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 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 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: Entanglement witness for two qubits, i.e. if X=  x etc But not for e.g. two 8-dimensional systems: just define 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, 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 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 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 dichotomic observables Two parties Two measurements per party Two outcomes per measurement Maximal violation in quantum physics: S=2  2 (Clauser, Horne, Shimony, Holt 1969)

Warm-up: assume two qubits The figure of merit: Trace distance: bound on the prob of distinguishing 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? Theorem: two dichotomic observables A, A’ can be simultaneously block-diagonalized with blocks of size 1x1 or 2x2.               “”“”“”“”

Multiple scenarios       “”“”“”“” We have derived       “  ” 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 result       “  ” F S 2222 2 1/41/21 2 qubits Fidelity: tight Trace distance: not tight

Summary of results on D(S) S D(S) 2222 1/  2  3/2 2 qubits tight Arbitrary d, pure states, achievable. Arbitrary d, any state, scenario (  ), not tight 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