Measurements for Quantum Communication

using Linear Optics

*Peter van LoockPhilippe Raynal

Norbert Lütkenhaus *

National Institute ofInformatics (NII), Tokyo, Japan

* QIT group, University Erlangen-Nürnberg, Germany

Overview I

Motivation: why quantum information ?applications in communication and computation

unavailable without exploiting quantum effects

Implementation: why optical quantum information ?light is an optimal medium for communication

Optical implementation: linear versus nonlinear optics,continuous versus discrete quantum variables

Resources: nonorthogonal states, entangled states,

Protocols: quantum teleportation, dense coding,entanglement distillation, quantum error correction,quantum cryptography, quantum cloning,quantum logic, quantum computing

Classical versus QuantumCommunication

BobAlice

quantum channel

classical channel

Classical communication: the restrictions imposed by quantum theoryon the transmission of classical bits using quantum signal states are described by

∑−≤ )ˆ()ˆ():(aa

SpSBAI ρρaρ

(Holevo bound)

Holevo can always be approached via quantum coding, extending Shannon to the quantum realm

Quantum communication: turn the supposed restrictions into a virtue and benefit from quantum features such as nonorthogonality and entanglement

Quantum Information Toolbox

quantum state preparation

(arbitrary) quantum states(joint) measurements

entangled states

(local) unitary transformations U

...and for quantum communication, include classical and quantum channelsconnecting the participants Alice, Bob, Eve, etc.

Quantum Communication

Applications: Quantum Key DistributionAuthentication, Secret Sharing…

Sub-Protocols:Quantum Teleportation

Entanglement Distillation

Entanglement Swapping

*** ***

****** ******

U

Dense Coding: all swapped

U U

Quantum Repeater!

Quantum Optics Toolbox (single photons, discrete variables)

Resources: signal and entangled states built fromparametric down-conversion, single photon sources, ... nonlinear optics, )2(χ

1sin0cos αα ϕie+occupation number qubit: (one mode)

01sin10cos αα ϕie+dual-rail qubit: (two modes)

e.g. polarization of single photon:

100,010,001multiple-rail qudit:

αα ϕ sincos ie+↔(e.g. three modes)

Processing: preferably linear optics, nonlinear interaction hard to obtain,photon counting, polarization rotations

)3(χ

Quantum Optics Toolbox (many photons, continuous variables)

Resources: signal and entangled states created via optical parametric amplification, Kerr effect, … coherent-state sources, … Schrödinger cat states, …

nonlinear optics, )3(),2(χ

16/1,2/],[, ≥∆∆=+= pxipxipxa

potentially bright beams!

Quadratures behave like position and momentum of an harmonic oscillator

coherent state

x

psqueezed state

vacuum

r pep ∆=∆ + 2vacuum

r xex ∆=∆ − 2pinclude linear opticsto make entanglementfrom squeezing!

x

)3(),2(χ

Processing: linear optics, preferably Gaussian operations,homodyne detection, feedforward techniques (phase-space displacements)

Example:

Entanglement Swappingx p

Alice Claire Bob

1 2 3 4

continuous var.:

1 2 3 4teleportation of entanglement,non-maximum entanglementpreserved for perfect EPR channel

creates entanglement betweensystems that never interacted

P. van Loock and S.L. Braunstein,PRA 61, 010302(R) (2000) basic ingredient of a quantum repeater,

combined with entanglement purification

for qubits, half of the Bell measurement resultsturn non-maximum into maximum entanglement

experiment

O. Glöckl et al., quant-ph/0302068

Entanglement-generating Circuits

example qubits: ( )1...110...002

1GHZ +=discrete var.:

H

0

0

0

0

GHZC-NOTHadamard transform H: C-NOT gate:

baaba ⊕→( )102

11 +→

( )102

11 −→

contin.var.: Inputs: zero-position eigenstates position0

Replace Hadamard by a Fourier transform F: momentumpositionposition

21F xpydyx ixye === ∫π

momentum position 00F =C-NOT gate:

xxxdx ...1becomesGHZ ∫πyxxyx +→

Optical Implementation?

