Physics is becoming Physics is becoming too difficult for too difficult for physicists.physicists.— — David Hilbert David Hilbert

(mathematician)(mathematician)

Simple Harmonic OscillatorSimple Harmonic Oscillator

Credit: R. Nave (HyperPhysics)Credit: R. Nave (HyperPhysics)

2-Particle wave functions2-Particle wave functions► 2 Particles, each moving in 1 2 Particles, each moving in 1

D, i.e. along the X axisD, i.e. along the X axis► Then Then configuration spaceconfiguration space is 2 is 2

dimensional: xdimensional: x11, x, x22..► Here each particle is in a Here each particle is in a

superposition of two wave superposition of two wave packets, say packets, say ψψ(x) and (x) and φφ(x).(x). e.g. they may have just gone e.g. they may have just gone

through a double slit.through a double slit.► Position of each particle is Position of each particle is

independent of the other, so independent of the other, so we can factorise wave we can factorise wave function:function: ΨΨ(x(x11,x,x22) = ) =

[ [ ψψ(x(x11)+)+φφ(x(x11) ][ ) ][ ψψ(x(x22)+)+φφ(x(x22) ]) ] or |or |ΨΨ = ( | = ( |ψψ+|+|φφ ) )( ( ||ψψ+|+|φφ ) ) Particle 1 XParticle 1 X

Part

icle

2 X

Part

icle

2 X

φφ(x)(x) ψψ(x)(x)

2-Particle wave functions2-Particle wave functions► Now particle positions are Now particle positions are

correlatedcorrelated..► Particles are still in a Particles are still in a

superposition of two possible superposition of two possible positions…positions…

► ……but position of each but position of each particle depends on the particle depends on the other: other: if particle 1 went through if particle 1 went through

right slit, particle 2 went right slit, particle 2 went through left, and vice versa.through left, and vice versa.

► Wave function is composite:Wave function is composite: ΨΨ(x(x11,x,x22) = ) =

ψψ(x(x11) ) φφ(x(x22)) + + φφ(x(x11) ) ψψ(x(x22)) or |or |ΨΨ = | = |ψψ||φφ + | + |φφ||ψψ NB: NB: twotwo products! products!

Particle 1 XParticle 1 X

Part

icle

2 X

Part

icle

2 X

Applications of EntanglementApplications of Entanglement

► Tests of non-locality Tests of non-locality (Bell inequalities etc)(Bell inequalities etc)

► Quantum Quantum cryptographycryptography

► Quantum computingQuantum computing► Quantum Quantum

teleportationteleportation► Interface between Interface between

quantum and quantum and classical worldsclassical worlds

Einstein, Podolsky & Rosen Einstein, Podolsky & Rosen (1935)(1935)

► ““Is Quantum Mechanical Is Quantum Mechanical Description of reality Description of reality complete?”complete?”

► Create entangled pair of Create entangled pair of particles with correlated particles with correlated position and momentum:position and momentum: Measure momentum of particle A, Measure momentum of particle A,

tells you automatically tells you automatically momentum of particle Bmomentum of particle B

Measure position of particle B, Measure position of particle B, tells you position of Atells you position of A

► Let particles separate and Let particles separate and measure mom of A and pos of B measure mom of A and pos of B at the same time:at the same time: Apparently give Apparently give xx and and pp of both of both

particles (just before particles (just before measurement)measurement)

Breaks uncertainty principle?Breaks uncertainty principle?► No practical use as No practical use as

measurements disrupt measurements disrupt pp of A of A and and xx of B! of B!

Bell InequalitiesBell Inequalities

► Measurements at non-Measurements at non-orthogonal angles for orthogonal angles for local theory must local theory must satisfy certain satisfy certain inequalities in their inequalities in their probabilities of results, probabilities of results, which Bohm states which Bohm states violate.violate.

