Entanglement in Quantum Information Science

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Entanglement in Quantum Information Science
Imperial College London
Abingdon, 12th July 2003

• Martin Plenio
• Imperial College London

Local Collaborators D. Browne, J. Hartley, S.
Scheel, S. Virmani Non-local UK collaborators
K. Audenaert (Bangor)
S.F. Huelga (Hertfordshire)
I.
Walmsley, C. Silberhorn (Oxford) Non-local
spatially separated collaborators 02/03 J.
Eisert (Potsdam), J.I. Cirac (München), R.F.
Werner (Braunschweig)
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Abingdon, 12th July 2003
Imperial College London
Areas we are thinking about

• Mathematical methods in quantum information
  science

• Identify and develop of useful tools from Matrix
  Analysis, Optimization Theory
  Collaborations outside IC Audenaert (Bangor),
  Eisert (Potsdam), Werner (Braunschweig)

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Abingdon, 12th July 2003
Imperial College London
Areas we are thinking about

• Mathematical methods in quantum information
  science

• Identify and develop of useful tools from Matrix
  Analysis, Optimization Theory
  Collaborations outside IC Audenaert (Bangor),
  Eisert (Potsdam), Werner (Braunschweig)

• Theory of entanglement as a resource

• Manipulate, Quantify Provide abstract
  protocols
• All of the above for infinite dimensional
  systems ? Light modes, Cold gases,
  condensed matter systems

Collaborations outside IC Cirac (Munich), Eisert
(Potsdam)
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Abingdon, 12th July 2003
Imperial College London
Areas we are thinking about

• Mathematical methods in quantum information
  science

• Identify and develop of useful tools from Matrix
  Analysis, Optimization Theory
  Collaborations outside IC Audenaert (Bangor),
  Eisert (Potsdam), Werner (Braunschweig)

• Theory of entanglement as a resource

• Manipulate, Quantify Provide abstract
  protocols
• All of the above for infinite dimensional
  systems ? Light modes, Cold gases,
  condensed matter systems

Collaborations outside IC Cirac (Munich), Eisert
(Potsdam)

• Practical implementation of quantum information
  processing

• System Ion traps, CQED specific
  Light modes
• Condensed matter systems
• General Novel non-standard approaches to QIP
  eg QIP supported by noise

Collaborations outside IC Eisert (Potsdam),
Huelga (Hertfordshire), Walmsley (Oxford)
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Abingdon, 12th July 2003
Imperial College London
Areas we are thinking about

• Mathematical methods in quantum information
  science

• Identify and develop of useful tools from Matrix
  Analysis, Optimization Theory
  Collaborations outside IC Audenaert (Bangor),
  Eisert (Potsdam), Werner (Braunschweig)

• Theory of entanglement as a resource

• Manipulate, Quantify Provide abstract
  protocols
• All of the above for infinite dimensional
  systems ? Light modes, Cold gases,
  condensed matter systems

Collaborations outside IC Cirac (Munich), Eisert
(Potsdam)

• Practical implementation of quantum information
  processing

• System Ion traps, CQED specific
  Light modes
• Condensed matter systems
• General Novel non-standard approaches to QIP
  eg QIP supported by noise

Collaborations outside IC Eisert (Potsdam),
Huelga (Hertfordshire), Walmsley (Oxford)

• Applications of quantum information science to
  other areas of physics

• Statistical physics, condensed matter systems,
  QFT, black holes

Collaborations outside IC Eisert (Potsdam),
Werner (Braunschwieg)
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Abingdon, 12th July 2003
The vision . . .
Prepare and distribute pure-state entanglement
Local preparation
A
B
Entangled state between distant sites
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. . . and the reality
Decoherence will degrade entanglement
Local preparation
Noisy channel
Can Alice and Bob repair the damaged
entanglement?
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Abingdon, 12th July 2003
The three basic questions of a theory of
entanglement
Provide efficient methods to

• decide which states are entangled and which are
  disentangled (Characterize)
• decide which LOCC entanglement manipulations are
  possible and provide the protocols to
  implement them (Manipulate)
• decide how much entanglement is in a state and
  how efficient entanglement manipulations can
  be (Quantify)

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Imperial College London
Abingdon, 12th July 2003
. . . and what about experiments?
Theory of entanglement is usually purely abstract
All results assume availability of unlimited
experimental resources
For example accessibility of all QM allowed
operations
BUT
Doesnt match experimental reality very well!
Develop theory of entanglement under
experimentally accessible operations
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Abingdon, 12th July 2003
Discrete systems
If you can implement a particular single qubit
rotations then you can generally do any single
bit rotation.
Only single qubit operations possible
Single qubit rotations A two-qubit operation
No entanglement
Everything is possible
Its difficult to find an interesting intermediate
class that is experimentally well motivated.
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From discrete systems . . .
. . . to infinite dimensional, continuous-variable
systems
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Quantum Continuous Variable Systems

• Harmonic oscillators, light modes or cold
  atom gases.

• canonical variables with commutation relations

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Characteristic function

• Characteristic function (Fourier transform of
  Wigner function)

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General CV states too general Restrict to
Gaussian states

• A state is called Gaussian, iff its
  characteristic function (or its Wigner function)
  is a Gaussian
• Gaussian states are completely determined by
  their first and second moments
• Are the states that can be made experimentally
  with current technology (see in a moment)

coherent states squeezed states (one and two
modes) thermal states
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Abingdon, 12th July 2003
CV entanglement of Gaussian states

• Separability Distillability Necessary and
  sufficient criterion known for M x N systems
  Simon, PRL 84, 2726 (2000), Werner and Wolf,
  PRL 86, 3658 (2001), G. Giedke, Fortschr. Phys.
  49, 973 (2001)
• These statements concern Gaussian states, but
  assume the
• availability of all possible operations (even
  very hard ones).