encode qubit into single-photon state of two modes: discrete variables:

C-NOT gate:

either via strongcross Kerr nonlinearity or probabilisticallyvia linear optics usingauxiliary photons

Hadamard by 50 : 50 beam splitter:

( )10102

110 +→ ( )10102

110 −→,

Or directly: send a nonclassical state through N-splitter

output is purenonmaximallyentangled statefor N > 2 (not GHZ !)

for example, one-photon state

W)1...000...0...0100...100(10...100 ≡++→

N

Optical Implementation?

continuous variables: Replace C-NOTs by beam splitters

apply N-splitter to a zero-momentumand N – 1 zero-position eigenstates

However, this corresponds to unphysical states infinitely squeezed in the quadratures x and p

Finite squeezing: apply N-splitter to a momentum-squeezedand N – 1 position-squeezed states

P. van Loock and S.L. Braunstein, PRL 84, 3482(2000)

output is purenonmaximallyentangled statefor any N

genuinelyN-partyentangled ?

Yes! Quantum teleportationpossible between any pairwith the help of the remaining parties

Summary Irelatively cheap resources: squeezed light and linear optics;in principle, even one single squeezer suffices to makeentanglement; efficient, unconditional, imperfect

relatively simple feasible protocols, employing Gaussianoperations such as linear optics, homodyne detection and feedforward; for projection measurements onto themaximally entangled basis of arbitrarily many modes,invert entanglement-generating circuit

this is in contrast to the single-photondiscrete-variable case ... next part of the poster

important exception: entanglement distillation requiresunfeasible non-Gaussian operations

Quantum information with continuous variablesS.L. Braunstein and P. van Loock,Review of Modern Physics, to appear

Overview II

Motivation: Measurements for quantum communicationusing tools solely from linear optics

Background: what is known ? ... No-Go statements for some projective measurements;any measurement possible asymptotically

Extensions: active linear devices; generalized measurements

New method: include detection mechanism into state transformations

Criteria: directly applicable to exact state discrimination

complete projection measurement

Manipulation with linear opticsbeam splitter

a2

c1 c2

−

=

2

1

2

1

21

aa

rttr

cc

0

122

=−

=+∗∗ trtr

tr

a1

phase shifter exclude squeezing transformations:++= aBaAcaec iϕ=ϕ

a c

linear network

aUc =U unitary

2

1

a

a

2

1

c

c

.

.

.

.

.

.

ϕ

ϕ

Any U can be realized: M. Reck et al., Phys. Rev. Lett. 73, 58 (1994)

PLUS: photon counting or homodyne detection …

Projection measurements

Some require signal interaction (joint measurements)

• non-linear interaction (e.g. cross Kerr effect) too weak for given signal strength

• interferencereadily available via linear optics

Canonical representationof Bell measurement:

H

01sin10cos αα ϕie+αα ϕ sincos ie+↔

„dual-rail qubit“

Why Bell measurement?

a) Basic tool forteleportation, entanglement swapping, quantum repeater

b) Gottesman/ChuangGHZ state + Bell measurement + one-qubit operations universal computing

+Φ

Bell

+ΦBell

U(l)

U(k)

Bell

Ω

Bell

U(l,k)

V(k,l)

Bell measurement for teleportation (discr.var.)

Innsbruck Experiment

−

=

1

1

1

1

1111

21

ba

dc

( )BABA

↔↔+2

1SSe αα ϕ sincos i+↔

A B

a1 a2 b1b2

c1 c2 d1 d2PBS

BS

PBS ResultU(i)

−

=

2

2

2

2

1111

21

ba

dc

S

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )[ ] 00

00

00

22

22

21

2122

122112

12

1

212121

122121

21

212121

122121

21

++++++++

++++++++

++++++++

−±−→±=↔↔±

−→−=↔−↔

−→+=↔+↔

dcdcbaba

dccdbaba

ddccbaba

ASAS

ASAS

ASAS distinct click pattern(two separate clicks)

Analysis:

undistinguishable click pattern 50% success rate

Bell measurementfor teleportation (contin.var.)

