Engineering entanglement:How and how much?
Alfred U’renPablo Londero
Konrad BanaszekSascha Wallentowitz
Matt AndersonChristophe DorrerIan A. Walmsley
The Center for Quantum
Information
Objectives
Develop “Quantum Toolbox” of elementary protocols
Determine resources needed for each element
• Manipulating quantum fields
• Scaling issues for QIP readout based on experiment
Quantum field theoretic model of resources
Engineering indistinguishability and entanglement
Approaches
• Developed engineered photon Sources
• Experimentally demonstrated resource scaling for Interference-based information processing
Outcomes
A quantum computer
InputClassical
information
OutputClassical
information
• Resources for preparing and reading register are important
The structure of quantum fields
ˆ E (+) x,t( ) = φλ x,t( )ˆ a λλ∑
Quantum field
Mode function Particle annihilation operator
Quantum state
Mode amplitude Vacuum
Quantum state characterized by classical and quantum parts
Size of computer
Number of Particles
Field-theoretic view Provides a natural measure of resources
ψ = c nλ{ }( )
nλ{ }
∑ ) a λ( )
nλ
λ=1
N
∏ vac†
Detection of quantum systems via particle counting
Particle physics
Quantum Computation Optics
Atomic physics
Generating Entangled States
Entangled state: multi-mode, multi-particle
ψ =12
ˆ a 1↑† ˆ a 2→
† +ˆ a 1→† ˆ a
2↑†
( )vac
λ =αβ; α ∈1,2( ),β ∈ ↑,→( )
• N-particles• 2N-modes (inc. hyper-entangled states)
• 2N pathways for creating particles in 2N modes• Non-observed degrees of freedom must be identical
Coincidence detection implies input photons are
entangled
mode engineering: Distinguishing information destroys interference
Braunstein-Mann Bell-state analyzer
Bell-state measurements are a requirement for teleportation, a
computational primitive
Classical mode structure
)(ωI)(tI
)(ωϕ
)(tϕ
Wavelength (nm)Time (fs)
Inst
anta
neou
s po
wer
Spec
tral
den
sity
A. Baltuska et al, Opt. Lett. 23, 1474 (1998)
Even a single photon can have a complicated shapee.g. localized in space and time
t2 ω2 ≥14
Classical mode structure
ωp
ωs
ωi
Q
Q
ωs
ωi
Generation of entangled photons
Spontaneous parametric downconversion generates pairs of photons that may be entangled in frequency, time of emission and polarization
Pulsed pump
Signal photon spectrum
Idler photon spectrum
Type-I and II quasi-phase matching inNonlinear wave guides
Pump Envelope
Phase-Matching Function
Product of
One-Photon
Fock States
ψ = d ωs∫∫
d ωi
φ ωs
, ωi
( )α ωs
+ ωi
( ) ωs SIGNAL
ωi IDLER
ωi
ωs
ωi
ωs
Generation of entangled photons
Interfering the two-photon state with itself
+ e iθ
ωx
ωy
λ/4
θ
BBO
Generation of entangled photons
Supply two pathways for the generation of a pair of photons with no distinguishing information in the
unmeasured degree of freedom
Spectral entanglement is robust against decoherenceBut Bell measurements difficult
Type II BBO, centered at 800nm (shows typicalspectral correlations present in SPDC.
S=1.228
Type II ADP, centered at 800nm (note thatspectral correlations have been eliminated)
Type II BBO, centered at 1600nm (note thatspectral correlations have been eliminated).
S=0
S=0
By appropriately choosing:
i) the crystal materialii) the central wavelengthiii) the pump bandwidthiv) the crystal length
it is possible to engineer a two-photonstate with zero spectral correlation.
Engineering the entropy of entanglement
Generating Correlated, unentangled photons
Why no entanglement? How to attain positive correlation?
