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