H

Hadamard Fourier transform:

positionmomentummomentum21F pxydyp ipye === ∫π

( ) ( )yxyxyx −+→2

1

2

1

BABn

n nn∑− ,1 2 λλA

a1 b1BS

xp

c1 d1

C-NOT beam splitter:

Caltech Experiment

ResultD(x,p)

Se.g.

Sα

No-Go Statements and beyond Is there any linear network that distinguishes always all four Bell states unambiguously?

NO, not even using conditional dynamics and auxiliary photons!N. Lütkenhaus, J. Calsamiglia, K.-A. Suominen, PRA 59, 3295 (1999)

What is the best we can do without conditional dynamics and without auxiliary photons?50%, that is just what has been done in Innsbruck.

J. Calsamiglia, N.Lütkenhaus, Appl. Phys. B 72, 67 (2001)Why? Is it because we project onto entangled states?

NO, there are even separable states onto which we cannot project exactly!

UQubit 1Qubit 2Auxil.

U(k) U(k,l) …

0 1 20

1

2

±±

±±

qutrit product states, e.g. ( )

BBA100

21

±

( )BBA

0100010012

1±

represented in single-photon states,e.g.

A. Carollo, G.M. Palma, C. Simon and A. Zeilinger,PRA 64, 22318 (2001)

Asymptotic ImplementationsE. Knill, R. Laflamme, G. Milburn Nature 409, 49 (2001) „KLM“

Asymptotic perfect implementation of C-NOT gate(on polarization qubits)

Tools:• linear optics• photon counters• conditional dynamics• n entangled auxiliary photons

Success probability:

HCircuit for Bell-Measurement

Parity

Sign

( )( )( )( )

BABABA

BABABA

BABABA

BABABA

011100

001100

110110

100110

21

21

21

21

→−

→+

→−

→+

111+

−n

No-Go theorems indicate whenever finite resources and less sophisticated tools do not allow for arbitrarily high efficiency!

Probabilistic quantum logic with linear optics

(Knill, Laflamme, Milburn, Nature)

ϕ HH ϕ

0100100000100001

−

⊗Ι×

−

×

−

⊗Ι11

11

1000010000100001

1111

21

21 Sufficient to implement phase

gate on occupation-number qubits

phase gate on occupation-number qubitcombined with• Gottesman/Chuang trick:

replace two-qubit gate by entangled input and Bell measurement• probabilistic implementation of Bell measurement on occupation number qubits

Probabilistic Bell measurement for occupation number qubits

( ) ∑=

−−

+⊗+

n

j

jnjjnj

n0

11 100110 βαInput:

Φ

10 βα +

U

sink

10 βα +

U(k)

( )⊗+++ knkknk

10101000

βαStep 1: find k photons

11+

=n

PfailureFailure for k=0 or k= (n+1)

Step 2: Project onto ( )kk

0110002

1 ± Independent of k Fourier transform

( ) 1010 selection)( βαβα + →⊗±⇒ +++

kUknkn

Dephasing approach to quantum state discrimination via linear optics: idea

ρ

iHe−Dephasing

iHe−

Goal: state discrimination via linear optics with subsequent photon detections

Model: replace photon detections by dephasing

ρstill formally a quantum object simpler formulation

is mixture diagonal in Fock basis

Dephasing approach to quantum state discrimination via linear optics: formalism

Distinguishable with linear optics?

inρ AS ⊗=χ

signal state auxiliary state

χχρ =inˆAuxiliary modes:

N modes: TNaaaa )ˆ,...,ˆ,ˆ( 21=

linear mode mixing:

,χχ aHaiH e

+−=HHH χχρ =ˆHρ

Dephasing:

,ˆ...1'ˆ 1aDai

HaDai

NH eeddM

++−∫= ρϕϕρ iijijD ϕδ='ˆ Hρ

Distinguishable?

