Download Towards a Tight Finite Key Analysis for BB84

Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Towards a Tight Finite Key Analysis for BB84
The Uncertainty Relation for Smooth Entropies
Marco Tomamichel
joint work with Charles Ci Wen Lim, Nicolas Gisin and Renato Renner
Institute for Theoretical Physics, ETH Zurich
Group of Applied Physics, University of Geneva
[arXiv: 1103.4130, 2011]
Vienna, July 2011
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
1
Entropic Uncertainty Relations
Heisenberg’s Uncertainty Principle
Variance vs. Shannon Entropy
Entropic Uncertainty Relation
Quantum Memory
2
Uncertainty Relation for Smooth Entropies
The Uncertainty Relation for Smooth Entropies
Guessing Probability
Smooth Min-entropy
Smooth Max-entropy
3
Application to Quantum Key Distribution
Protocol and Security
Numerical Results
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Heisenberg’s Uncertainty Principle
Fresh from Wikipedia:
In quantum mechanics, the Heisenberg uncertainty
principle states by precise inequalities that certain
pairs of physical properties, such as position and
momentum, cannot be simultaneously known to
arbitrarily high precision. That is, the more precisely
one property is measured, the less precisely the other
can be measured.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Heisenberg’s Uncertainty Principle
Fresh from Wikipedia:
In quantum mechanics, the Heisenberg uncertainty
principle states by precise inequalities that certain
pairs of physical properties, such as position and
momentum, cannot be simultaneously known to
arbitrarily high precision. That is, the more precisely
one property is measured, the less precisely the other
can be measured.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Heisenberg’s Uncertainty Principle
Fresh from Wikipedia:
In quantum mechanics, the Heisenberg uncertainty
principle states by precise inequalities that certain
pairs of physical properties, such as position and
momentum, cannot be simultaneously known to
arbitrarily high precision. That is, the more precisely
one property can be measured, the less precisely the
other can be measured.
Think of it as a gedankenexperiment. No quantum states will
be harmed (i.e. measured, forced to collapse) during this talk!
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Robertson’s Uncertainty Relation
A common formalization of the uncertainty principle is due to
Robertson:
1
σX σZ ≥ hψ|[X̂ , Ẑ ]|ψi,
2
where
X̂ and Ẑ are two observables,
[X̂ , Ẑ ] = X̂ Ẑ − Ẑ X̂ is their commutator,
|ψi is the state of the system before measurement,
and σX and σZ are the standard deviations of the two
potential measurement outcomes.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Inadequacies of Robertson’s Relation
1
σX σZ ≥ hψ|[X̂ , Ẑ ]|ψi
2
The lower bound on the uncertainty in general depends on the
state |ψi, which might be unknown.
The standard deviation is not always a good measure of the
uncertainty about the measurement outcome.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Uncertainty as Shannon Entropy
The Shannon entropy of a random variable X , H(X ), is a
functional of the probability distribution over outcomes,
Pr[X = x], and not the outcomes themselves.
H(X ) :=
X
Pr[X = x] log2
x
Marco Tomamichel
1
.
Pr[X = x]
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Uncertainty as Shannon Entropy
The Shannon entropy of a random variable X , H(X ), is a
functional of the probability distribution over outcomes,
Pr[X = x], and not the outcomes themselves.
H(X ) :=
X
Pr[X = x] log2
x
1
.
Pr[X = x]
The entropies of the distributions on the previous slide are
H
=1
Marco Tomamichel
and H
≈ 3.
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Entropic Uncertainty Relation
The entropic uncertainty relation gives a lower bound on the
sum of the entropies of the two possible measurements in
terms of the overlap of the measurements, c.
Deutsch, Maassen/Uffink 1988
H(X ) + H(Z ) ≥ log2
1
c
2
with c := max hx|zi ,
x,z
where |xi and |zi are the eigenvectors of the observables X̂
and Ẑ .
For general positive operator valued measurements (POVMs)
with elements {Mx } for X and {Nz } for Z , the overlap is
p p 2
c := max Mx Nz .
x,z
Marco Tomamichel
∞
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Quantum Memory
What happens when we allow quantum memory?
A
|ψi ∼ |01i−|10i
Marco Tomamichel
B
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Quantum Memory
What happens when we allow quantum memory?
|ψi ∼ |01i−|10i
A
σX
X

