Quantum Communication Harry Buhrman CWI & University of Amsterdam
Physics and Computing Computing is physical Miniaturization quantum effects Quantum Computers ) Enables continuing miniaturization ) Fundamentally faster algorithms
Overview Introduction Quantum Mechanics & Notation. Deutsch-Jozsa Algorithm Distributed Deutsch-Jozsa: Communication Complexity Physics: Non-locality & Detection Loophole Quantum Fingerprinting
Quantum Mechanics
Quantum Mechanics Most complete description of Nature to date Superposition principle: particle can be at two positions at the same time Interference: particle in superposition can interfere with itself
Superposition Classical Bit: 0 or Quantum Bit: Superposition of 0 and Bit 0
Classical Bit: 0 or Superposition Quantum Bit: Superposition of 0 and α 0 +β Bit 0 0 Qubit
Qubit α 0 + β α 0 + β amplitudes Rule: α + β =, α, β are complex numbers.
Measurement α 0 +β Projection on the 0 axis or axis. 0 Rule: observe 0 with probability α observe with probability β after measurement qubit is 0 or
Example ψ = 0 +
Example ψ = 0 + Measuring ψ: Prob [] = / Prob [0] = /
Example ψ = 0 + Measuring ψ: Prob [] = / Prob [0] = / After measurement: with prob / ψ = 0 with prob / ψ =
Quantis QUANTUM RANDOM NUMBER GENERATOR Although random numbers are required in many applications, their generation is often overlooked. Being deterministic, computers are not capable of producing random numbers. A physical source of randomness is necessary. Quantum physics being intrinsically random, it is natural to exploit a quantum process for such a source. Quantum random number generators have the advantage over conventional randomness sources of being invulnerable to environmental perturbations and of allowing live status verification. Quantis is a physical random number generator exploiting an elementary quantum optics process. Photons - light particles - are sent one by one onto a semi-transparent mirror and detected. The exclusive events (reflection - transmission) are associated to "0" - "" bit values.
Two Qubits α 00 + α 0 + α 0 + α 4 3 α + α + α 3 + α 4 = Prob[00] = α, Prob[0] = α, Prob[0] = α 3, Prob[] = α 4,
n Qubits Prob[observing j] = α j
Dirac Notation norm vector complex conjucate transpose
inner product inner product between ai and bi
inner product()
Evolution
Evolution. Postulate: the evolution is a linear operation. quantum states maped to quantum states & implies that operation is Unitary length preserving rotations. U U * = I. (U * : complex conjugate, transpose)
Hadamard Transform = H = * H I H H = = 0 0 *
Hadamard on 0 = = = + 0 H 0 + = ψ 0 interference H
Hadamard on 0 = = = + 0 H 0 + = ψ 0 interference H
Hadamard on = = = + 0 H H 0 = ψ
Hadamard on = = = + 0 H H 0 = ψ
Hadamard on n qubits inner product modulo
Quantum Algorithms
Black Box Model Input N (= n ) bits (variables): X = X X X 3 X N Compute some function F of X: F(X) {0,} Only Count how many bits/variables of X are read.
Deutsch-Jozsa Problem Promise on X: () For all i: X i = (0) or (constant) () {i X i = } = {j X j = 0} (balanced) Goal: determine case () or () Classical: N/ + probes. Quantum: probe.
Quantum query Querying X i General query:
Deutsch-Jozsa Algorithm () () (3)
Deutsch-Jozsa cont. measure state Constant: see with prob. Balanced: see with prob. 0
Quantum Algorithms Deutsch-Jozsa Simon s algorithm Shor s factoring algorithm Grover s search algorithm Impossibility results
Alice and Bob
Communication? qubits
Communication? qubits Theorem [Holevo 73] Can not compress k classical bits into k- qubits
Communication Complexity Classical bits X = x x x N Y = y y y N Goal: Compute some function F(X,Y) {0,} minimizing communication bits.
Communication Complexity F(X,Y) Classical bits X = x x x N Y = y y y N Goal: Compute some function F(X,Y) {0,} minimizing communication bits.
