The adventures of Alice, Bob & Eve in the Quantumland

1. The adventures of Alice, Bob & Eve in the Quantumland Stefano Mancini University of Camerino, Italy

2. Outline (part 2) • Useful tools about information and measurement theory • A funny idea: the quantum money! • Quantum Key Distribution (BB84 protocol) • Eavesdropping and security

3. What is Information? • The difference between what you know now and what you knew before • “Information in communication theory relates not to what you do say, but what you could say.” – Shannon & Weaver 1949

4. How Much Information? • Amount of information depends on initial uncertainty • Zero for certainties • Additive for two independent events • Logarithm function is only choice

5. Entropy The Entropy measures uncertainty: Logarithm to base 2 gives bits • Example: binary entropy h(p)=-p log p-(1-p) log (1-p) • Coin flip has uncertainty of 1 bit!

6. Conditional Entropy • Uncertainty in X, now that you know Y • Difference in entropies: H(X|Y) = H(X,Y) – H(Y) • Must be positive, you cannot be more uncertain about Y alone, than you are about X and Y

7. Mutual Information • The difference between what you didn’t know before and what you don’t know now • Uncertainty in X – Uncertainty in X given Y H(X:Y) = H(X)-H(X|Y) = H(X) + H(Y) – H(X,Y) Mutual Information of X and Y

8. A picture of Entropies H(X,Y) H(X) H(Y) H(X:Y) H(X|Y) H(Y|X)

9. The Qu(antum)bit • A Hilbert space isomorphic to C2 is a qubit • Let |0>, |1> two orthonormal vecs, then • States of qubit and Bloch sphere • Let X,Y,Z be the Pauli operators • Let us identify |0>, |1> with Z’s eigenstates

10. Quantum Measurements

11. Quantum Measurements • Quantum measurement is an irreversible process! • Measuring Z on states prepared on its basis {|0>,|1>} would not disturb it • Measuring X on states prepared on its basis {|+>,|->} would not disturb it • Measuring Z on states prepared on X basis {|+>,|->} would project it into {|0>,|1>} with Pr=1/2 • Measuring X on states prepared on Z basis {|0>,|1>} would project it into {|+>,|->} with Pr=1/2

12. Density operator • Generalization of “pure” quantum state • Tr()=1; 0 • Always obtainable by tracing out environment of larger quantum state • Von Neumann entropy

13. Entangled states • Superpositions of two “non-local” states, e.g. • The EPR programme…. • Bell’s inequality: | E(a,b)+E(a,b’)+E(a’,b)-E(a’,b’) |  2

14. Fundamental Theorems • NO-Cloning Theorem. An unknown quantum state cannot be copied. • Theorem. In any attempt to distinguish between two non-orthogonal quantum states, information gain is only possible at expenses of introducing some disturbance.

15. Generalized measurements

16. Trying to distinguish non-orthogonal states

17. Trying to distinguish non-orthogonal states

18. A funny idea: the quantum money (Wiesner 1969) 101101 Money 101101 V 45°135° H H Bank record For N photons in a bank note, a duplicate has only a prob (3/4)N of passing the bank’s verification

19. Quantum Key Distribution BB84 protocol • Alice uses two random bits a and a’ to prepare the state of a qubit |yaa’>: |y00 >=|0> |y10 >=|1> |y01 >=|+> |y11 >=|-> • Alice sends the qubit to Bob through a quantum channel. Since Alice hasn’t revealed a’, Eve can only guess the basis and in the wrong case she disturbes the qubit. However also Bob does not know a’.

20. Bob measures the qubit in the basis X or Z as determined by a random bit b’ which he creates on his own (0-Z, 1-X). Let Bob’s measurement result be b (0-positive eigenvalue, 1-negative eigenvalue). • Alice publicly announces a’ through a public classical channel. • The above procedure is repeated 4n times. Then Alice and Bob by a discussion over a public channel discard all bits except those for which a’=b’ (raw key, approx 2n bits). • Alice selects n bits (of her 2n) at random and publicly announces the selection. Then Alice and Bob compare the values of these check bits to establish the error rate (or Eve’s presence). • Eventually the remaining n bits are the sifted key.

21. a’ b b=b’ implies perfect correlations # usable bits _____________________________ R= (# transmitted qubits)+(# transmitted bits) R= 1/6 BB84 is probabilistic!

22. What is gained by using qubits? Example: a simple intercept-resend strategy Z (|0>, p=1/2) Z (p=1/2) Z (|0>, p=1/4) |0> X (p=1/2) Z (|1>, p=1/4)

23. What is gained by using qubits? • For a simple intercept-resend eavesdropping,the prob that Eve is present and Alice and Bob choose n uncorrupted (coincident) bits for the check is (3/4)n which goes to zero as n goes to infinity • In this simple example Eve gets 0.5 bits [H(a:e)=0.5] of info per bit in the sifted key for an induced QBER of 25% [d=0.25] • We expect H(a:e) is an increasing function of d, nevertheless, provided d0 Alice and Bob would be able to outwit Eve (ideal situation of no noise in the channel!)

24. Information Reconciliation & Privacy Amplification Quite generally it is difficult for Alice and Bob to distinguish the effect of Eve intrusion from that of the noise. Suppose at some point Alice, Bob and Eve perform measurements with outcomes , ,  with P(,,), then: Theorem (Csiszar & Korner): For a given P(,,) Alice and Bob can establish a secret key (using only EC and PA) iff H(: )>H(:) or H(:)>H(:)

25. Information Reconciliation (IR) Alice and Bob publicly compare a randomly chosen subset of the key in order to estimate the error rate. If the conditions of CK’s theorem are satisfied they use standard Error Correction procedure. Example. Alice and Bob apply an iterative process of comparing the parity of a publicly chosen random subset of their data. If disagreement is found a bisective search is applied to locate and correct the error.

26. Privacy Amplification (PA) Alice and Bob distill from the key a smaller set of bits whose correlation with Eve is below a desired threshold and constitutes the effective secret key. Example. Suppose that there are n bits in the key and Eve has at most l bits of it. A hash function h should be chosen randomly f:{0,1}n--->{0,1}n-l-s with s>0. At the end the key x will be mapped into f(x) such that Eve’s expected information is <2-s.