B
σZ
Z
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Quantum Memory
What happens when we allow quantum memory?
|ψi ∼ |01i−|10i
A
σX
X

B
−σX
σZ
Z
Marco Tomamichel
X

−σZ
Z
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Quantum Memory
What happens when we allow quantum memory?
|ψi ∼ |01i−|10i
A
σX
X

B
−σX
σZ
Z
X
−σZ

Z
For this example H(X |B) = H(Z |B) = 0 while c = 1/2.
Hence, the following does not hold in general:
H(X |B) + H(Z |B) log2
Marco Tomamichel
1
.
c
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Uncertainty Relation for Quantum Memory
An uncertainty relation is
possible if we introduce
an additional quantum
memory, E .
The monogamy of
entanglement helps.
X o
A
/Z
ρABE
B
E
Berta et al. 2010, Coles et al. 2010
H(X |E ) + H(Z |B) ≥ log2
Marco Tomamichel
1
.
c
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Main Tool
The uncertainty relation for smooth entropies:
Tomamichel/Renner 2011
For any state ρABE , ε ≥ 0 and POVMs {Mx } and {Nz } on A:
ε
ε
Hmin
(X |E ) + Hmax
(Z |B) ≥ log2
1
,
c
p p 2
c = max Mx Nz ∞ .
x,z
This generalizes previous results for the Shannon/von
Neumann entropy.
It has direct applications in quantum cryptography.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Guessing Probability
Let X be a random variable correlated to a memory E. We denote
by pguess (X |E ) the probability that X is guessed correctly using the
optimal strategy with access to E.
E is empty: We pick the most probable event and
pguess (X ) = max Pr[X = x] .
x
E is classical: We pick the most probable event given the
state of our memory and
X
pguess (X |E ) =
Pr[E = e] max Pr[X = x|E = e] .
e
Marco Tomamichel
x
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Guessing Probability
Let X be a random variable correlated to a memory E. We denote
by pguess (X |E ) the probability that X is guessed correctly using the
optimal strategy with access to E.
E is quantum: The state of the joint system is of the form
X
ρXE =
Pr[X = x] |xihx| ⊗ ρxE ,
x
where ρxE is the state of the memory when x is measured. The
guessing probability is
X
pguess (X |E ) = sup
Pr[X = x] tr Fx ρxE ,
{Fx } x
where the optimization is over all POVM’s {Fx } on the
quantum memory.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Smooth Min-Entropy
The min-entropy is defined as
Renner 2005, König/Renner/Schaffner 2009
Hmin (X |E ) := − log pguess (X |E ) .
ε (X |E ), results from a
The smooth min-entropy, Hmin
maximization of the min-entropy over an ε-neighborhood of
the density operator of the state.
It quantifies how many random bits that are independent of
the memory E can be extracted from X .
Renner/König 2005
ε
`secr ≈ Hmin
(X |E ) .
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Smooth Max-Entropy
ε (Z |B), quantifies how many
The smooth max-entropy, Hmax
bits of additional information about Z are needed to
reconstruct it from B.
Renes/Renner 2010
ε
`enc ≈ Hmax
(Z |B) .
If Z = Z1 . . .Zn is a bit string and B = Z10 . . .Zn0 is classical,
ε (Z . . .Z |Z 0 . . .Z 0 ) ≈ nh(δ), where δ is chosen such
then Hmax
1
n 1
n
that the fraction of errors that Z 0 has on Z is smaller than δ
with high probability.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
BB84 Type Protocol
Alice encodes a random bit into a qubit in one of two bases,
either X or Z, chosen at random. The X bits will be used to
extract a key, while the Z are used to check security.
She sends the qubit over a public channel to Bob, while the
eavesdropper, Eve, may interfere as she wishes.
Bob measures the system randomly either in the X or Z basis.
Alice and Bob sift the strings containing their binary
measurement outcomes so that they contain n bits where
both used X, denoted X1 . . . Xn , and k bits where they both