Beam splitter solution for Bell measurement

Dephased states are:

( )001100111100110021'ˆ , +=+Ψ Hρ

( )011001101001100121'ˆ , +=−Ψ Hρ

( 020002002000200041'ˆ , +=±Φ Hρ

)0002000200200020 ++

H Beam Splitter

Exact state discrimination

Exact distinguishability of states kχ

( ) lk ≠∀= ,0'ˆ'ˆTr Hl,Hk, ρρ

,0~

=+

lkcDcie χχ jijijj D ϕδϕ ∆=∆∀ ~,

ϕ0

0...

~

'''=

=∆

∆∂∆∂∆∂

∂+

ϕϕϕϕ

χχ

jjj

lkn cDcie

,',,0ˆˆˆˆ

,,0ˆˆ

'' jjcccc

jcc

ljjjjk

ljjk

∀=

∀=

++

+

χχ

χχ

lk ≠∀

analytic function of ∆

normalordering

lk ≠∀

,'',',,0ˆˆˆˆˆˆ

,',,0ˆˆˆˆ

,,0ˆˆ

''''''

''

jjjcccccc

jjcccc

jcc

ljjjjjjk

ljjjjk

ljjk

∀=

∀=

∀=

+++

++

+

χχ

χχ

χχ

aUc =U unitary

higher-order conditions break offfor finite number !

of necessary and sufficient conditionsfor exact distinguishability of stateswith a fixed array of linear optics

kχ

conditional-dynamics solutionafter detecting one mode j

( ) ( )lkn

cc ljjknn

≠∀≥∀

=+

,0

,0ˆˆ χχ

P. van Loock and N. Lütkenhaus, PRA 69, 012302 (2004)

Application: Bell measurement

One of the necessary conditions:existence of a mode

24132211 bvbvavavc +++=such that

0ˆˆ =+lk cc χχlk≠∀

iv complex

Signal states:

( )( ) 0

0

221121

4/3

122121

2/1

++++

++++

±=

±=

baba

baba

χ

χ

Proof of No-Go-Theorem:

04,2

03,2

04,1

03,1

04,3

02,1

34124321

34124321

34124321

34124321

23

22

24

21

23

22

24

21

=+−+−==

=−++−==

=−−+==

=+++==

=+−−==

=−−+==

∗∗∗∗

∗∗∗∗

∗∗∗∗

∗∗∗∗

vvvvvvvvlk

vvvvvvvvlk

vvvvvvvvlk

vvvvvvvvlk

vvvvlk

vvvvlk

∗∗

∗∗

=

−=

=

=

4321

4321

24

23

22

21

vvvv

vvvv

vv

vv Not even one mode exists thatcan be dephased (photon counting)

not even conditional dynam. helps:

U

only trivial solution!

Auxiliary photons don´t help…

0ˆˆ =+lk SccSlkc ≠∀∃ ˆ

Assume that orthogonal states cannot be exactly distinguishedkS

Auxiliary systems cannot help to prevent violation of first-order condition;for only one mode detected in a conditional-dynamics scheme, this applies to any order !

lSkAlSSk

lSkAlkAA

lk

ScSAcASccSAA

ScSAcASSAccA

cc

+∗+

+∗+

+

++

+=

αββ

βαα

χχ

2

2klδ 0

AS kk ⊗=χAdd ancilla modes

DecomposeSA ccc βα +=

S

A

cc

linear combination of modesaux.system

US

A

iforhave fixed photon number

kSA

Summary II

• dephasing approach yields necessary and sufficient conditionsfor complete projection measurements with linear optics

• for bounded photon number:finite hierarchies of conditionssolution for fixed linear array readily decidable

• homodyning: e.g. continuous-variable Bell measurement works with simple beam splitter

• generalized measurements (POVM’s), Naimark extensionvon Neumann measurements with linear opticsquantitative statements on success rate, error rate etc.upper bounds on performance as function of auxiliary resources