Entanglement in Quantum Information Processing

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Entanglement in Quantum Information Processing
25 April, 2004 Les Houches
Samuel L. Braunstein University of York
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Classical/Quantum State Representation
Bit has two values only 0, 1 Information is
physical
BITS ? QUBITS
Many qubits leads to ...
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Fast Quantum Computation
(Shor)
(Grover)
(slide with permission D.DiVincenzo)
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Computational Complexity
Computational complexity how the time to
complete an algorithm scales with
the size of the input. Quantum computers add
a new complexity class BQP
For machines that can simulate each other
in polynomial time.
Shor, 35th Proc. FOCS, ed. Goldwasser (1994)
p.124
Bernstein Vazirani, SIAM J.Comput. 25, 1411
(1997).
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Picturing Entanglement
Pure states are entangled if
(picture from Physics World cover)
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Computation as Unitary Evolution
Any unitary operator U may be simulated by a set
of 1-qubit and 2-qubit gates. e.g., for a
1-qubit gate
Barenco, P. Roy. Soc. Lond. A 449, 679 (1995).
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Entanglement as a Resource
Theorem Pure-state quantum algorithms may be
efficiently simulated classically, provided
there is a bounded amount of global
entanglement. Jozsa Linden,
P. Roy. Soc. Lond. A 459, 2011 (2003).

Vidal, Phys. Rev. Lett. 91, 147902 (2003).
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Entanglement as a Prerequisite for Speed-up
Naively, to get an exponential speed-up, the
entanglement must grow with the size of the
input.

• Caveats
• Converse isnt true, e.g., Gottesman-Knill
  theorem
• Doesnt apply to mixed-state computation, e.g.,
  NMR
• Doesnt apply to query complexity, e.g., Grover
• Not meaningful for communication, e.g.,
  teleportation

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Gottesman-Knill theorem

• Subgroups of Pn have compact descriptions.

stabilizes
.
Gottesman, PhD thesis, Caltech (1997).
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Entanglement as a Prerequisite for Speed-up
Naively, to get an exponential speed-up, the
entanglement must grow with the size of the
input.

• Caveats
• Converse isnt true, e.g., Gottesman-Knill
  theorem
• Doesnt apply to mixed-state computation, e.g.,
  NMR
• Doesnt apply to query complexity, e.g., Grover
• Not meaningful for communication, e.g.,
  teleportation

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Mixed-State Entanglement
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Test for Mixed-State Entanglement
? negative eigenvalues in ?
entangled.
Peres, Phys.Rev.Lett. 77, 1413 (1996). Horodecki3,
Phys.Lett.A 223, 1 (1996).
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Liquid-State NMR Quantum Computation
(figure from Nature 2002)
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NMR Quantum Computation (1997 - )
Selected publications Nature (1997),
Gershenfeld et al., NMR scheme Nature (1998),
Jones et al., Grovers algorithm Nature
(1998), Chuang et al., Deutsch-Jozsa
alg. Science (1998), Knill et al., Decoherence Na
ture (1998), Nielsen et al., Teleportation Natu
re (2000), Knill et al., Algorithm
benchmarking Nature (2001), Lieven et
al., Shors algorithm But mixed-state
entanglement and hence computation is
elusive. Physics Today (Jan. 2000), first
community-wide debates ...
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Does NMR Computation involve Entanglement?
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Braunstein et al, Phys.Rev.Lett. 83, 1054 (1999).
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Can there be Speed-Up in NMR QC?
For Shors factoring algorithm, Linden and
Popescu showed that in the absence of
entanglement, no speed-up is possible with
pseudo-pure states.
Caveat Result is asymptotic in the number
of qubits (current NMR experiments involve lt
10 qubits).
For a non-asymptotic result, we must move away
from computational complexity, say to query
complexity.
Linden Popescu, Phys.Rev.Lett. 87, 047901
(2001).
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Entanglement as a Prerequisite for Speed-up
Naively, to get an exponential speed-up, the
entanglement must grow with the size of the
input.

• Caveats
• Converse isnt true, e.g., Gottesman-Knill
  theorem
• Doesnt apply to mixed-state computation, e.g.,
  NMR
• Doesnt apply to query complexity, e.g., Grover
• Not meaningful for communication, e.g.,
  teleportation

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Grovers Search Algorithm
Suppose we seek a marked number from
satisfying
Grover, Phys.Rev.Lett. 79, 4709 (1997).
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Can there be Speed-up without Entanglement?
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Braunstein Pati, Quant.Inf.Commun. 2, 399
(2002).
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Entanglement as a Prerequisite for Speed-up
Naively, to get an exponential speed-up, the
entanglement must grow with the size of the
input.

• Caveats
• Converse isnt true, e.g., Gottesman-Knill
  theorem
• Doesnt apply to mixed-state computation, e.g.,
  NMR
• Doesnt apply to query complexity, e.g., Grover
• Not meaningful for communication, e.g.,
  teleportation

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Entanglement in Communication Teleportation
Braunstein et al, J.Mod.Opt. 47, 267
(2000) Braunstein et al, Phys.Rev.Lett. 88,
097904 (2002)
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Summary
The role of entanglement in quantum information
processing is not yet well understood. For pure
states unbounded amounts of entanglement are a
rough measure of the complexity of the underlying
quantum state. However, there are exceptions
For mixed states, even the unentangled state
description is already complex. Nonetheless,
entanglement seems to play the same role (for
speed-up) in all examples examined to-date, an
intuition which extends to few-qubit systems. In
communication entanglement is much better
understood, but there are still important open
questions.
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Entanglement in communication

• The role of entanglement is much better
  understood,
• but there are still important open questions
• Theorem
• additivity of the Holevo capacity of a
  quantum channel.
• additivity of the entanglement of
  formation.
• strong super-additivity of the entanglement
  of formation.
• If true, then we would say that
• wholesale is unnecessary!
• We can buy entanglement or Holevo capacity retail.

Shor,
quant-ph/0305035 some key steps by Hayden,
Horodecki Terhal, J. Phys. A 34, 6891 (2001).
Matsumoto, Shimono
Winter, quant-ph/0206148.
Audenaert Braunstein, quant-ph/030345