title

From Quantum Tomography to Quantum Error Correction:

playing games with the information in atoms and photons

Aephraim Steinberg Dept. of Physics, University of Toronto

Acknowledgments

Acknowledgments

U of T quantum optics & laser cooling group:

PDFs: Morgan Mitchell Marcelo Martinelli Optics: Kevin Resch (  Vienna) Jeff Lundeen Chris Ellenor (  Korea) Masoud Mohseni (  Lidar ) Reza Mir Rob Adamson Atom Traps: Stefan Myrskog Ana Jofre Samansa Maneshi Jalani Fox Mirco Siercke Salvatore Maone (  real world)

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

OUTLINE

•

Introduction:

OUTLINE

Photons and atoms are promising for QI.

Need for real-world process characterisation and tailored error correction.

•

Can there be nonlinear optics with <1 photon?

- Using our "photon switch" to test Hardy's Paradox.

•

Quantum process tomography on entangled photon pairs - E.g., quality control for Bell-state filters.

- Input data for tailored Quantum Error Correction.

•

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

•

Summary / Coming attractions… (Optimal discrimination of non-orthogonal states… Tunneling-induced coherence between lattice sites… Coherent control of quantum chaos… Quantum computation in the presence of noise…)

Quantum Information

What's so great about it?

Quantum Information

What's so great about it?

If a classical computer takes input |n> to output |f(n)>, an analogous quantum computer takes a state |n>|0> and maps it to |n>|f(n)> (unitary, reversible).

By superposition, such a computer takes  n |n>|0> to  n |n>|f(n)>; it calculates f(n) for every possible input simultaneously.

A clever measurement may determine some global property of f(n) even though the computer has only run once...

A not-clever measurement "collapses" n to some random value, and yields f(that value).

The rub: any interaction with the environment leads to "decoherence," which can be thought of as continual unintentional measurement of n.

The Rub

What makes a quantum computer?

What makes a computer quantum?

If a quantum "bit" is described by two numbers: |



> = c 0 |0> + c 1 |1>, then n quantum bits are described by 2 n coeff's: |



> = c 00..0

|00..0>+c 00..1

|00..1>+...c

11..1

|11..1>; this is exponentially more information than the 2n coefficients it would take to describe n independent (e.g., classical) bits. It is also exponentially sensitive to decoherence. Photons are ideal carriers of quantum information-- they can be easily produced, manipulated, and detected, and don't interact significantly with the environment. They are already used to transmit quantum-cryptographic information through fibres under Lake Geneva, and soon through the air up to satellites. Unfortunately, they don't interact with each other very much either! How to make a logic gate?

PART 1: Can we build a two-photon switch?

Photons don't interact (good for transmission; bad for computation) Nonlinear optics: photon-photon interactions Generally exceedingly weak.

Potential solutions: Cavity QED Better materials (10 10 times better?) Measurement as nonlinearity (KLM) Novel effects (slow light, EIT, etc) Interferometrically-enhanced nonlinearity

Entangled photon pairs

(spontaneous parametric down-conversion)

The time-reverse of second-harmonic generation.

A purely quantum process (cf. parametric amplification) Each energy is uncertain, yet their sum is precisely defined.

Each emission time is uncertain, yet they are simultaneous.

Is SPDC really the time-reverse of SHG?

(And if so, then why doesn't it exist in classical e&m?) The probability of 2 photons upconverting in a typical nonlinear crystal is roughly 10

-10

(as is the probability of 1 photon spontaneously down-converting).

Quantum Interference

Suppression/Enhancement of Spontaneous Down-Conversion

(57% visibility)

PART 1a: Applications of 2-photon switch

N.B.: Does not work on Fock states!

Have demonstrated controlled-phase operation.

Have shown theoretically that a polarisation version could be used for Bell-state determination (and, e.g., dense coding)… but not for projective Bell measurements.

Present "application," however, is to a novel test of QM.

"Interaction-Free Measurements"

(AKA: The Elitzur-Vaidman bomb experiment)

C D

Consider a collection of bombs so sensitive that is guarranteed to trigger it.

but differ in their behaviour in Bomb absent: Only detector C fires

no way

other than that Bomb present: "boom!" C 1/2 1/4 D 1/4

C + D + Hardy’s Paradox D C O + BS2 + I + I BS2 O W BS1 + BS1 -

Outcome Prob

D+ e- was in D + and C and C + 1/16 C + and C 9/16 D + and D 1/16 Explosion 4/16 e + e -

Hardy's Paradox: Setup

Det. A Det. B CC PBS 50-50 BS2 50-50 BS1 GaN Diode Laser V CC DC BS H Switch (W) DC BS Cf. Torgerson

et al.

, Phys. Lett. A.

204

, 323 (1995)

Conclusions when both "dark" detectors fire Probabilities e- in e+ in

0

e- out 1 1 e+ out 1 -

1

0 1 0

Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear out these predictions.

The Real Problem

• The danger of errors grows exponentially with the size of the quantum system.

• Without error-correction techniques, quantum computation would be a pipe dream.

• A major goal is to learn to completely characterize the evolution (and decoherence) of physical quantum systems in order to design and adapt error-control systems.

• The tools are "quantum state tomography" and "quantum process tomography": full characterisation of the density matrix or Wigner function, and of the "

$

uperoperator" which describes its time-evolution.