► Experiments (1980s) Experiments (1980s) confirm QM prediction, confirm QM prediction, i.e. violation of Bell i.e. violation of Bell inequalitiesinequalities

QubitQubit► Classical computer: store data as binary digits (bits) 0 or 1.Classical computer: store data as binary digits (bits) 0 or 1.► Quantum computer: store data in 2-level systems (Qubits) with Quantum computer: store data in 2-level systems (Qubits) with

eigenstates |0eigenstates |0 & |1 & |1 (or |g (or |g & |e & |e for ground and excited) for ground and excited) e.g. spin up & spin down; ground & 1e.g. spin up & spin down; ground & 1stst excited state of oscillator, etc. excited state of oscillator, etc.

► Quantum logic gates change state of qubit eigenstates.Quantum logic gates change state of qubit eigenstates. E.g. quantum not gate:E.g. quantum not gate:

► If qubit is in superposition, output is entangled:If qubit is in superposition, output is entangled:

0101

1000

outputinput

1100010 baba

Quantum computingQuantum computing► In principle qubit-based computers allow massively parallel In principle qubit-based computers allow massively parallel

calculations (each element of superposition acts as a separate calculations (each element of superposition acts as a separate process).process).

► Current state-of-the-art: 4-qubit superconducting chip from Current state-of-the-art: 4-qubit superconducting chip from University of California, Santa Barbara (UCSB)University of California, Santa Barbara (UCSB)

► Problem is “decoherence”, i.e. effective “measurement” of Problem is “decoherence”, i.e. effective “measurement” of quantum state by unwanted correlation with environment.quantum state by unwanted correlation with environment.

► Target for practical use: ~100 qubits (few years time?)Target for practical use: ~100 qubits (few years time?)► Aim is to produce faster algorithms than possible on classical Aim is to produce faster algorithms than possible on classical

computers, for a given size of problem N (e.g. number of bits in computers, for a given size of problem N (e.g. number of bits in input data).input data).

► Algorithms developed as exercise in theory, e.g.Algorithms developed as exercise in theory, e.g. 1994: Shor’s algorithm: factorises large integers (would break 1994: Shor’s algorithm: factorises large integers (would break

many common high-security cryptography systems). t many common high-security cryptography systems). t log(N) log(N)3 3 for for quantum computer, vs t quantum computer, vs t exp(log N /3) for best classical exp(log N /3) for best classical algorithm.algorithm.

1996: Grover’s algorithm: database search t 1996: Grover’s algorithm: database search t N N1/21/2 quantum vs t quantum vs t N classicalN classical

Quantum cryptographyQuantum cryptography

► To send secure message from A to B, even if message is intercepted To send secure message from A to B, even if message is intercepted by evesdroppers:by evesdroppers:

► Choose private key (very long binary number: for complete security, Choose private key (very long binary number: for complete security, needs to be as long as the message.)needs to be as long as the message.)

► Use to encode messageUse to encode message► Send message to BSend message to B► B must get key without any possibility of evesdropping.B must get key without any possibility of evesdropping.► Quantum Key Distribution (QKD) allows this, or rather, allows Eve to Quantum Key Distribution (QKD) allows this, or rather, allows Eve to

be unambiguously detected.be unambiguously detected.

G. V

on A

sche / C

UP

G. V

on A

sche / C

UP

Quantum Key DistributionQuantum Key Distribution► Key idea: if Eve reads the key, Key idea: if Eve reads the key,

she makes a measurement and she makes a measurement and hence disturbs the message. hence disturbs the message. Hence even if she forwards Hence even if she forwards message on, her interference message on, her interference can be detected.can be detected.

► Encode and publically transmit Encode and publically transmit message only when key message only when key transmission has been verified.transmission has been verified.

► Quantum transmission uses Quantum transmission uses photon polarization state to photon polarization state to encode 0 or 1, e.g.encode 0 or 1, e.g.