Equality Classical bits X = x x x N F(X,Y) = iff X=Y Y = y y y N
Equality Classical bits X = x x x N Y = y y y N F(X,Y) = iff X=Y N bits necessary and sufficient: C(EQ) = N
Quantum Communication Complexity F(X,Y) {0,} qubits X = x x x N Y = y y y N F(X,Y) = iff X=Y Question: Can qubits reduce communication for certain F s?
Qubits Can Reduce Cost Theorem [B-Cleve-Wigderson 98] EQ (X,Y) = iff X=Y Promise (X,Y) = N/ or 0 Need Ω(N) classical bits. Hamming Distance Can be done with O(log(N)) qubits.
Reduction to D-J X X.. X N Y Y.. Y N Z Z.. Z N (X,Y) = N/ Z is balanced (X,Y) = 0 Z is constant
The quantum protocol Finishes Deutsch-Josza Algorithm
Cost Alice sends n= log(n) qubits to Bob Bob sends n= log(n) qubits to Alice Total cost is *log(n)
Classical Lower Bound
Lower Bound * Theorem [Frankl-Rödl 87] S,T families of N/ size sets {,,N} for all s,t in S,T : s t N/4 then: S * T 4 0.96N *$50 problem of Erdös
Lower Bound Theorem [Frankl-Rödl 87] S,T families of N/ size sets {,,N} for all s,t in S,T : s t N/4 then: S * T 4 0.96N protocol solving EQ in N/00 bits induces S and T satisfying: S * T 4 0.99N
Sets and Strings interpret X X N as set x: { i X i = } only use X with exactly N/ ones N N log(n) of such strings N/ N (X,Y) N/ x y N/4
Rectangle Claim if - (X,Y) induce conversation M and -(X,Y ) induce conversation M then -(X,Y ) and (X,Y) also induce conversation M
N/00 Bit Protocol (X,X ) (X,X ).. (X N, X N) M M. M N/00
N/00 Bit Protocol (X,X ) (X,X ).. (X N, X N) M M. M N/00 M k induced by N / N/00 = 0.99N inputs: (X,X ) (X 0.99N, X 0.99N )
N/00 Bit Protocol (X,X ) (X,X ).. (X N, X N) M M. M N/00 M k induced by N / N/00 = 0.99N inputs: (X,X ) (X 0.99N, X 0.99N ) all (X i,x j ) induce M k : (X i, X j ) N/ x i x j N/4
N/00 Bit Protocol (X,X ) (X,X ).. (X N, X N) M M. M N/00 M k induced by N / N/00 = 0.99N inputs: (X,X ) (X 0.99N, X 0.99N ) all (X i,x j ) induce M k : (X i, X j ) N/ x i x j N/4 S = T = {X, X,,X 0.99N } S * T 4 0.99N
other quantum algorithms
Quantum Algorithm 0 0 0 0 0 O U 0 U... O T U T k i = α i i T Black Box queries Prob [output = ] = α i all i that end in > / 3 { < /3 Prob [output = 0] = - Prob [output = ]
Generalization 0 0 0 0 0 O U 0 U... O T U T
Grover s Algorithm Find i such that X i = OR(X,,X N ) Classical Probabilistic: N/ queries Quantum: O( N) queries No promise!
Non-Disjointness Goal: exists i such that X i = and Y i =? Grover X X.. X N Y Y.. Y N Z Z.. Z N
Disjointness Bounded Error probabilistic Ω(N) bits [Kalyanasundaram-Schnitger 87] Grover s algorithm + reduction O(log(N)* N) qubits [BCW 98] O( N) qubits [AA 04] Ω ( N) lower bound [Razborov 03]
Apointment Scheduling qubits Quantum: n qubits communication Classical: n bits communication
Other Functions Exponential gap [Raz 99] O(log(N)) with qubits, Ω (N /4 ) bits classically. partial Domain, bounded error Computing certificates for total AND-OR relation [Buhrman-Cleve-deWolf- Zalka 99/Klauck 000] O(N /3 ) qubits, Ω(N) bits randomized total Domain, zero error
back to physics
Einstein Podolsky Rosen paradox
EPR Pair Bob Alice if Alice measures: 0/ with prob. ½ if Bob measures: 0/ with prob. ½
EPR Pair Bob Alice Alice measures: 0 state will collapse!
