Concentration of Measure Effects in Quantum Information. Patrick Hayden (McGill University)

Transcription

1 Concentration of Measure Effects in Quantum Information Patrick Hayden (McGill University)

2 Overview Superdense coding Random states and random subspaces Superdense coding of quantum states Quantum mechanical encryption

3 Information theory A practical question: How to best make use of a given communications resource? A mathematico-epistemological question: How to quantify uncertainty and information? Shannon: Solved the first by considering the second. A mathematical theory of communication [1948] The

4 A challenge to the physicists John Pierce [1973]: I think that I have never met a physicist who understood information theory. I wish that physicists would stop talking about reformulating information theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases.

5 Superdense coding To send i 2 {0,1,2,3} ¾ i Time Ψ 0 i 1 ebit + 1 qubit 2 cbits Ψ i i

6 More generally... Superdense coding: To send i 2 {1,,d 2 } Ψ (d) i U i Time log d ebits + log d qubits 2 log d cbits i

7 Quantifying uncertainty Entropy: H(X) = - x p(x) log 2 p(x) Proportional to entropy of statistical physics Term suggested by von Neumann (more on him later) Can arrive at definition axiomatically: H(X,Y) = H(X) + H(Y) for independent X, Y, etc. Operational point of view

8 Mixing quantum states: The density operator Draw φ x i with probability p(x) Perform a measurement { 0i, 1i}: Probability of outcome j: q j = x p(x) hj φ x i 2 = x p(x) tr[ jih j φ x ihφ x ] Á 1 i Á 2 i Á3 i Á 4 i = tr[ jih j ρ ], where ½ = x p(x) Á x ihá x Outcome probability is linear in ρ

9 Properties of the density operator ρ is Hermitian: ρ = [ x p(x) φ x ihφ x ] = x p(x) [ φ x ihφ x ] =ρ ρ is positive semidefinite: hω ρ ωi = x p(x) hω φ x ihφ x ωi 0 tr[ρ] = 1: tr[ρ] = x p(x) tr[ φ x ihφ x ] = x p(x) = 1 Ensemble ambiguity: I/2 = ½[ 0ih0 + 1ih1 ] = ½[ +ih+ + -ih- ]

10 The density operator: examples Which of the following are density operators?

11 Quantifying uncertainty Let ρ = x p(x) φ x ihφ x be a density operator von Neumann entropy: H(ρ) = - tr [ρ log ρ] Equal to Shannon entropy of ρ eigenvalues Analog of a joint random variable: ρ AB describes a composite system A B H(A) ρ = H(ρ A ) = H( tr B ρ AB )

12 Quantifying uncertainty: Examples H( φihφ ) = 0 H(I/2) = 1 H(ρ σ) = H(ρ) + H(σ) H(I/2 n ) = n H(pρ (1-p)σ) = H(p,1-p) + ph(ρ) + (1-p)H(σ)

13 Surprises in high dimension Choose a random pure quantum state: φ 2 R C d A C d B What can we expect of φ? (d A d B ) On average, states are highly entangled Lubkin, Lloyd, Page, Foong & Kanno, Sanchez-Ruiz, Sen

14 Concentration of measure A n θ A n ~ exp[-n f(θ)] S n LEVY: Given an η-lipschitz function f : S n! R with median M, the probability that a random x 2 R S n is further than ε from M is bounded above by exp (-nε 2 C/η 2 ) from some C > 0.

15 Application to entropy Choose a random pure quantum state: φ 2 R C d A C d B (d A d B ) P H(φ A )

16 Random subspaces S ½ C d A C d B 1) Choose a fine net N of states on S. 2) P( Not all states in N highly entangled ) N P( One state isn t ) 3) True for sufficiently fine N implies true for all of S. THEOREM: There exist subspaces of dimension Cd A d B α 3 /(log d A ) 3, all of whose states have entanglement at least log d A - α - 2β. The probability that a random subspace does goes to 1 with d A d B.

17 In qubit language In a bipartite system of n by n+o(n) qubits, there exists a subspace of 2n o(n) qubits in which all states have at least n o(1) ebits of entanglement. The subspace of nearly maximally entangled states is almost as big as the whole system!

18 Compare to pairs of qubits The subspace spanned by two or more Bell pairs always contains some product states. (No subspaces of entangled states, let alone maximally entangled states.) C 2 C 2

19 What s this good for? Superdense coding: To send i 2 {1,,d 2 } U i Time Ψ d i i

20 What s this good for? Superdense coding: To send maximally entangled φi U φ Time Ψ d i φi Asymptotically, an arbitrary 2 qubit maximally entangled quantum state can be communicated using 1 qubit and 1 ebit.

21 What s this good for? Superdense coding: To send arbitrary Ái 2 C k Ψ d i U E(Á) Time E( φi ) ~E(φ) There exists a subspace S of near-full size containing only nearly maximally entangled states. Asymptotically, an arbitrary 2 qubit quantum state can be communicated using 1 qubit and 1 ebit.

22 What s this good for? Superdense coding: To send arbitrary Ái 2 C k Ψ d i U E(Á) Time 1 ebit + 1 qubit 2 cbits E -1 ~ φi There exists a subspace S of near-full size containing only nearly maximally entangled states. Asymptotically, an arbitrary 2 qubit quantum state can be communicated using 1 qubit and 1 ebit.

23 Recurse: 1 ebit + 1 qubit 2 qubits 2 ebits + (1 ebit + 1 qubit) 4 qubits 4 ebits + (3 ebits + 1 qubit) 8 qubits 2 r -1 ebits + 1 qubit 2 r qubit Send an unbounded amount of quantum information with a single qubit? NO!

24 Knowledge is power??? Oblivious Alice: 1 ebit + 1 qubit 1 qubit All you need in this life is ignorance and confidence, and then success is sure. -Mark Twain! Non-Oblivious Alice: 1 ebit + 1 qubit 2 qubits

25 Credit where credit is due Accidental quantum information theorists? Milman and Schechtman. Asymptotic theory of finite dimensional normed spaces. Springer-Verlag, Others: Gowers, Gromov, Ledoux, Szarek, Talagrand

26 Another application: One-time pad Message " Shared key ! bit of key per bit of message necessary and sufficient [Shannon49]

27 Private quantum channels " # d!c! Sk k T! k {, } k! 1 K,n Eavesdropper learns nothing: n 1! ( ) n S k # = " 0 k = 1 Lower bound on key length: n! d 2 ( log n! 2log d) [BR,AMTW 2000]

28 Relax security criterion R! A physical operation is ε-randomizing if for all states, R (" ) #. I $! d % d CONSEQUENCE: Given ε>0, there exists a choice of unitaries {U k }, k=1,,n such that the map n 1 ( )! " R # = U k # U k n k = 1 is ε-randomizing, with Cd log d n =. 2! HLSW03

29 Approximate PQC Can encrypt a quantum state using 1 secret random bit per encrypted qubit asymptotically.! Sk k T! k E Security: { p," } Eve( )#! i i $ I ln 2 = E! = 1 poly( l) 1 bit of key/qubit # 1 2 = "!l (1+2α) bits of key/qubit

30 Conclusions General rule: Random states and subspaces exhibit extremal behaviour Entanglement Communication Error-correction Many effects do not have good low-dimensional analogues Tip of the iceberg: encryption, state identification, data hiding, secret sharing. Direction application of the concentration of measure phenomenon to problems in communication