Qkd and de finetti theorem

Quantum Key Distribution
and de Finetti’s Theorem
Matthias Christandl
Institute for Theoretical Physics, ETH Zurich
June 2010
Matthias Christandl Quantum Key Distribution and de Finetti’s Theorem

Overview
Introduction to Quantum Key Distribution
Two tools for proving security:
De Finetti’s Theorem
Post-Selection Technique
Summary

Alice und Bob want to communicate in secrecy, but their
phone is tapped.
Alice
Eve
Bob
phone
If they share key (string of secret random numbers),
Alice Eve Bob
message
+key
--------------
cipher cipher
- key
--------------
message
cipher is random and message secure (Vernam, 1926)

key is as long as the message
Shannon (1949): this is optimal
secret communication ˆ= key distribution
possible key distribution schemes:
Alice and Bob meet ⇒ impractical
Weaker level of security
assumptions on speed of Eve’s computer
(public key cryptography)
assumptions on size of Eve’s harddrive
(bounded storage model)
Use quantum mechanical eﬀects
(Bennett & Brassard 1984, Ekert 1991)

Quantum mechanics governs atoms and photons
Spin-1
2
system: points on the sphere
cos θ
eiϕ
sin θ
= cos θ|0 + eiϕ
sin θ|1 ∈ C2
unit of information, the quantum bit or ”qubit”
we measure a qubit along a basis
if basis is {|0 , |1 }, we obtain ’0’ with probability cos2
θ.
in general: express qubit in basis and consider
|amplitude|2
.

The state of two qubits: |ψ ∈ C2
⊗ C2
, ψ||ψ = 1
|ψ = ψ00|0 ⊗ |0 + ψ01|0 ⊗ |1 + ψ10|1 ⊗ |0 + ψ11|1 ⊗ |1
= ψ00|0 |0 + ψ01|0 |1 + ψ10|1 |0 + ψ11|1 |1
each qubit is measured in basis {|0 , |1 }
measurement basis {|0 |0 , |0 |1 , |1 |0 , |1 |1 }
obtain ’ij’ with probability |ψij |2
.

Entangled state of two qubits 1√
2
(|0 A|0 B + |1 A|1 B)
New basis
|+ =
1
√
2
(|0 + |1 ) |− =
1
√
2
(|0 − |1 )
easy calculation
1
√
2
(|0 A|0 B + |1 A|1 B) =
1
√
2
(|+ A|+ B + |− A|− B)
Alice and Bob measure in basis {|0 , |1 } ⇒ same result
Alice and Bob measure in basis {|+ , |− } ⇒ same result
Converse is true, too:
same measurement result ⇒ they have state
1√
2
(|0 A|0 B + |1 A|1 B)
Alice and Bob can test whether or not they have the
state 1√
2
(|0 A|0 B + |1 A|1 B)!

Assume that Alice and Bob have the state
|φ AB = 1√
2
(|0 A|0 B + |1 A|1 B)
and measure in the same basis.
Can someone else guess the result?
No! The measurement result is secure!
Total state of Alice, Bob and Eve
|ψ ABE = |φ AB ⊗ |φ E ,
because Alice and Bob have a pure state
Eve is not at all correlated with Alice and Bob!
Monogamy of entanglement

The Quantum Key Distribution Protocol
Distribution
Alice
Eve
Bob
glass fibre
1
1 1

Distribution
Alice
Eve
Bob
glass fibre
1 2
1 2 1 2

Distribution
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n

Distribution
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n
Measurement with {|0 , |1 } or {|+ , |− }
0 1 1 0 0 1
FE

FE

FE

FE

FE

FE


Distribution
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n
0 1 1 0 0 1
FE

FE

FE

FE

FE

FE

Error-free? |φ AB
?
= 1√
2
(|0 A|0 B + |1 A|1 B )
phone

Distribution
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n
0 1 1 0 0 1
FE

FE

FE

FE

FE

FE

Error-free? |φ AB
?
= 1√
2
(|0 A|0 B + |1 A|1 B )
phone
If YES: key. If NO: no key

Proof works as long as |Ψ n
ABC = |ψ ⊗n
ABE .
Alice Bob
But why should Eve prepare such a state?
Why not the following?
Alice Bob
We can assume: π|Ψ n
ABC = |Ψ n
ABC for all π ∈ Sn.
Goal: two methods that reduce second to ﬁrst case!

