Quantum information & computationtph.tuwien.ac.at/~svozil/publ/2005-stpoelten-pres.pdfClassical & quantum correlations and the Boole-Bell inequalities 2 Quantum computation No-cloning

Quantum informationQuantum computation

ResourcesConclusions

Quantum information & computationhttp://tph.tuwien.ac.at/∼svozil/publ/2005-stpoelten-

pres.pdf

Karl Svozil

Institut für Theoretische Physik, University of Technology Vienna,Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria

[email protected]

Nov. 8, 2005

Karl Svozil Quantum information & computation



1 Quantum informationBasics & differences to classical informationQuantum state evolution: one-to-oneMach-Zehnder interferometer“Quantum mindbogglers”Classical & quantum correlations and the Boole-Bellinequalities

2 Quantum computationNo-cloning (no-copy) theoremVisions of parallelism & interferenceDeutsch algorithm: parity of a function of one bitQuantum cryptography & man-in-the-middle attacks

3 Resources

4 Conclusions




Basics & differences to classical informationQuantum state evolution: one-to-oneMach-Zehnder interferometer“Quantum mindbogglers”Classical & quantum correlations and the Boole-Bell inequalities

Basics & differences to classical information

I Elementary unit of classical information is the classical bit(“cbit”) which is in one of the two classical states “0” or “no” or“false” and “1” or “yes” or “true,” respectively.

I Elementary unit of quantum information is the quantum bit(“qubit”) which can be in a coherent superposition

|Ψ〉 = a0|0〉+ a1|1〉, with |a0|2 + |a1|2 = 1

of the classical states “0” and “1.”I A single qubit “embodies” two classically contradictory states at

once. This is the basis of “quantum parallelism.”I n single qubits “embody” 2n classically contradictory states at

once. A linear increase of quantum information is associatedwith an exponential increase of embodied classically states –“quantum parallelism.”





Representations of cbits & qubits

I Representation of the two cbits as orthogonal vectors in atwo-dimensional vector space:

0 ≡(

01

), 1 ≡

(10

).

I Representation of qubits as normalized vectors in atwo-dimensional vector space:

|Ψ〉 = a0|0〉+ a1|1〉 ≡(

a1a0

), with |a0|2 + |a1|2 = 1.





Quantum state evolution: one-to-one

I Classical reversible computation associated with permutations ofthe classical states associated with permutation matrices (only asingle entry “1” per row & column, else “0”).

I Inbetween measurements, quantum states follow reversibledeterministic, unitary state evolution:

|Ψlater〉 = U|Ψformer〉.

I U is a unitary matrix: UU† = U[(U∗)T ] = 1;i.e., U† = U−1. Here,

I “∗” stands for “complex conjugate,”I “T ” stands for “transposition,” andI “†” stands for “hermitean conjugate” (“= ∗&T ”), respectively.





Quantum state evolution: Examples

I The identity defined by |0〉 → |0〉, |1〉 → |1〉: I2 =

(1 00 1

),

I the “not” defined by |0〉 → |1〉, |1〉 → |0〉:

X =

(0 11 0

),

I the Hadamard “H =√

I2,” defined by|0〉 → (|0〉+ |1〉)(1/

√2), |1〉 → (|0〉 − |1〉)(1/

√2):

H =1√2

(1 11 −1

)with

√I2 ·

√I2 = I2,

I the√not: 1

2

(1 + i 1− i1− i 1 + i

)with

√not

√not = not.





Mach-Zehnder interferometer

L��

�D1

c

S1

S2

aϕP

d

D2

e

b M

M

Mach-Zehnder interferometer. A single quantum (photon, neutron,electron etc) is emitted in L and meets a lossless beam splitter(half-silvered mirror) S1, after which its wave function is in acoherent superposition of b and c . In beam path b a phase shiftershifts the phase of state b by ϕ. The two beams are thenrecombined at a second lossless beam splitter (half-silvered mirror)S2. The quant is detected at either D1 or D2, corresponding to thestates d and e, respectively.





Mach-Zehnder interferometer cntd.

L��

�D1

c

S1

S2

aϕP

d

D2

e

b M

M

S1 : a → (b + ic)/√

2 ,P : b → be iϕ ,

S2 : b → (e + id)/√

2 ,

S2 : c → (d + ie)/√

2 .

a → ψ = i(

e iϕ + 12

)d +

(e iϕ − 1

2

)e.

ϕ = 0, i.e., there is no phase shift at all: a → id , and the emittedquant is detected only by D1.ϕ = π: a → −e, and the emitted quant is detected only by D2.For general phase shift ϕ:

PD1(ϕ) = |(d , ψ)|2 = cos2(ϕ

2) , PD2(ϕ) = |(e, ψ)|2 = sin2(

ϕ

2) .





