Transcript in PPT
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