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Experimental generationand characterisation of private
states
Experimental generationand characterisation of private
states
Paweł HorodeckiWydział Fizyki Technicznej i
Matematyki Stosowanej, Politechnika Gdańska
Karol HorodeckiInstytut Informatyki, Uniwersytet
Gdański
Konrad BanaszekRafał Demkowicz-DobrzańskiMichał KarpińskiWydział FizykiUniwersytet Warszawski
Krzysztof DobekWydział Fizyki, Uniwersytet Adama Mickiewicza w Poznaniu
1p2
³j$ $ i + jl l i
´
Bell’s inequalitiesBell’s inequalities+
-
A: a = 45o
A’: a = 0oB: b = 22.5o
B’: b = 67.5o
+
-1p2
³j00i + j11i
´
hAB i + hA0B i + hAB 0i ¡ hA0B 0i = 2p
2
Clauser-Horne-Shimony-Holt inequality¯¯hAB i + hA0B i + hAB 0i ¡ hA0B 0i
¯¯¯ · 2
is violated!
Quantum cryptographyQuantum cryptography
A1: a = 45o
A2: a = 0o
B1: b = 22.5o
B2: b = 67.5o
For each photon pair Alice and Bob select randomly measurement bases…
A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991)
B3: b = 0o
Perfect correlations one-time padPerfect correlations one-time pad
Security testSecurity test
…and compare measurements over a public channel afterwards.
Entangled photon pairsEntangled photon pairsP. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999)
Output state:
j$ $ i + jl l ij©+ i /
Correlation measurementsCorrelation measurements
b
0a
45a
Entanglement monogamyEntanglement monogamy
• Even when the pair has been prepared by Eve…
• …if Alice and Bob verify that the systems arrived in a maximally entangled pure state…
• …measurement results will be known only to Alice and Bob.
Eve
Alice
Bob
j©+ iAB = 1p2
³j00iAB + j11iAB
´
Statistical mixtureStatistical mixture
Define:
Equally weighted mixture:
Eve
Alice
Bob
Density matrixDensity matrix
Maximally entangled state
Statistical mixture12
³j©+ i AB h©+ j + j©¡ i AB h©¡ j
´
Correlations between measurement outcomes in the key basis
0
BBBB@
12 ¢ ¢ 1
2¢ ¢ ¢ ¢¢ ¢ ¢ ¢12 ¢ ¢ 1
2
1
CCCCA
0
BBBB@
12 ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ 1
2
1
CCCCA
j©+ i AB h©+ j
Security tested by the violation of Bell’s inequalities(If trusting quantum theory, could be also tested by measurements in the basis.)(j0i § j1i )=
p2
j00ij01ij10ij11i
Noisy entanglementNoisy entanglement
How much secure key can be extracted from a noisy state?
0
BBBB@
12 ¢ ¢ 1
4¢ ¢ ¢ ¢¢ ¢ ¢ ¢14 ¢ ¢ 1
2
1
CCCCA
DistillationDistillation
M…
…N
Distillable entanglement:
C. H. Bennett et al., Phys. Rev. Lett. 76, 722 (1996)
Example IExample I
A B
A’
B’
Shield states:
enable Alice and Bob to distinguish locally and generate
the key using the standard strategy. HenceE D = 1
Example IIExample II
A B
A’
B’
• States and cannot be discriminated unambiguously using local operations and classical communication.
• Distillable entanglement bounded by log-negativity:
What if
jª ¡ i = 1p2(j01i ¡ j10i )
EavesdroppingEavesdropping
AA’
B B’
The worst case scenario: all the noise is controlled by Eve
K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim,Phys. Rev. Lett. 94, 160502 (2005)
E
%AA0B B 0 = T rE
³jª i AA0B B 0E hª j
´
Alice Eve channelAlice Eve channel
Alice measures an outcome a with a probability
Eve infers a from the conditional state of her subsystem E:
%(a)E = 1
paT rA0B B 0
³Ahajª ihª jaiA
´
pa = T rA0B B 0E
³Ahajª i hª jaiA
´
Holevo quantity:
X (A : E ) = S
ÃX
apa%(a)
E
!
¡X
apaS
³%(a)
E
´
Key rateKey rate
Key rate
E
Mutual information
Holevo quantity
For Example II, Eve’s subsystem contains no information about outcomes of Alice’s measurement on her qubit, hence KD = 1.