used Z, denoted Z1 . . . Zk .
If the security criterion is satisfied, they extract ` bits of
shared secret key, using classical post-processing (data
reconciliation and privacy amplification).
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
B1
B2
B3
B4
..
.
AN−1
BN−1
AN
BN
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
X1 o
Z1 o
X2 o
X3 o
Xn o
Zk o
B1
B2
B3
B4
..
.
/ X0
AN−1
BN−1
AN
BN
/ X0
n
/ Z0
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
Marco Tomamichel
1
/ Z0
1
/ X0
2
/ X0
3
k
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
X1 o
Z1 o
X2 o
X3 o
Xn o
Zk o
B1
B2
B3
B4
..
.
/ X0
AN−1
BN−1
AN
BN
/ X0
n
/ Z0
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
1
/ Z0
1
/ X0
2
/ X0
3
k
ε
`secr ≈ Hmin
(X1 . . .Xn |E )
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
X1 o
Z1 o
X2 o
X3 o
Xn o
Zk o
B1
B2
B3
B4
..
.
/ X0
AN−1
BN−1
AN
BN
/ X0
n
/ Z0
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
1
/ Z0
1
/ X0
2
/ X0
3
k
ε
ε
`secr ≈ Hmin
(X1 . . .Xn |E ) ≥ n − Hmax
(Ẑ1 . . .Ẑn |Ẑ10 . . .Ẑn0 )
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
X1 o
Z1 o
X2 o
X3 o
Xn o
Zk o
B1
B2
B3
B4
..
.
/ X0
AN−1
BN−1
AN
BN
/ X0
n
/ Z0
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
1
/ Z0
1
/ X0
2
/ X0
3
k
ε
ε
`secr ≈ Hmin
(X1 . . .Xn |E ) ≥ n − Hmax
(Ẑ1 . . .Ẑn |Ẑ10 . . .Ẑn0 )
≈ n 1 − h(δ)
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Proof Sketch
E
X1 o
Z1 o
X2 o
X3 o
Xn o
Zk o
B1
B2
B3
B4
..
.
/ X0
AN−1
BN−1
AN
BN
/ X0
n
/ Z0
A1
A2
A3
A4
..
.
ρA1 ...AN B1 ...BN E
1
/ Z0
1
/ X0
2
/ X0
3
k
ε
ε
`secr ≈ Hmin
(X1 . . .Xn |E ) ≥ n − Hmax
(Ẑ1 . . .Ẑn |Ẑ10 . . .Ẑn0 )
X
k
1
0
≈ n 1 − h(δ) ≈ n 1 − h
Zi ⊕ Zi
k
i=1
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Secure Key Rate
The extractable -secure key per block of size N = n + k is
` ≤ n 1 − h(Qtol + µ) − 3 log(3/) − leakEC
p
µ ≈ 1/k · ln(1/) is the statistical deviation from the
tolerated channel noise, Qtol .
k is the number of test bits used for statistics.
leakEC ≈ nh(Qtol ) is the information about the key leaked
during error correction.
The achievable key rate, `/N, deviates from its optimal
asymptotic value, 1 − 2h(Qtol ), only by (probably
unavoidable) terms due to finite statistics.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Numerical Results
10
10
10
10
0
-1
-2
-3
10
3
10
4
10
5
10
6
10
7
Plot of the expected key rate as function of the block size n for
channel bit error rates Q ∈ {1%, 2.5%, 5%} (from left to right).
The security rate is fixed to /` = 10−14 .
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Comparison to Scarani/Renner 2008
0
10
Asymptotic limit, Q=1.0%
Asymptotic limit, Q=2.5%
Asymptotic limit, Q=5.0%
-1
10
-2
10
3
10
4
10
5
10
6
10
7
10
The plots show the rate `/N as a function of the sifted key size
N = n + k and a security bound of = 10−10 .
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Conclusion and Outlook
The improved finite key bounds are due to the simplicity of
the proof via the uncertainty relation.
No tomography of single quantum systems is necessary.
Instead, the min-entropy of the whole string can be bounded
directly.
Security against general attacks comes for free — no De Finetti
or Post-Selection necessary.
This proof technique can (hopefully) be applied to other
problems in quantum cryptography.
As pointed out by Hayashi/Tsurumaru (arXiv:1107.0589,
yesterday), the key rates can be improved when we allow a
dynamic protocol that chooses a different ` in each run.
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84
Entropic Uncertainty Relations
Uncertainty Relation for Smooth Entropies
Application to Quantum Key Distribution
Thank you for your attention.
Any questions?
Marco Tomamichel
Towards a Tight Finite Key Analysis for BB84