Alternative representations

Alternatively, the action of a lossless beam splitter may bedescribed by the unitary matrix(

i√

R(ω)√

T (ω)√T (ω) i

√R(ω)

)=

(i sinω cosωcosω i sinω

).

A phase shifter in two-dimensional Hilbert space is represented byeither the unitary matrix

diag(e iϕ, 1

)or diag

(1, e iϕ)

.





“Interaction-free” measurement

L��

�D1

c

S1

S2

a

P

d

D2

e

b M

M

Case #1: Suppose, P is not a beam splitter, but a perfect absorber.Then, the beam path b is blocked entirely, leaving open only beampath c , resulting in a 50:50 chance that detectors D1 and D2 fire.Case #2: Suppose, P is a transparent medium (no absorber): sinceϕ ≡ 0: only D1 fires.Hence: if we want to know whether or not an absorber is in beampath b, then whenever D2 fires (in 1/4 of all cases), we know thatthe absorber is present although the quant “has not touched it.” Weals say that “no interachtion has taken place between the absorber &the quant.” Has it not ;-)SINGLE QUANT (QUBIT) EFFECT!!!!!





“Delayed choice” measurements

L��

�D1

c

S1

S2

a

P

d

D2

e

b M

M

Suppose we block beam path b with an absorber at P only after thequant has “passed” the first 50:50 mirror at S1 and is “somewhereinbetween S1 and P .”Would this make any difference as compared to blocking the path bbeforehand; i.e., before the quant has “passed” the first 50:50 mirrorat S1?Guess what happens ;-)SINGLE QUANT (QUBIT) EFFECT!!!!!





General setup

I Two measurement directions a and b of two dichotomicobservables with values “-1” and “1” at two spatially separatedlocations.

I The measurement direction a at “Alice’s location” is unknownto an observer “Bob” measuring b and vice versa.

I A two-particle correlation function E (θ) with θ = |a − b| isdefined by averaging the product of the outcomesO(a)i ,O(b)i ∈ −1, 1 in the ith experiment; i.e.,E (θ) = (1/N)

∑Ni=1 O(a)iO(b)i .





Classical correlations for two-particle “perfectly correlated”state

Assume uniform distribution of (opposite) “angular momentum” ofthe two particles; Alice measuring along angle a, Bob measuringalong b:

E (a, b) = A+(a,b)−A−(a,b)2π = 2A+(a,b)−2π

2π == 2

π |a − b| − 1 = 2πθ − 1

a ab b

−

−+

+

+

−

−

+





Quantum correlations for two-particle singlet state

E (θ) = 3/[j(j + 1)]C (θ)

with non-normalized

C(θ) = 〈J = 0, M = 0 | α · JA ⊗ β · JB | J = 0, M = 0〉

=X

m,m′

〈00 | jm, j − m〉〈jm′, j − m′ | 00〉 ×

×A〈jm |B 〈j − m | α · JA ⊗ β · JB | jm′〉A | j − m′〉B

=X

m,m′

〈00 | jm, j − m〉〈jm′, j − m′ | 00〉 ×

×〈jm | α · JA | jm′〉〈j − m | β · JB | j − m′〉

=X

m,m′

(−1)j−m(−1)j−m′

2j + 1〈jm | JA

z | jm′〉〈j − m | β · JB | j − m′〉

=X

m,m′

(−1)j−m(−1)j−m′

2j + 1mδmm′〈j − m | β · JB | j − m′〉

=X

m

m(−1)2j−2m

2j + 1〈j − m | β · JB | j − m〉 =

12j + 1

X

m

−m2βz = − 12j + 1

cos θ

jX

m=−j

m2 for 0 ≤ θ ≤ π

= − j(j + 1)

3cos θ for 0 ≤ θ ≤ π .





Two-particle correlations cntd.

��

��

��

��

��

��

�+1

−1

0π/2 π

θ

E

Ec(θ)

Eqm(θ)

Es(θ)

r

More anti-coincidences of detector clicks between 0 < θ < π/2;more coincidences of detector clicks between π/2 < θ < π;same-as-classical and quantum for θ = 0, π/2, π.





Boole-Bell-type inequalities

I Would you believe that (i) it rains in Vienna with probability80%; (ii) it rains in Budapest with probability 80%; (i)&(ii) itrains in Vienna & jointly in Budapest with probability 0.1% ?Exactly when would you start believing me?

I Around 1860 Boole: “conditions of possible experience” (in“Laws of Thought”)

I Around 1965 Bell: similar inequalities as classical bounds forprobabilities of joint events.