EPR Pair Bob Alice Alice measures: 0 state will collapse! Bob s state has changed! he will also measure 0
EPR Pair Bob Alice Alice measures: state will collapse!
EPR Pair Bob Alice Alice measures: state will collapse! Bob s state has changed! he will also measure
Einstein: nothing, including information, can go faster than the speed of light, hence quantum mechanics is incomplete
Communication? bits 0 with EPR pairs 0 + Can not compress k bits into k- bits
Teleportation
Teleportation α 0 + β 0 0 +
Teleportation Φ Φ Classical bits: b b
Teleportation b b Φ Φ Classical bits: b b
Teleportation Φ Φ Classical bits: b b b b
Teleportation Φ U b b Φ Classical bits: b b b b
Teleportation Φ α 0 + β Classical bits: b b b b
Alice s protocol H
Alice s protocol H
Alice s protocol H
Alice s protocol H
Alice s protocol H
Bob s protocol Alice with prob. ¼ in one of: Bob do nothing bit flip phase flip bit flip and phase flip
Quantum Communication Complexity F(X,Y) F(X,Y) {0,} Classical bits X = x x x n 0 With EPR pairs Y = y y y n 0 + Question: Can EPR pairs reduce communication for certain F s?
Teleportation Qubit Model can be simulated by EPR model. Teleport qubit at cost of classical bits and EPR pair. EPR-pairs can reduce communication cost: use qubit protocol + teleportation
Apoinment Scheduling EPR-pairs bits Quantum: n bits communicatie Classical: n bits communicatie
EPR en information EPR-pairs Alice can not send information to Bob, but she can save information for certain communication problems
application to experimental physics
Bell inequalities and nonlocality
Non locality k (>) parties each party i has part of an entangled state φ> receives input x i performs measurent M x i outputs measurement value o i Induces correlations: P Q (o o k x x k ) no communication!
0 Quantum Setup EPR-pair 0 + x x M x M x o induces correlations: P Q (o o x x ) o
Non locality Question: Can these correlations be reproduced classically?
Local hidden var. model Classical setup Each party has: copy of random bits (shared randomness) input x i Performs computation (protocol) Oututs o i Induces correlations: P C (o o k x x k )
Classical Setup r r..r k shared randomness r r..r k x computation x computation o induces correlations: P C (o o x x ) o
Non locality If for every protocol: P C (o o k x x k ) P Q (o o k x x k ) Non locality Requires State + measurements to obtain P Q Prove that for every classical lhv protocol: P C (o o k x x k ) P Q (o o k x x k )
Examples of states parties: EPR pair: Bell inequalities 3 parties GHZ state: Mermin state: n parties
non locality experiments
0 Quantum Setup EPR-pair 0 + x M x o induces correlations: P Q (o o x x ) x M x o
Quantum Setup 0 EPR-pair 0 + x detector source detector x M x M x o induces correlations: P Q (o o x x ) o
Quantum Setup 0 EPR-pair 0 + x detector source detector x M x M x o induces correlations: P Q (o o x x ) o
prob. -η detection loophole sometimes detector(s) don t click Alice and/or Bob don t have an output can only test correlations when both Alice and Bob have an output Classical non clicking: classical lhv protocol sometimes no output only check whenever there is an output η = detector efficiency = prob. of clicking small η allows for lhv protocols prob. η
example r r k r k+ r k x=x x k shared randomness correlation P(o xy) η = -k r r k r k+ r k y=y y k if x x k r r k No Click if x x k = r r k assume y = r k+ r k output P(o xy) if y yk rk+ rk No Click if y y k = r k+ r k assume x = r r k output P(o xy)
