DECOHERENCE AND QUANTUM INFORMATION

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DECOHERENCE AND QUANTUM INFORMATION
JUAN PABLO PAZ Departamento de Fisica, FCEyN
Universidad de Buenos Aires, Argentina paz_at_df.uba
.ar
Paraty August 2007
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Lecture 1 2 Decoherence and the quantum origin
of the classical world Lecture 2 3 Decoherence
and quantum information processing, noise
characterization (process tomography)

• 1, 2 3. Decoherence, an overview.
• Hadamard-Phase-Hadamard and the origin of the
  classical world! Information transfer from the
  system to the environment.
• Bosonic environment. Complex environments. Spin
  environments (some new results)
• 3 . Decoherence in quantum information. How to
  characterize it?
• Quantum process tomography (some new ideas)

Colaborations with W. Zurek (LANL), M. Saraceno
(CNEA), D. Mazzitelli (UBA), D. Dalvit (LANL), J.
Anglin (MIT), R. Laflamme (IQC), D. Cory (MIT),
G. Morigi (UAB), S. Fernandez-Vidal (UAB), F.
Cucchietti (LANL),
Current/former students D. Monteoliva, C. Miquel
(UBA), P. Bianucci (UBA, UT), L. Davila (UEA,
UK), C. Lopez (UBA, MIT), A. Roncaglia (UBA), C.
Cormick (UBA), A. Benderski (UBA), F. Pastawski
(UNC), C. Schmiegelow (UNLP, UBA)
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DECOHERENCE AN OVERVIEW
SUMMARY SOME BASIC POINTS ON DECOHERENCE

• DECOHERENCE AND THE QUANTUM-CLASSICAL TRANSITION

• POINTER STATES W.Zurek, S. Habib J.P. Paz, PRL
  70, 1187 (1993), J.P. Paz W. Zurek, PRL 82,
  5181 (1999)
• TIMESCALES J.P. Paz, S. Habib W. Zurek, PRD
  47, 488 (1993), J. Anglin, J.P. Paz W. Zurek,
  PRA 55, 4041 (1997)
• CONTROLLED DECOHERENCE EXPERIMENTS Zeillinger et
  al (Vienna) PRL 90 160401 (2003), Haroche et al
  (ENS) PRL 77, 4887 (1997), Wineland et al (NIST),
  Nature 403, 269 (2000).

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DECOHERENCE AN OVERVIEW
LAST DECADE MANY QUESTIONS ON DECOHERENCE WERE
ANSWERED

• NATURE OF POINTER STATES QUANTUM SUPERPOSITIONS
  DECAY INTO MIXTURES WHEN QUANTUM INTERFERENCE IS
  SUPRESSED. WHAT ARE THE STATES SELECTED BY THE
  INTERACTION? POINTER STATES THE MOST STABLE
  STATES OF THE SYSTEM, DYNAMICALLY SELECTED BY THE
  ENVIRONMENT W.Zurek, S. Habib J.P. Paz, PRL
  70, 1187 (1993), J.P. Paz W. Zurek, PRL 82,
  5181 (1999)
• TIMESCALES HOW FAST DOES DECOHERENCE OCCURS?
  J.P. Paz, S. Habib W. Zurek, PRD 47, 488
  (1993), J. Anglin, J.P. Paz W. Zurek, PRA 55,
  4041 (1997)
• INITIAL CORRELATIONS THEIR ROLE, THEIR
  IMPLICTIONS L. Davila Romero J.P. Paz, Phys Rev
  A 54, 2868 (1997), MORE REALISTIC PREPARATION OF
  INITIAL STATE (FINITE TIME PREPARATION, ETC) J.
  Anglin, J.P. Paz W. Zurek, PRA 55, 4041 (1997)
• DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS W.
  Zurek J.P. Paz, PRL 72, 2508 (1994), D.
  Monteoliva J.P. Paz, PRL 85, 3373 (2000).
• CONTROLLED DECOHERENCE EXPERIMENTS S. Haroche
  et al (ENS) PRL 77, 4887 (1997), D. Wineland et
  al (NIST), Nature 403, 269 (2000), A. Zeillinger
  et al (Vienna) PRL 90 160401 (2003),
• ENVIRONMENT ENGENEERING Cirac Zoller, etc
  see Nature 412, 869 (2001) (review of PRL work
  published by UFRJ group on environment
  engeneering with ions)
• SPIN ENVIRONMENTS ASK CECILIA CORMICK (POSTER
  NEXT WEEK).

