Entanglement Creation in Open Quantum Systems

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Entanglement Creation in Open Quantum Systems
Milano, December 1, 2006

• Fabio Benatti
• Department of Theoretical Physics
• University of Trieste

In collaboration with R. Floreanini
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Outline

• Open quantum dynamics dissipation and
  decoherence
• Entanglement in open quantum systems
  creation and its persistence

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Open Quantum Dynamics

• qubits (S) in interaction with an environment
  (E) heat bath, external classical noise
• weak coupling limit Lindblad equation

NOISE
FRICTION
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Phenomenological Coefficients

• Kossakowski matrix
• dissipative generator after ergodic mean

R
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Physical ConsistencyComplete Positivity

• necessary for the physical consistency of
• equivalent to complete positivity of
• for all entangled states of
• any n-level ancilla

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One open 2-level atom

• S(ystem)E(nvironment)
• Lindblad equation for 1 qubit density matrices

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Two open 2-level atoms

• S(1)S(2)E no direct S(1)--S(2) interaction

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Lindblad for two open atoms
Environment induced interaction
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Can without direct interaction
generate entanglement ?
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Sufficient Condition (F.B., R. Floreanini, M.
Piani, PRL 2003)

• an initial 2 qubit separable state
• gets entangled as soon as if

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Idea for proof

• use partial transposition to check
  whether
• as well the the maps
  the maps
• form a semigroup with generator

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need not be positive
need not preserve the positivity of
need not be positive
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and
get entangled
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Particular Case

• choose
• sufficient condition becomes

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Example
If ,
gets entangled for small times
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Two atoms in a scalar thermal field in
equilibrium at inverse temperature

• qubit 1 and qubit 2 linearly coupled to
• full Hamiltonian

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Kossakowski matrix D explicitly calculable
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Can entanglement created irreversibly survive
decoherence?
YES Moreover, entangled states can remain
entangled asymptotically and even become more
entangled F.B., R. Floreanini (Int. J. Quant.
Inf. 2006)
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Quantifying EntanglementConcurrence

• 2 qubits entanglement content (Wootters 1998)
• spectrum(R)
• concurrence

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Any initial state goes into
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Asymptotic Concurrence

• initial state
• concurrence
• asymptotic gain

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Two atoms separated by a distance L

• interaction Hamiltonian
• smearing functions

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Kossakowski Matrix
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Controlling Entanglement Creation

• separable initial state
• sufficient condition
• with
  it becomes
• separable if