AA’
B B’
ShieldShield
Without the shield:
The complete state:
key security
T rA0B 0
³%AA0B B 0
´=
0
BBBB@
12 ¢ ¢ 1
4¢ ¢ ¢ ¢¢ ¢ ¢ ¢14 ¢ ¢ 1
2
1
CCCCA
AB
00
01
10
11
Double photon pairsDouble photon pairs
Output state:A’
AB’
B
j©+ iAB j©+ iA0B 0
Experimental setupExperimental setup
Quantum state tomographyQuantum state tomography
Projective qubit measurements:
¾x; ¾y; ¾z
Four-qubit POVM:
M i = j§ iA i j§ iA0i j§ iB i j§ iB 0i
34 = 81 measurement bases
34 x 24 = 1296 event types
: number of events i
Probability of an outcome i:
X
ini ¼5 £ 105
Density matrix estimate
%
Maximum likelihood reconstructionMaximum likelihood reconstruction
L(%) = p(fnigj%) =Q
i [p(ij%)]n i
– number of events i
Likelihood function:
Maximum-likelihood estimatemaximizes
Probability of an outcome i:
Z. Hradil, Phys. Rev. A 55, R1561 (1997)
ML: ParametrisationML: Parametrisation
PRO: - Guaranteed positivity
CON: - Impractical in higher dimensions (>6 qubits)- Underestimates errors, difficult to include
uncertainty of the measuring device (Monte Carlo simulations)
- Biased towards low-rank matrices for undersampled data
K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi,Phys. Rev. A 61, 010304(R) (1999)
Task: maximize
with a constraint
Ensuring positivity:
Bayesian approachBayesian approach
A priori distributionA posteriori:
Estimate:
PRO:
CON: - Difficult to normalise probability distribution- A priori distribution not well defined
- Clear statistical interpretation- Provides uncertainty- No numerical optimisation
K. Audenaert and S. Scheel, New J. Phys. 11, 023028 (2009)
• Gaussian approximation
• Truncated to positive definite density operators
State reconstructionState reconstructionK. Dobek. M. Karpiński, R. Demkowicz-Dobrzański, K. Banaszek,
and P. Horodecki, Phys. Rev. Lett. 106, 030501 (2011)
Privacy characterisationPrivacy characterisation
Distillable entanglement: ED 0.581(4)
A posteriori ensemble
Warning:
Key (cqq scenario): K 0.690(7)
More conservative
Distillation protocolDistillation protocol
Measure qubits A′B′ in the same basis.
50% 50%
Identical outcomes
Opposite outcomes
Single-copy distillationSingle-copy distillation
averageidentical outcomes A
′B′
K = 0.354(5)
Reduced density matrix
K = 0
opposite outcomes A′B′
conditional
Cryptographic keyCryptographic key
Optimal strategy
Raw key: 3716 bits
Error correction
2726 bits
Privacy amplification
Secure key: 2164 bits
Distillation-based approach
Raw key: 1859 bits
1300 bits
Secure key: 650 bits
%AA0B B 0=
0
BBB@
S00;00 S00;01 S00;10 S00;11S01;00 S01;01 S01;10 S01;11S10;00 S10;01 S10;10 S10;11S11;00 S11;01 S11;10 S11;11
1
CCCA
Witnessing privacyWitnessing privacy
ab
00011011
00 01 10 11
Complete density matrix for key and shield subsystems:
Si j ;kl = A0B 0hij j%AB A0B 0jkliA0B 0
where
Key boundKey bound
K³%AA0B B 0
´
¾AB =12
0
BBB@
p+ ¢ ¢ c+¢ p¡ c¡ ¢¢ c¡ p¡ ¢
c+ ¢ ¢ p+
1
CCCA
where
p+ = jjS00;00 + S11;11jj
c+ = jjS00;11 + S11;00jj
¸ K³¾AB
´In general:
p¡ = jjS01;01 + S10;10jj
c¡ = jjS01;10 + S10;01jj
K. Horodecki et al., IEEE Trans. Inf. Theory 54, 2621 (2008); ibid. 55, 1898 (2009).
Single witnessSingle witness
w =¯¯¯D
12(¾x
A ¾xB ¡ ¾y
A ¾yB ) UA0B 0
E¯¯¯
Suppose we have measured
where UyU · I
K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012)
w =¯¯T r[(S00;11 + S11;00)UA0B 0]
¯¯¯
· jjS00;11 + S11;00jj = c+
This provides an estimate
Single witnessSingle witness
12
0
BBB@
p+ ¢ ¢ c+¢ p¡ c¡ ¢¢ c¡ p¡ ¢
c+ ¢ ¢ p+
1
CCCA
c+ ¸ w
Positivity of implies that ¾AB
p+ ¸
w
Take the worst-case scenario for p¡ ; c¡
K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012)
K
Two observablesTwo observables
wx =¯¯¯D¾x
A ¾xB UA0B 0
E¯¯¯
12
0
BBB@
p+ ¢ ¢ c+¢ p¡ c¡ ¢¢ c¡ p¡ ¢
c+ ¢ ¢ p+
1
CCCA
We have:
c+ + c¡ ¸ wx
p§ = 12(1 § wz)
wz =¯¯¯D¾z
A ¾zB I A0B 0
E¯¯¯
c¡ · p¡
K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012)
ConclusionsConclusions
• Experimental demonstration of the separation between distillable entanglement and cryptographic key contents
• Practical comparison of quantum state reconstruction methods for a noisy multiqubit state
• Full privacy analysis based on the reconstructed state
• Evaluation of highly non-linear information theoretic quantities
• Implementation of a simple entanglement distillation protocol
• Witnessing privacy with few observables
• Multiple degrees of freedom?