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• SYSTEM ENVIRONMENT INTERACTION CREATES A
  RECORD OF THE STATE OF THE SYSTEM IN THE
  ENVIRONMENT

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• WHAT SHOULD YOU DO IF YOU WANT TO LEARN ABOUT
  THE SYSTEM? (THE INITIAL STATE, FOR EXAMPLE)

• LOOK AT THE ENVIRONMENT!
• HOW MUCH CAN YOU LEARN ABOUT THE SYSTEM BY
  LOOKING AT THE ENVIRONMENT? TOOL MUTUAL
  INFORMATION (NOT ALWAYS USEFUL BUT)

I(A,B)S(A)S(B)-S(A,B)

• FLOW OF INFORMATION BETWEEN SYSTEM AND
  ENVIRONMENT AND ITS RELATION WITH CLASSICALITY
  (A. Roncaglia J.P.P, 2007)
• A ONE of the oscillators of the environment (a
  band centered around a certain frequency) B
  SYSTEM

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I(A,B)S(A)S(B)-S(A,B)
Environmental fraction

• THE SYSTEM BECOMES CLASSICAL WHEN THE
  INFORMATION ABOUT IT IS STORED REDUNDANTLY IN
  THE ENVIRONMENT. RECOVER THE STATE OF THE SYSTEM
  ONLY WHEN YOU MEASURE THE COMPLETE ENVIRONMENT.
  OTHERWISE YOU ONLY KNOW WHERE THE CAT IS (NOT THE
  PHASE!)

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1. Classical random walk
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2. Quantum walk algorithm A quantum coin (spin
1/2) and a quantum walker (moving in a ring with
N sites) It could be a useful subroutine
The coin (spin) and the walker become entangled.
The state of the walker after t-iterations is
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Classical and quantum walks have rather different
properties
Quantum walker spreads faster than
classical! Reason? Quantum interference (more
later!)
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NOTE Quantum walks on graphs have been proposed
as potentially useful quantum subroutines Review
J. Kempe, Contemp Phys 44, 307 (2003) Proposed
in D. Aharonov, A. Ambainis, J. Kempe, U.
Vazirani, Proc 33. ACM STOC-2001, 50-59 There are
very few algorithms that use quantum walks as a
central piece N. Shenvi, J. Kempe B.
Whaley, PRA 67 052307 (2003) (DISCRETE) A.
M. Childs et al, Proc 35 ACM STOC-2003, 59-68
(CONTINUOUS)
Key to the potential advantadge of quantum
walks? Use the quantum nature of the walk, that
allows for faster spreading over the graph (this
enables, for example, exponentially faster hiting
times)
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What happens if the coin (or walker) interacts
with an environment?
(2002, 2003 V. Kendon, B. Tregenna, H. Carteret,
T. Brun, A. Ambainis, etc)
Simple model to simulate coupling to a spin
environment (NMR)
(G.Teklemarian, et al PRA67, 062316 (2003))
Feature model can be exactly solved (analytic
solutions available _at_ C.Lopez J.P.P, PRA 68,
052305 (2003)
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RESULTS Decoherence and quantum-classical
transition for quantum walks Fixe time, vary
system-environment coupling strength
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QUANTUM WALKS IN PHASE SPACE
Quantum walk with decoherence in the coin looks
like random walk. Effect Diffusion along the
position direction in phase space.
Lesson I from decoherence studies a) Environment
couples to spin. b) Spin couples with walker
via U (displacement operator). d) U is
diagonal in momentum. Then momentum states are
pointer states!
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QUANTUM WALKS IN PHASE SPACE
Lesson II from decoherence If the phase space
picture of decoherence (position diffusion) is
correct, we can make a prediction Superposition
of two positions should be robust agains
decoherence!
No decoherence
Full decoherence
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QUANTUM WALKS IN PHASE SPACE
Quantum coherence in this Schrodinger cat state
is robust against decoherence. No entropy is
produced from the decay of the coherent
superposition
Mixture of two positions
Superposition of two positions
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QUANTUM WALKS IN PHASE SPACE
A surprise Entropy curves cross each other. Why?
Some decoherence may be useful!
Fix time, vary coupling strength
Superposition of two positions
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USE WIGNER FUNCTIONS TO REPRESENT THE STATE AND
THE EVOLUTION OF QUANTUM COMPUTERS see C.
Miquel, J.P.P M. Saraceno, Phys Rev A 65
(2002), 062309
See paper for other interesting (useful?)
analogies between quantum algorithms and quantum
maps
Phase space representation of Grovers algorithm
Ather approaches to Discrte Wigner functions in
NxN grids with interesting connection with
stabilizer formalism Wootters et al C.Cormick,
A. Roncaglia, M. Saraceno, E. Galvao, J.P.P, et
al.
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EVOLUTION OF QUANTUM OPEN SYSTEMS
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EVOLUTION OF QUANTUM OPEN SYSTEMS
EXAMPLE CONSIDER A DECOHERENCE MECHANISM THAT
SUPRESSES OFF DIAGONAL TERMS IN 0,1 BASIS
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EVOLUTION OF QUANTUM OPEN SYSTEMS
MORE GENERAL NOISY CHANNEL
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HOW TO FIGHT AGAINST DECOHERENCE
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HOW TO FIGHT AGAINST DECOHERENCE

