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Quantum capacity of a dephasing channel with memory

Quantum capacity of a dephasing channel with memory. A. D’Arrigo. MATIS CNR-INFM, Catania & DMFCI Universita’ di Catania. G. Benenti. CNISM CNR-INFM & CNCS Università dell’Insubria. G. Falci. MATIS CNR-INFM, Catania & DMFCI Universita’ di Catania. Palermo CEWQO, 4th June 2007.

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Quantum capacity of a dephasing channel with memory

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  1. Quantum capacity of a dephasing channel with memory A. D’Arrigo MATIS CNR-INFM, Catania & DMFCI Universita’ di Catania G. Benenti CNISM CNR-INFM & CNCS Università dell’Insubria G. Falci MATIS CNR-INFM, Catania & DMFCI Universita’ di Catania Palermo CEWQO, 4th June 2007

  2. Quantum systems as channel • Motivation • Quantum • capacity • degradable • and • forgetful • channels • Dephasing • channel Markovian model Spin-Boson model Utilizing quantum states to reliable transmit: • 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. Low frequency noise in solid state devices (Lorentzian noise, 1/f noise) • Fluctuating birefringence in optical fiber longer then noise characteristic correlation time time interval elapsed between successive carriers Question:can memory enhance capacity? Macchiavello and Palma, 2002. • Why dephasing channel? • Quantum systems with A. Wallraff et al., 2005. Why memory channel? • Motivation • Quantum • capacity • degradable • and • forgetful • channel • Dephasing • channel Markovian model Spin-Boson model Noisy Channel Memory

  4. TrE U S map N Two problems: entangled! E TrS conjugate map Quantum noisy channel coding theorem Sending Quantum Information • Motivation • Quantum • capacity • degradable • and • forgetful • channels • Dephasing • channel Markovian model Spin-Boson model Coherent information: Nielsen and Schumacher 1996 (1) Ic is not subadittive! The limit is mandatory The theorem holds for memoryless Channel! (2) Barnum, Nielsen and Schumacher 1998

  5. degradability and forgetfulness TrE U S N entangled! E TrS Forgetfulness: Kretschmann and Werner, 2002 • coding on the first N: • successive N-blocks • are uncorrelated! Noise correlation decay exponentially • let N to ∞: 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 alwaysdegradable 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 ≤ 1memory 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->∞ memory > N=100 N=6 N=4 memoryless QN/N Memory enhances quantum capacity N=2 memoryless Upper bound to any rate achievable by QECCs D’Arrigo, 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 isn’t maximized by runp Maximization becomes a hard task!

  9. Gaussian Model N=10 N=6 Ic/N N=4 N=2 memoryless Results: t=0 -> 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! D’Arrigo, 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.

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