BB84 Protocol:BB84 Protocol:1.1. Randomly generate digits of Randomly generate digits of

keykey2.2. Randomly choose to encode Randomly choose to encode

each digit using either Q or U each digit using either Q or U states, & transmitstates, & transmit

3.3. Bob randomly chooses Q or U Bob randomly chooses Q or U states to decode. If he gets it states to decode. If he gets it wrong, gets random answer as wrong, gets random answer as U states are 50:50 U states are 50:50 superpositions of Q states and superpositions of Q states and vice-versavice-versa

4.4. Alice & Bob exchange list of Alice & Bob exchange list of choice of Q vs U states (via choice of Q vs U states (via public channel).public channel).

5.5. Keep only cases where they Keep only cases where they randomly made the same randomly made the same choice (in which case the Bob choice (in which case the Bob should get the digit Alice sent, should get the digit Alice sent, barring interference).barring interference).

6.6. Check subsample of digits for Check subsample of digits for interference (Eve or bad interference (Eve or bad transmission).transmission).

Digit 0 1

“Q” states

“U” states

QKD detailsQKD details

►What if Eve makes a copy and “measures” What if Eve makes a copy and “measures” that, while forwarding original?that, while forwarding original? Not possible: “Quantum no-cloning theorem” Not possible: “Quantum no-cloning theorem”

shows that it is impossible to make a perfect shows that it is impossible to make a perfect copy of an arbitrary unknown quantum state.copy of an arbitrary unknown quantum state.

►Can you really do this?Can you really do this? YES! There are now several commercial YES! There are now several commercial

systemssystems►Photons transmitted by optical fibre or free-space Photons transmitted by optical fibre or free-space

transmissiontransmission►Transmission lengths currently > 100 kmTransmission lengths currently > 100 km

A Quantum object you can A Quantum object you can see!see!

► Mechanical resonator with a Mechanical resonator with a fundamental frequency of 6 fundamental frequency of 6 GHzGHz Expected to be in ground state Expected to be in ground state

for T < 0.1 Kfor T < 0.1 K► Quantum properties Quantum properties

demonstrated by electrical demonstrated by electrical coupling to a qubit.coupling to a qubit. Ground and first excited statesGround and first excited states Superposition of the twoSuperposition of the two Exchange of quantum with qubitExchange of quantum with qubit

► Andrew Cleland & John Andrew Cleland & John Martinis’s team at UCSB Martinis’s team at UCSB (O’Connell et al, (O’Connell et al, NatureNature 2010) 2010)

► Recognized as “Breakthrough Recognized as “Breakthrough of 2010” by of 2010” by ScienceScience magazine magazine

AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967

Dilatational resonator(Quantum Drum, Q=260)(Quantum Drum, Q=260)

Piezo-electric slab (~ 10Piezo-electric slab (~ 101414 atoms) driven/measured by capacitor plates: atoms) driven/measured by capacitor plates:

22ffrr 1/1/(L(LmmCCss), C), Css-1-1= C= Cmm

-1-1 +C +C00-1-1

AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967

Coupled qubit–resonator.

► Qubit has Qubit has tunable energy tunable energy difference: vary difference: vary around energy around energy quantum of quantum of oscillator.oscillator.

► Repeat ~1000 Repeat ~1000 measurements measurements to calculate to calculate probability of probability of excited state excited state ((PPee))

AD O’Connell et al. Nature 000, 1-7 (2010) doi:10.1038/nature08967

Qubit–resonator swap oscillations

► Excite Qubit with microwave XExcite Qubit with microwave X pulse. pulse.► Turn on interaction with resonator for time Turn on interaction with resonator for time by tuning qubit energy by tuning qubit energy

to close to resonator frequency (offset = to close to resonator frequency (offset = ) ) ► Measure Measure PPee vs (vs (,,): the quantum is shunted between qubit & ): the quantum is shunted between qubit &

resonator and back:resonator and back: ||ΨΨ = cos(t/ph)|e|0 + sin(t/ph)|g|1► After After phph quantum is in resonator; after quantum is in resonator; after phph/2 we have entangled /2 we have entangled

state.state.

TheoryTheory ExperimentExperiment