I Pitowsky & others: geometric interpretation as “inside–outside”conditions with regards to faces of correlation polytopes: Takeall possibilities of classical events. Take their joints. Interpretthe entries in the truth tables as vectors in a vector space.These vectors form the vertices of a “correlation polytope”formed by the convex sum. The surface of this polytoperepresents all classical probability distributions. The faces ofthis polytope form the inside–outside relations. They arerepresented by Boole-Bell inequalities.





Kochen-Specker theorem & quantum “meaning”

I “It is impossible to consistently (re)construct an entire set ofquantum properties from its parts. Therefore, a comprehensivelist of ‘elements of physical reality’ cannot exist.”Simon Kochen and Ernst P. Specker, “The Problem of HiddenVariables in Quantum Mechanics,” Journal of Mathematics andMechanics 17(1), pp.59-87 (1967)Review in Karl Svozil, “Quantum Logic,” (Springer,Singapore,1998)

I Feynman: “Nobody understands quantum mechanics.” (in “TheCharacter of Physical Law”)

I Is it useless to even think about possible interpretations of theformalism; even more so to go beyond the quantum? Will thehuman mine ever transcend the quantum world? Stronganti-rationalist tendencies (Bohr, Heisenberg,... versus Einstein,Schrödinger, De Broglie, ...).




No-cloning (no-copy) theoremVisions of parallelism & interferenceDeutsch algorithm: parity of a function of one bitQuantum cryptography & man-in-the-middle attacks

Reversible, one-to-one computation

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The lowest “root” represents the initial state interpretable asprogram. Forward computation represents upwards motion througha sequence of states represented by open circles. Different symbolspi correspond to different initial states, that is, different programs.a) One-to-one computation. b) Many-to-one junction which isinformation discarding. Several computational paths, movingupwards, merge into one. c) One-to-many computation is allowedonly if no information is created and discarded; e.g., in copy-typeoperations on blank memory.





No-cloning (no-copy) theorem

I Ideally, a perfect qcopy device A, acting upon an arbitrary stateψ and some arbitrary blank state b, would do this:

ψ ⊗ |b〉 ⊗ |Ai 〉 −→ ψ ⊗ ψ ⊗ |Af 〉.

I Suppose it would copy the two “quasi-classical” state “+” and“−” accordingly:

|+, b,Ai 〉 −→ |+,+,Af 〉, |−, b,Ai 〉 −→ |−,−,Af 〉.

I By the linearity of quantum mechanics, the state1√2(|+〉+ |−〉) is copied according to

1√2(|+〉+ |−〉)⊗ |b,Ai 〉 −→

1√2(|+,+,Af 〉+ |−,−,Af 〉)

6= 1√2(|+〉+ |−〉)⊗ 1√

2(|+〉+ |−〉)⊗ |Ai 〉.





Visions of parallelism & interference

I A single qubit “embodies” two classically contradictory states atonce. This is the basis of “quantum parallelism.”

I n single qubits “embody” 2n classically contradictory states atonce. A linear increase of quantum information is associatedwith an exponential increase of embodied classically states –“quantum parallelism.”

I The information in N qubits can be coded in a “distributed”(“entangled”) manner, such that measurement of a single qubit“destroys” this information and makes a readout impossible.

I Encoding of a classical decision problem byI “folding” a quantum state as a coherent superposition of all

(contradictory) classical casesI processing this coherent superposition; and finallyI “unfolding” the processed state properly such that a readout of

the unfolded state presents the solution to the decision problem(equivalent to a state identification).





Deutsch algorithm: parity of a function of one bit

f 0 1f0 0 0f1 0 1f2 1 0f3 1 1

Table: The binary functions of one bit considered in Deutsch’s problem.





Interlude: definition of elementary unitary operations onsingle bits

I X =

(0 11 0

)is the not-operator

I H = 1√2

(1 11 −1

)is the normalized Hadamard matrix





Three step strategy

I First step: unfold the quantum bitI Second step: process the quantum bitI Third step: read out the quantum bit

To preserve reversibility of information processing, start with twobits instead of one (undorgoing an irreversible functionaltransformation f ):

Uf (|x〉|y〉) = |x〉|y ⊕ f (x)〉





Deutsch algorithm: parity of a function of one bit cntd.