• TO CORRECT ERRORS YOU MUST CHARACTERIZE THEM
  KNOW YOUR ERRORS!
• MOST GENERAL EVOLUTION OF A QUANTUM OPEN SYSTEM
  CAN BE WRITTEN IN KRAUSS FORM (Exercise Think
  about this!)
• THUS, WE NEED TO FIND OUT WHAT ARE THE KRAUSS
  OPERATORS CHARACTERIZE A QUANTUM CHANNEL

• WHAT TO DO IF YOU DONT KNOW THE KRAUSS OPERATORS?

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HOW TO FIGHT AGAINST DECOHERENCE

• ARE DOMINANT KRAUSS OPERATORS ONE QUBIT ERRORS?
• WHAT IS THE WEIGHT OF THE IDENTITY IN THE KRAUSS
  REPRESENTATION?

• COEFFICIENTS DEPEND UPON THE BASIS WE CHOOSE

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HOW TO CHARACTERIZE A QUANTUM CHANNEL
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HOW TO CHARACTERIZE A QUANTUM CHANNEL
WHAT DOES THE AVERAGE FIDELITY MEASURES?
EFFICIENT ESTIMATION OF AVERAGE FIDELITY IS
POSSIBLE! (Klappenecker Roetteler 2005 see C.
Dankerts Thesis quant-ph/0512217)
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HOW TO CHARACTERIZE A QUANTUM CHANNEL
THIS SEEMS TO BE VERY HARD! HOW DO YOU INTEGRATE
OVER ALL HILBERT SPACE IN REAL LIFE? HOWEVER
MOREOVER THIS CAN BE COMPUTED AS A SUM OVER
d(d1) STATES!
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HOW TO CHARACTERIZE A QUANTUM CHANNEL
ESTIMATE AVERAGE FIDELITY AVERAGE OVER d(d1)
STATES TO FIND AN ESTIMATE (WITH PRECISION
INDEPENDENT OF d) WE NEED TO SAMPLE A RANDOM
SUBSET OF THE STATES OF THE MUBS! (it is
efficient if we are OK with a d-independent
accuracy)
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HOW TO CHARACTERIZE A QUANTUM CHANNEL
SEE POSTER BY F. PASTAWSKI FOR A GENERALIZATION
TO DETERMINE (EFFICIENTLY) ANY COEFFICIENT IN THE
EXPANSION OF THE CHANNEL (SELECTIVE EFFICIENT
PROCESS TOMOGRAPHY)
NEED A CLEAN SOURCE OF STATES FROM MUBs (can be
efficiently produced)
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PHOTONIC QUANTUM PROCESS TOMOGRAPHY



SEE POSTER BY CH. SCHMIEGELOW FOR AN EXPLANATION
OF PHOTONIC IMPLEMENTATION (ALL GATES NEEDED TO
PREPARE THE d(d1) STATES ARE REALIZED WITH
VARIATIONS OF PHASE SHIFTERS, QWP HWP).
RESULT OBTAIN ALL DIAGONAL COEFFICIENTS OF THE
CHANNEL
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HOW TO CHARACTERIZE A QUANTUM CHANNEL
METHOD TO DETERMINE ANY COEFFICIENT IN THE
EXPANSION OF THE CHANNEL (SELECTIVE EFFICIENT
PROCESS TOMOGRAPHY). AVERAGE SURVIVAL PROBABILITY
OVER A SAMPLE OF STATES FROM MUBs IN AN
INTERACTION OF THE FORM
C
NEED ONE EXTRA QUBIT (BETTER THAN OTHER
METHODS..). SEE F. PASTAWSKIs POSTER