InconsistentWith general operations I can make
any state Impractical Experimentally, cannot
access all operations
Programme
Develop theory of entanglement under Gaussian
operations.
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Characterization of Gaussian operations

• For all general Gaussian operations, a
  dictionarywould be helpful that links the
• physical manipulation that can be done in an
  experiment to
• the mathematical transformation law

J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89,
097901 (2002) J. Eisert and M.B. Plenio, Phys.
Rev. Lett. 89, 137902 (2002) J. Eisert, S. Scheel
and M.B. Plenio, Phys. Rev. Lett. 89, 137903
(2002) G. Giedke and J.I. Cirac, Phys. Rev. A 66,
032316 (2002) B. Demoen, P. Vanheuverzwijn, and
A. Verbeure, Lett. Math. Phys. 2, 161 (1977)
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Abingdon, 12th July 2003
Gaussian operations can be implemented easily!

• Gaussian operations Map any Gaussian state to a
  Gaussian state

• In a quantum optical setting

• Application of linear optical elements
• Beam splitters
• Phase plates
• Squeezers

Addition of vacuum modes

• Measurements
• Homodyne measurements
• Photon detection (vacuum outcome)

• Applications of Gaussian states and operations
• Unconditional teleporation
• Continuous-variable quantum key distribution

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Gaussian manipulation of entanglement

• What quantum state transformations can be
  implemented under natural constraints?

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Gaussian manipulation of entanglement

• What quantum state transformations can be
  implemented under natural constraints?

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Gaussian manipulation of entanglement

• Is there a local quantum operation such
  that
• ?

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The general theorem

• Necessary and sufficient condition for the
  transformation of pure Gaussian states under
  Gaussian local operations (GLOCC)

under GLOCC
if and only if (componentwise)
A
B
A
B
G. Giedke, J. Eisert, J.I. Cirac, and M.B.
Plenio, Quant. Inf. Comp. 3, 211 (2003)
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What can you do without squeezers?
Question Given a Gaussian state of n modes,
described by covariance matrix g, is there an
array of beamsplitters and phase plates such
that it can be turned into an entangled
state. Answer
M.M. Wolf, J. Eisert and M.B. Plenio, Phys. Rev.
Lett. 90, 047904 (2003)
Question Given a mixed Gaussian state of 2
modes, described by covariance matrix g, when
can it be transformed into a state with
covariance matrix g, by Gaussian local
operations. Answer Necessary and sufficient
conditions can be given.
J. Eisert and M.B. Plenio, Phys. Rev. Lett. 89,
097901 (2002)
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Abingdon, 12th July 2003
Gaussian entanglement distillation on mixed states
Homodyne measurements
General local unitary Gaussian operations (any
array of beam splitters, phase shifts and
squeezers)
A1
B1
A2
B2
Symmetric Gaussian two-mode states r

• Characterised by 20 real numbers
• When can the degree of entanglement be increased?

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Abingdon, 12th July 2003
Gaussian entanglement distillation on mixed states

• The optimal iterative Gaussian distillation
  protocol can be identified
• Do nothing at all (then at least no
  entanglement is lost)!

J. Eisert, S. Scheel and M.B. Plenio, Phys. Rev.
Lett. 89, 137903 (2002)

• Even for the most general scheme with N-copy
  Gaussian inputs the best is to do nothing
• Challenge for the preparation of entangled
  Gaussian states over large distances as
  there are no quantum repeaters based on
  Gaussian operations (cryptography).

G. Giedke and J.I. Cirac, Phys. Rev. A 66, 032316
(2002)
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The end of the story?
But

• are there feasible Gaussian operations
  that map non-Gaussian states onto
  (approximately)
• pure
• entangled
• Gaussian
• states in an iterative (all-optical) procedure?

D. Browne, J. Eisert, S. Scheel, and M.B. Plenio,
Phys. Rev. A 68, . (2003) or quant-ph/0211173
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Distillation by leaving the Gaussian regime once
(Gaussian) two-mode squeezed states
Transmission through noisy channel
Initial step non-Gaussian state
(Gaussian) mixed states
Gaussifier
(Gaussian) two-mode squeezed states
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Abingdon, 12th July 2003
Initial Non-Gaussian step The Procrustean chop
Photon detectors distinguish vacuum state ( no
click)from the rest (click)
A1
B1
A2
B2

• Starting from a two-mode squeezed states using
  beam splitters and photon detectors, but
  keeping the non-vacuum output
  contribution

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Gaussification A single step
Photon detectors distinguishing the vacuum
state ( no click)from the rest (click)
5050 beam splitters
A1
B1
A2
B2

• The state is kept in case of the vacuum outcome,
  otherwise discarded
• This output state is the input for the next step

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Gaussification A single step
For all the details watch out for the Oxford
talks of
Jens Eisert on Tuesday
Dan Browne on Thursday
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Summary and Conclusions

• Reviewed theory of quantum entanglement both
  for discrete and continuous systems.

• Standard approach unconcerned with practical
  feasibility

• For discrete systems no separation between
  feasible and infeasible operations that is
  natural and interesting exists

• In CV systems such a separation exists and I
  presented the development of such a theory

Future Develop entanglement theory of Gaussian
CV systems Apply to theoretical problems and
support work on a possible experimental
demonstration of Gaussifier.