I Start with |0〉|0〉 [or rather (X⊗ X)(|0〉|0〉) for convenience];I then “unfold” with the two Hadamards H⊗H;I then apply Uf

12 [|0〉|0⊕ f (0)〉 - |0〉|1⊕ f (0)〉 - |1〉|0⊕ f (1)〉 + |1〉|1⊕ f (1)〉]

f0: ψ112 (|0〉|0〉 - |0〉|1〉 - |1〉|0〉 + |1〉|1〉)

f1: ψ212 (|0〉|0〉 - |0〉|1〉 - |1〉|1〉 + |1〉|0〉)

f2: -ψ212 (|0〉|1〉 - |0〉|0〉 - |1〉|0〉 + |1〉|1〉)

f3: -ψ112 (|0〉|1〉 - |0〉|0〉 - |1〉|1〉 + |1〉|0〉)

Table: State evolution of Uf (H⊗H)(X⊗ X)(|0〉|0〉) for the four functionsf0, f1, f2, f3. X and H stand for the not operator and the (normalized)Hadamard transformation.





Third step: Readout & state identification in Deutsch’s case

I The encoding Ansatz Uf (H⊗H)(X⊗ X)(|0〉|0〉), resulting inthe two different states

|ψ1〉 = ±12(|0〉−|1〉)(|0〉−|1〉) ≡ ±1

2((1,−1)⊗(1,−1))T = ±1

2(1,−1,−1, 1)T

for f0 as well as f3, and

|ψ2〉 = ±12(|0〉+|1〉)(|0〉−|1〉) ≡ ±1

2((1, 1)⊗(1,−1))T = ±1

2(1,−1, 1,−1)T

for f1 as well as f2.I Finally, application of two additional Hadamard-transformations

for each one of the two bits yields a representation in thestandard computational basis; i.e.,

(H⊗H)Uf (H⊗H)(X⊗X)(|0〉|0〉) =

{|1〉|1〉 ≡ (0, 0, 0, 1)T for f (0) = f (1),|0〉|1〉 ≡ (0, 1, 0, 0)T for f (0) 6= f (1).





Other speedups incl. factoring & database search

I Finding the period of a function, related to prime factorization,related to RSA encryption “Shor’s” algorithm.

I Finding whether or not a function acquires “1” on an argumentspace or is “0” everywhere (“database search”).

I General parity cannot be substantially sped up.





What may and may not be possible

I Speedup for all problems translatable into state identificationproblems.

I Speedup questionable for problems which are classicallyrecursion theoretic hard, such as the Ackermann or the BusyBeaver function.

I Still no quantum speedup for NP-complete problems.





History

1970 Stephen Wiesner, “Conjugate coding:” noisy transmission oftwo or more “complementary messages” by using singlephotons in two or more complementary polarizationdirections/bases.

1984 BB84 Protocol: key growing via quantum channel & additionalclassical bidirectional communication channel

1989 First realization by Bennett et al. at 1989 IBM YorktownHeights, 1993 by Gisin across Lake Geneva, 2003-presentDARPA Network Boston (permanent real-time).





BB84 Protocol

from [BBBSS92]





Man-in-the-middle attack: requiring classical authorization(merely “key growing”)

iiicq iiicqbox-in-the-middlefake “Bob” fake “Alice”

Eve

BobAlice

copy ormisinform

from http://arxiv.org/abs/quant-ph/0501062





Techniques & gadgets

I Photon sources: faint laser pulses, photon pairs generated byparametric downconversion, photon guns, . . .

I Quantum channels: single-mode fibers, free-space links, . . .I Single-photon detection: photon counters, . . .I (Quantum) Random number generators: calcite prism, . . .




Resources

I David Mermin’s qc lecture: very good, very popular:http://people.ccmr.cornell.edu/∼mermin/qcomp/CS483.html

I Up-to-May 2005 collection of findings & open questions:M. Arndt et al., “Quantum Physics from A to Z”http://www.arxiv.org/abs/quant-ph/0505187

I Older, very influential article by Schrödinger (the “cat” papers):E. Schrödinger, “Die gegenwärtige Situation in derQuantenmechanik”, Naturwissenschaften 23, pp.807-812;823-828; 844-849 (1935).http://wwwthep.physik.uni-mainz.de/∼matschul/rot/schroedinger.pdf

I Quantum “measurement” paradoxes:L. Vaidman, Z. Naturforsch. 56 a, 100-107 (2001)http://arxiv.org/abs/quant-ph/0102049

I Staging quantum cryptography with chocolate ballshttp://arxiv.org/abs/physics/0510050

I Up-to-date discussion on (subscribable abstracts):http://arxiv.org/archive/quant-ph




Conclusions

I Single quantum concepts and their experimental realization.I Novel phenomena which go beyond the classical field.I Possible application in quantum information processing.I With growing integration & miniaturization, thechnology will

“reach the quanta” soon.I Many open questions, active & fascinating research field!




Thank you for your attention!


Documents

Quantum information & computationtph.tuwien.ac.at/~svozil/publ/2005-stpoelten-pres.pdfClassical & quantum correlations and the Boole-Bell inequalities 2 Quantum computation No-cloning