PART 2: State and process tomography

Density matrices and superoperators

One photon: H or V. State: two coefficients

( )

C V Density matrix: 2x2=4 coefficients C HV C C VH VV Measure intensity of horizontal intensity of vertical intensity of 45o intensity 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.

Part 2a: Two-photon Process Tomography

Detector A Two waveplates per photon for state preparation HWP QWP HWP QWP PBS SPDC source "Black Box" 50/50 Beamsplitter Argon Ion Laser QWP HWP QWP HWP Two waveplates per photon for state analysis PBS Detector B

Hong-Ou-Mandel Interference

r r + t t How often will both detectors fire together?

r 2 +t 2 = 0; total destructive interference.

If the photons begin in a symmetric state, no coincidences.

The only

antisymmetric

state is the singlet state |HV> – |VH>, in which each photon is unpolarized but the two are orthogonal.

This interferometer is a "Bell-state filter," needed for quantum teleportation and other applications.

Our Goal: use process tomography to test this filter.

16 input states

“Measuring” the superoperator

Coincidencences Output DM Input } } } HH HV etc.

VV 16 analyzer settings VH

“Measuring” the superoperator

Input Output DM Superoperator HH HV VV VH Input Output etc.

“Measuring” the superoperator

Input Output DM Superoperator HH HV VV VH Input Output etc.

Testing the superoperator

LL =

input state Predicted N photons = 297 ± 14

Testing the superoperator

LL =

input state Predicted N photons = 297 ± 14

BBO two-crystal downconversion source.

Argon Ion Laser Detector A HWP QWP HWP QWP PBS "Black Box" 50/50 Beamsplitter QWP HWP QWP HWP PBS Detector B

Observed N photons = 314

So, How's Our Singlet State Filter?

Bell singlet state:  = (HV-VH)/√2

1/2 -1/2 -1/2 1/2

Observed   , but a different maximally entangled state:

Model of real-world beamsplitter

multi-layer dielectric AR coating

1

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



birefringent element + singlet-state filter + birefringent element Singlet filter

Comparison to ideal filter

Measured superoperator, in Bell-state basis: Superoperator after transformation to correct polarisation rotations: A singlet-state filter would have a single peak, indicating the one transmitted state.

Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors.

Part 2b: Tomography in Optical Lattices

Part I: measuring state populations in a lattice…

Time-resolved quantum states

Setup for lattice with adjustable position & velocity

Atoms oscillating

final vs midterm, both adjusted to 70 +/- 15 final vs midterm, both adjusted to 70 +/- 15

Series1 -1 -0.2

corr midterm

Also Atoms oscillating

final vs midterm, both adjusted to 70 +/- 15 final vs midterm, both adjusted to 70 +/- 15

Series1 -1 -0.2

corr midterm

Oscillations in lattice wells

Ground-state population vs. time bet. translations

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

Quantum state reconstruction

p



t



t x Wait… p



x x Shift… Initial phase space distribution Q(0,0) = Pg W(0,0) =



(-1)n Pn Measure ground state population

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

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

Exp't:"W" or [P

g

-P

e

](x,p)

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

W(x,p) for 80% excitation

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

Atom superoperators

sitting in lattice, quietly decohering…

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

being shaken back and forth resonantly Initial Bloch sphere

QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.

Coming attractions… State Discrimination

•

Non-orthogonal quantum states cannot be distinguished with certainty.

•

This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications.

•

Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do.

H-polarized photon 45o-polarized photon

Theory: how to distinguish non orthogonal states optimally

Step 1: Repeat the letters "POVM" over and over.

Step 2: Ask Janos, Mark, and Yuqing for help.

The view from the laboratory: A measurement of a two-state system can only yield two possible results.

If the measurement isn't guaranteed to succeed, there are three possible results: (1), (2), and ("I don't know").

Therefore, to discriminate between two non-orth.

states, we need to use an expanded (3D or more) system. To distinguish 3 states, we need 4D or more.

A test case

Consider these three non-orthogonal states: Projective measurements can distinguish these states with

certainty

to

two

no more than 1/3 of the time.

(No more than one member of an orthonormal basis is orthogonal of the above states, so only one pair may be ruled out.) But a unitary transformation in a 4D space produces: …and these states can be distinguished with certainty up to 55% of the time

Experimental layout

(ancilla)

Success!

"Definitely 3" "Definitely 2" "Definitely 1" "I don't know" The correct state was identified 55% of the time- Much better than the 33% maximum for standard measurements.

SUMMARY

•

Quantum interference allows huge enhancements of optical nonlinearities. Useful for quantum computation?

•

Two-photon switch useful for studies of quantum weirdness (Hardy's paradox, weak measurement,…)

•

Two-photon process tomography useful for characterizing (e.g.) Bell-state filters.

Next round of experiments on tailored quantum error correction (w/ D. Lidar

et al

.).

•

Wigner-function and Superoperator reconstruction also underway in optical lattices, a strong candidate system for quantum comp utation. Characterisation and control of decoherence expected soon.

•

Other work: Implementation of a quantum algorithm in the presence of noise; Optimal discrimination of non-orthogonal states; Tunneling-induced coherence; et cetera…