Quantum capacity of a dephasing channel with memory

1
Quantum capacity of a dephasing channel with
memory
A. DArrigo
MATIS CNR-INFM, Catania DMFCI Universita di
Catania
G. Benenti
CNISM CNR-INFM CNCS Università dellInsubria
G. Falci
MATIS CNR-INFM, Catania DMFCI Universita di
Catania
Palermo CEWQO, 4th June 2007
2
Quantum systems as channel
Utilizing quantum states to reliable transmit

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

• Classical data

Classical capacity of a quantum channel
Holevo 98 Schumacher and Westmoreland 98.

• Quantum information

Quantum capacity of a quantum channel
Barnum, Nielsen and Schumacher 98.

• quantum state transmission between different
  parts
• of a quantum computer
• distribution of entanglement among different parts

3
Why memory channel?

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channel
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

Noisy Channel
Memory
4
Sending Quantum Information

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

Coherent information
Ic is not subadittive!
(1)
Nielsen and Schumacher 1996
The limit is mandatory
The theorem holds for memoryless Channel!
(2)
Barnum, Nielsen and Schumacher 1998
5
degradability and forgetfulness
Degradability there exists a map T such that

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

Devetak and Shor, 2004
dephasing channels are always degradable
Ic is concave and subadditive in r
No limit in the channel uses is required!

• Computing Q using the double blocking strategy
• consider blocks of N L channel uses

forgetfulness depends on noise correlations
6
Markovian Model
One-use dephasing channel

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

There exists preferential basis such that
g dephasing factor
Kraus representation
N-use dephasing channel
where
memory
Ic is maximized by
Macchiavello and Palma, 2002
stationary Markov chain
propagator
0 m 1 memory factor
memory decays exponentially!
Forgetful channel
7
Markovian Model
Results

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spin-Boson
• model

where
QN/N converges!
H() is the binary Shannon entropy
and
Quantum Capacity
N-gt8
memory
gt
N100
N6
N4
memoryless
QN/N
Memory enhances quantum capacity
N2
memoryless
Upper bound to any rate achievable by QECCs
DArrigo, Benenti and Falci, cond-mat/0702014
8
Hamiltonian Model
Hamiltonian

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channel
• Dephasing
• channels
• Markovian
  
• model
• Spins-Boson
• model

where
Ic isnt maximized by runp
Maximization becomes a hard task!
9
Gaussian Model
Results
t0 -gt Decoherence free subspaces!

• Motivation
• Quantum
• capacity
• degradable
• and
• forgetful
• channels
• Dephasing
• channel
• Markovian
  
• model
• Spins-Boson
• model

Filters the noise effects!

• Assumptions
• bath correlation
• decay exponentially

• runp for input state

We found a numerical lower bound for Q!
Numerical results suggest Ic/n converges!
DArrigo, Benenti and Falci, cond-mat/0702014
10
Conclusion

• The coherent information in a dephasing channel
  with memory is maximized by separable input
  states
• Computed the quantum capacity Q for a Markov
  chain noise model
• Provided numerical evidence of a lower bound for
  Q in the case of a bosonic bath
• We are now interested in the behaviour of a
  memory channel that shows together relaxation and
  dephasing.