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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