KTP phase matchingfunction at 1.58m:
KTP spectral Intensity at 1.58m:
2. Multiple-source experiments:
Grice, U’Ren at al, Phys. Rev. A 64 63815 (2001)
Unwanted distinguishingInformation eliminated
Spectral uncorrelation⇓
1. Dispersion cancellation to all orders:
Erdmann et al, Phys. Rev. A 62 53810 (2000)
System immune to dispersion
⇓ Group velocity matching condition:
Rubin et al, Phys. Rev. A 56 1534 (1997)
Wave guide QPM downconversion
Towards a useful source of heralded photons
Compact NL structuresLow pump powers
Photons from independent sources will interfere
High repetition rates
STP operation
Conditioned generation
Generating downconversion economically
Economy figure of merit:
465mW 1250 kHzType-II 2mm BBO crystal
Weinfurter [2]62.710×
[1] Kwiat et al, Phys. Rev. A 48 R867 (1993)[2] Weinfurter et al, quant-ph/0101074 (2001)[3] Banaszek, U’Ren et al, Opt. Lett. 26 1367 (2001)
10 W 65 kHzType-I 10cm
KDP crystal
Kwiat, Steinberg [1]
76.510×
720 kHz22WType-I 1mm KTP QPM waveguide
Banaszek,
U’Ren,
Walmsley [3]
103.310×
HzR mmW⎛⎞⎜⎟•⎝⎠PUMP
POWER
COUNTSDOWNCOVERTERGROUP
Proposed Type II Polarization Entanglement Setup
FD: frequency doublerSWP DICH: short-wave-pass dichroic mirrorKTP II WG: waveguideLWP DICH: long-wave-pass dichroic mirrorPBS: polarizing beam splitterPOL1 and POL2: polarizersDET1 and DET2: detectors
+Ψ −Ψ
+Φ −Φ
Applications to quantum-enhanced precision measurement
Accuracy doubling in phase measurement using local entanglement only
No nonclassical light enters probed region -enhanced accuracy for lossy systems
e.g. near-field microscopy
Possibility for efficient wave-based computation
Classical quantum
Particles WavesEntangledParticles
Science, January 2000
Computations based on quantum interference
Scaling Criticisms
“Exponential overhead required for measurement”
Definition of distinguishable detector modes
• Each state of the system mapped to a specific space-time mode
Particle-counting readout
Equivalence of single-particle QIP and CWIP
• Single-particle systems do not scale poorly in readout
- Binary coding possible even for single particle systems(No increase in number of detectors or particles required over entangled register)- No advantage to using several different degrees of freedom
• Collective manipulations on several particles cannot be made efficiently through a single -particle degree of freedom (implications for error-correcting protocols)
Issues in single-particle quantum manipulation
• There’s nothing quantum about single particle processors w/ counting readout, even using several degrees of freedom
H
H
HX
H
H
ga
• Each line represents a single qubit. • H is a Hadamard transformation and X a bit-flip operation• ga is a controlled-NOT transformation acting on all bits simultaneously. • The top n qubits are measured at the end of the circuit.
Meyer-Bernstein-Vazirani Circuit
Anything better than Pentiums without QIP?
Since nowhere are the qubits entangled, they can be replaced by the modes of an optical field.
Implications for atomic and molecular-based QIP
Ahn et al., Science (2000)
Amitay et al., Chem. Phys. (2001)
Howell et al., PRA (2000)
Database search
Multilevel quantum simulator
Graph connectivity analysis
2Nx2N
NxN
2N 2N
? NxN
2N
?
Non-orthogonal Non-orthogonalorthogonal
N ln2 N
Coding
Particles N ln2 N(N)
CNOT gate Tesch and De Vivie-Riedle, CPL (2001)
• How to efficiently address the processor Hlibert space using only one or two degrees of freedom?
Summary: work to date
• New Methods developed for Generating entangled biphotons
• Model for resource analysis proposed based on experimental realization
Resources for single-particle readout scaling analyzed and experimentally verified
• Develop waveguide sources as “entanglement factories”
• make use of low decoherence rates of spectrally entangled biphotons
• Design classical implementation of MBV circuit
• Look at measures of nonclassicality based on scaling associated with quantum logic
Plan: future work