Quantum information with photons and atoms: from tomography to error correction

C. W. Ellenor, M. Mohseni, S.H. Myrskog, J.K. Fox, J. S. Lundeen, K. J. Resch,

M. W. Mitchell, and Aephraim M. Steinberg

Dept. of Physics, University of Toronto

Title slide

PQE 2003

U of T quantum optics & laser cooling group:PDF: Morgan Mitchell

Optics: Kevin Resch ( Wien) Jeff LundeenChris Ellenor ( Korea) Masoud MohseniReza Mir (Lidar)

Atom Traps: Stefan Myrskog Jalani FoxAna Jofre Mirco SierckeSalvatore Maone Samansa Maneshi

TBA: Rob Adamson

Theory friends: Daniel Lidar, Janos Bergou, John Sipe, Paul Brumer, Howard Wiseman

Acknowledgments

OUTLINE• Introduction:

Photons and atoms are promising for QI.Need for real-world process characterisation

and tailored error correction.No time to say more.

• Quantum process tomography on entangled photon pairs- E.g., quality control for Bell-state filters.- Input data for tailored Quantum Error Correction.

• An experimental application of decoherence-freesubspaces in a quantum computation.

• Quantum state (and process?) tomography on center-of-mass states of atoms in optical lattices.

• Coming attractions…

One photon: H or V.State: two coefficients ()CHCV

( )CHH

CHV

CVH

CVV

Density matrix: 2x2=4 coefficientsMeasure

intensity of horizontalintensity of verticalintensity of 45ointensity of RH circular.

Propagator (superoperator): 4x4 = 16 coefficients.

Two photons: HH, HV, VH, HV, or any superpositions.State has four coefficients.Density matrix has 4x4 = 16 coefficients.Superoperator has 16x16 = 256 coefficients.

Density matrices and superoperators

HWP

HWP

HWP

HWP

QWP

QWPQWP

QWPPBS

PBS

Argon Ion Laser

Beamsplitter"Black Box" 50/50

Detector B

Detector A

Two-photon Process TomographyTwo waveplates per photon

for state preparation

Two waveplates per photon for state analysis

SPDC source

0

1 104

2 104

3 104

4 104

5 104

6 104

0

20

40

60

80

100

0 10 20 30 40 50

22DecScan3

Position (microns)

Coincidences (per 50 s)

Hong-Ou-Mandel Interference

> 85% visibility for HH and VV polarizations

HOM acts as a filter for the Bell state:

= (HV-VH)/√2

Goal: Use Quantum Process Tomography to find the superoperator which takes in out

Characterize the action (and imperfections) of the Bell-State filter.

“Measuring” the superoperator

}

Output DM Input

HH

HV

VV

VH

}

}}

etc.

16 analyzer settings

16 input states

Coincidencences

“Measuring” the superoperatorInput Output DM

HH

HV

VV

VH

etc.

Superoperator

Input Output

“Measuring” the superoperatorInput Output DM

HH

HV

VV

VH

etc.Input Output

Superoperator

Testing the superoperatorLL = input state

Predicted Nphotons = 297 ± 14

Testing the superoperatorLL = input state

Predicted

ObservedHWP

HWP

HWP

HWP

QWP

QWPQWP

QWPPBS

PBS

Argon Ion Laser

"Black Box" 50/50Beamsplitter

Detector B

Detector A

BBO two-crystaldownconversionsource.

Nphotons = 297 ± 14

Nphotons = 314

So, How's Our Singlet State Filter?

Observed

Bell singlet state: = (HV-VH)/√2

Model of real-world beamsplitter

φ1 φ2

45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser)

birefringent element+

singlet-state filter+

birefringent element

Singlet filter

AR coating

multi-layer dielectric

PredictedBest Fit: 1 = 0.76 π

2 = 0.80 π

Model beamsplitter predicitons

φ1 φ2

Singlet filter

Observed

Predicted

Comparison to measured Superop

Predicted

H

H

x x H0

1 y y⊕f(x)

)1(f)0(f ⊕

210 −

We use a four-rail representation of our two physical qubits and encode the logical states 00, 01, 10 and 11 by a photon traveling down one of four optical rails numbered 1, 2, 3 and 4, respectively.

1234 0001

0010

0100

1000

Photon number basis

1st qubit 2nd qubit

11100100

Computational basis

OracleA A

Performing a quantum computation in a decoherence-free subspace

The Deutsch-Jozsa algorithm:

€

e iσ 2zσ 2zϕ

Modified Deutsch-Jozsa Quantum Circuit

1 y⊕f(x)

H x x H0

y ⊕ ⊕H

00 0001 ei 01

11 1110 ei 10

But after oracle, only qubit 1 is needed for calculation.Encode this logical qubit in either DFS: (00,11) or (01,10).

Error model and decoherence-free subspacesConsider a source of dephasing which acts symmetricallyon states 01 and 10 (rails 2 and 3)…

3

4

12

1

2

4

23

Experimental Setup

Oracle

Optional swap for choice of encoding

Preparation

Random Noise

MirrorWaveplate

Phase Shifter

PBS

Detector

2/λ

A

B

C

D

3/4

4/3

DJ experimental setup

3

C B CC CBB B

DFS Encoding Original encoding

Constant function

Balanced function

C

B

DJ without noise -- raw data

C B CC CBB B

DFS Encoding Original Encoding

C

B

Constant function

Balanced function

DJ with noise-- results

Tomography in Optical LatticesPart I: measuring state populations in a lattice…

Houston, we have separation!

x

p

W(0,0) = Σ(1)nPn

Wait… Shift…

Measure groundstate population

Quantum state reconstruction

Initial phase-space distribution

p

x

p

x

=x

x

pt

Q(0,0) = Pg

(More recently: direct density-matrix reconstruction)

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Quasi-Q (Pg versus shift) for a 2-state lattice with 80% in upper state.

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Exp't:"W" or [Pg-Pe](x,p)

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W(x,p) for 80% excitation

Coming attractions• A "two-photon switch": using quantum enhancement of two-photon nonlinearities for - Hardy's Paradox (and weak measurements) - Bell-state determination and quantum dense coding(?)• Optimal state discrimination/filtering (w/ Bergou, Hillery)• The quantum 3-box problem (and weak measurements)• Process tomography in the optical lattice

- applying tomography to tailored Q. error correction• Welcher-Weg experiments (and weak measurements, w/ Wiseman)• Coherent control in optical lattices (w/ Brumer)• Exchange-effect enhancement of 2x1-photon absorption

(w/ Sipe, after Franson)• Tunneling-induced coherence in optical lattices• Transient anomalous momentum distributions (w/ Muga)• Probing tunneling atoms (and weak measurements) … et cetera

Schematic diagram of D-J interferometer

1 2 3 41

2

3

4

1 2 3 4

Oracle00

01

10

11

“Click” at either det. 1 or det. 2 (i.e., qubit 1 low)indicates a constant function; each looks at an interferometer comparing the two halves of the oracle.

Interfering 1 with 4 and 2 with 3 is as effective as interfering1 with 3 and 2 with 4 -- but insensitive to this decoherence model.

x

pt

x

pt

x

p

Q(0,0) = Pg

W(0,0) = Σ(1)nPn

Wait… Shift…

Measure groundstate population

Quantum state reconstruction

Initial phase-space distribution

x

p

x

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Q(x,p) for a coherent H.O. state

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Theory for 80/20 mix of e and g