detection loophole All experiments that show non locality have η such that a lhv model exist! Solution: Design tests that allow small η test also useful to test devices that claim to behave non local (eg quantum crypto) No good tests known
η * definition η * is the maximum detector efficiency for which a lhv model exists. Goal: design correlation problem/test prove upper bounds on η *
from quantum communication complexity back to non locality
Monochromatic rectangles X,X set of inputs for Alice and Bob Rectangle R = A B, A X & B X Ris a-monochromatic if for all(x,x ) R D: F(x,x ) = a D = set of promise inputs R a = max {R D R is a-monochr.} R a yields lower bound on C(F)
Monochromatic rectangles X,X set of inputs for Alice and Bob Rectangle R = A B, A X & B X Ris a-monochromatic if for all(x,x ) R D: F(x,x ) = a D = set of promise inputs R a = max {R D R is a-monochr.} R a yields lower bound on C(F)
set of inputs that have a as output C ( F ) log D R a a
EPR-pairs Can Reduce Cost exponential gap [BCW 98] EQ (x,x ) = iff x =x Promise (x,x ) = n/ or 0 need Ω(n) classical bits. can be done with log(n)) bits +EPR-pairs. Protocol: distributed Deutsch-Jozsa Hamming Distance
EQ set of inputs that have as output D C ( F ) log R =.04n R 0.96n D = n hard comb. theorem due to Frankl & Rödl
non-locality test Promise (x,x ) = n/ or 0 Alice outputs log(n) bits o Bob outputs log(n) bits o correlation: x =x o = o D-J algorithm on EPR-pairs [BCT 99] [Massar 0]
0 DJ-test log(n) EPR-pairs 0 + promise (x,x ) = n/ or 0 x H+ph-flip o x =x o =o x H+ph-flip o log(n) bits
Monochromatic rectangles X,X set of inputs for Alice and Bob Rectangle R = A B, A X & B X Ris a-monochromatic if for all(x,x ) R D: D = set of promise inputs P(a x x )>0 R A = max {R D R is a-mon. a A} R A yields upper bound on η *
Bound on η * number of possible outputs = A * R A η d D a A is set of other outputs {b x P(a x)>0 & P(b x)>0} set of inputs x s.t. P(a x)>0
proof * R A η d D a lhv protocol is distribution of deterministic prot. Q i : for all x Alice & Bob yield admissible outcome, or at least one doesn t click [prob. η] exist Q j Alice & Bob yield admissible outcome on η fraction of a-inputs det. protocol Alice & Bob yield outcome on at most dr A of the inputs dr A / D a η
Application of bound
0 DJ-test log(n) EPR-pairs 0 + promise (x,x ) = n/ or 0 x H+ph-flip o x =x o =o x H+ph-flip o log(n) bits
Bound on η * for DJ-test number of possible outputs * R A η d D aa A= {a i a i } d = n R A 0.96n D aa = n set of inputs x s.t. P(aa x)>0
Bound on η * for DJ-test number of possible outputs η * R A n d = 0. 0 D aa A= {a i a i } n d = n R A 0.96n D aa = n set of inputs x s.t. P(aa x)>0
n parties
n party test [BvDHT 99] input party i: promise: output a i : detector:
n-party bound largest mon. rectangle R * d D η n inputs d = n R (n-/n) n D = nlog(n) number of possible outputs
n-party bound largest mon. rectangle n R η * d = D n inputs d = n R (n-/n) n D = nlog(n) number of possible outputs
error s DJ-test can be simulated classically with small error. n-party test is even robust against error! Can not be simulated classically with: error prob. < ½-/n and η * /n
Bounded error One way communication complexity Two Parties exponential separation between randomized classical en quantum communication complexity [B-YJK 04] party nonlocality test: error prob< ½-/n c and η * /n c
open problems construct party test: η * / n and prob. of error < /n quantum gives perfect correlation Other applications of non-locality tests? What is the relation between EPR assisted communication complexity and qubit communication complexity? What are the best bounds for total functions?