De Finetti’s Theorem (Diaconis and Freedman, 1980)
Drawing balls from an urn with or without replacement results
in almost the same probability distribution.
If k are drawn out of n, then
||Pk
−
i
pi Q×k
i ||1 ≤ const
k
n
.

Quantum generalisations by Størmer, Hudson & Moody, and
Werner et al.. (n = ∞)
Quantum De Finetti Theorem
Christandl, K¨onig, Mitchison, Renner, Comm. Math. Phys. 273, 473498 (2007)
Let |Ψ n
be a permutation-invariant state: π|Ψ n
= |Ψ n
for
all π ∈ Sn, then
||ρk
−
i
pi |ψ ψ|⊗k
i ||1 ≤ const
k
n

Alice and Bob select a random sample of pairs
(after pairs have been distributed!)
Alice BobAlice Bob
Quantum de Finetti
⇒ can use proof from before (tensor product)
⇒ proof of the security of Quantum Key Distribution!

Closer look: deviation from perfect key (due to quantum
de Finetti theorem)
≈ k/n
n: number of pairs that Eve distributed
k: number of bits of key
key rate r ≈ k/n ≈ ≈ 0 ⇒ not good enough
need replacement for de Finetti theorem
Renner’s exp. de Finetti theorem, involved, non-optimal

Statistical process Λ
Input: n-bit string
Output: success or failure
Lemma
If for any i.i.d. distribution
Prob[failure] ≤ ,
then
Prob[failure] ≤ (n + 1)
for any permutation-invariant distribution.
Typically, ≈ 2−αn
, in information-theoretic tasks.

Proof: P permutation-invariant distribution on n bits:
P =
n
k=0
pkQk.
Qk equal probability for all strings with k zeros.
Take P as n-fold i.i.d distribution, where ’0’ has probability r.
r*n k
p_k
Certainly, prn ≥ 1/(n + 1) ⇒ Qrn ≤ (n + 1)Pr .

Qrn ≤ (n + 1)Pr .
Prob[failure]P =
k
pkProb[failure]Qk
≤
k
pk(n + 1)Prob[failure]Pk/n
≤ (n + 1)

Quantum operation Λ
Input: n-qubit state
Output: success or failure (classical bit)
Post-selection Technique
Christandl, K¨onig, Renner, Phys. Rev. Lett. 102, 020504 (2009)
For any input |Ψ n
= |ψ ⊗n
Prob[failure] ≤ ,
then
Prob[failure] ≤ n3
for any state permutation-invariant state |Ψ .
Typically, ≈ 2−αn
, in information-theoretic tasks.

Distribution |Ψ n
ABC = |ψ ⊗n
ABE
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n
0 1 1 0 0 1
FE

FE

FE

FE

FE

FE

Error-free? phone
By assumption: Prob[failure] ≤

Distribution |Ψ n
ABC permutation-invariant (or general)
Alice
Eve
Bob
glass fibre
1 2 n
1 2 n 1 2 n
0 1 1 0 0 1
FE

FE

FE

FE

FE

FE

Error-free? phone
Post-selection tech.: Prob[failure] ≤ poly(n) ≈ poly(n)2−δ2n

security proof against the most general attacks
optimal security parameters
relevant in current experiments (since n ≈ 105
)
Eve’s best attack |Ψn
ABE = |ψABE
⊗n
conceptual and technical simpliﬁcation of security proofs
Other applications: Quantum Reverse Shannon Theorem
Berta, Christandl and Renner, arXiv:0912.3805

Summary
Alice BobAlice Bob
Quantum de Finetti
optimal security parameters
applications outside quantum cryptography:
quantum Shannon theory and quantum tomography

Qkd and de finetti theorem

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