( A' , B) E ( A, B' ) E ( A' , B' ) 2 • Two parties • Two measurements per party • Two outcomes per measurement • Maximal violation in quantum physics: S=22 Warm-up: assume two qubits The figure of merit: D( S ) max min UU 1 |S U u A uB S: the amount of violation of the CHSH inequality : the ideal state Trace distance: bound on the prob of distinguishing U: check only S=CHSH up to LU Solution: D( S ) 1 S / 2 2 2 1 Tight bound, reached by cos S 00 sin S 11 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 1x1 or 2x2. a a a ? ?C C ? P{a} ?C C ? ? P “a” ? P{a} ? C C ? b P{b} {b} “b” b b Multiple scenarios We have derived a “a” P{a} ? C C ? ? P “b” {b} b But after all, black-box it’s also possible to have ab “a,b” P{a} ? C C ? ? P {b} “a,b” ba 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 result “a,b” ab 22 P{a} ? C C ? P “a,b” {b} ba Fidelity: tight S F p 1 (1 p) (1 / 4) F (S ) S p 2 2 (1 p) 2 2 qubits F 2 1/4 ? 1/2 1 p(1,1) (1 p)I ( 2,2) / 4 Trace distance: not tight D(S ) 1 F (S ) Summary of results on D(S) D(S) 3/2 0.8 Arbitrary d, any state, scenario (a,b), not tight 1/2 0.4 Arbitrary d, pure states, achievable. 0 2 2.2 2.4 2 qubits tight 2.6 S 2.8 22 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