Spin chain model for correlated quantum channels

Transcription

1 Spin chain model for correlated quantum channels Davide Rossini Scuola Internazionale Superiore di Studi Avanzati SISSA Trieste, Italy in collaboration with: Vittorio Giovannetti (Pisa) Simone Montangero (Ulm) Italian Quantum Information Science Conference 2008 Camerino, Ducal Palace 24 th 29 th October 2008

2 Quantum memory channels Quantum channel: communication channel which can transmit quantum information n uses of the channel are typically employed Alice noise Bob

3 Quantum memory channels Quantum channel: communication channel which can transmit quantum information. n uses of the channel are typically employed Alice noise Bob Model: Alice encodes her message into a n long sequence of information carriers. Each of them interacts independently with an element of a n party environment. (self interacting) environment Single channel use

4 Quantum memory channels Quantum channel: communication channel which can transmit quantum information. n uses of the channel are typically employed Alice noise Bob Model: Alice encodes her message into a n long sequence of information carriers. Each of them interacts independently with an element of a n party environment. Uncorrelated channel Correlated channel Single channel use (self interacting) environment Bosonic modes V. Giovannetti, S. Mancini, 2005 Spins ½ M. Plenio, S. Virmani, 2007

5 Our model D. Rossini, V. Giovannetti, S. Montangero, arxiv: to appear in New J. Phys. We assume E to be an Ising spin 1/2 chain Correlated channel (self interacting) environment Single channel use Spins ½ M. Plenio, S. Virmani, 2007

6 Our model D. Rossini, V. Giovannetti, S. Montangero, arxiv: to appear in New J. Phys. We assume E to be an Ising spin 1/2 chain Each channel use (qubit) is dynamically coupled to one environmental spin through the coupling Hamiltonian with Correlated channel (self interacting) environment Single channel use Spins ½ M. Plenio, S. Virmani, 2007

7 The channel Initial state: system environment with, e.g. ground state of the Ising chain (x can assume configurations)

8 The channel Initial state: system environment with, e.g. ground state of the Ising chain with

9 The channel Initial state: system environment with, e.g. ground state of the Ising chain with Diagonalize Hamiltonian with Jordan Wigner transformation & Bogoliubov rotation JW fermions D. Rossini et al., 2007

10 The channel Lxy completely characterize the channel (but they are ) unital channel purely dephasing channel (Lxx = 1) Can we give an estimate of the quantum capacity of this channel?

11 Capacity of the channel? The Choi Jamiolkowski state of the map is the output density matrix obtained when sending through the channel half of the canonical maximally entangled state of the N level system S:

12 Capacity of the channel? The Choi Jamiolkowski state of the map is the output density matrix obtained when sending through the channel half of the canonical maximally entangled state of the N level system S: For forgetful channels the 1 way distillable entanglement of a bound for the quantum capacity: provides entropy of the channel

13 Capacity of the channel? The Choi Jamiolkowski state of the map is the output density matrix obtained when sending through the channel half of the canonical maximally entangled state of the N level system S: For forgetful channels the 1 way distillable entanglement of a bound for the quantum capacity: provides Q. Is our channel forgetful? Not yet proven, even if it is a reasonable assumption, at least for some values of the model parameters entropy of the channel

14 Channel entropy The channel entropy seems the most suitable estimator of the channel noise

15 Channel entropy The channel entropy seems the most suitable estimator of the channel noise Recall that the Choi Jamiolkowski state is the NxN matrix of elements Lxy/N Its von Neumann entropy, i.e. the channel entropy, is given by: > that is the entropy of environmental spin chain ground state after it has evolved through a random application of the perturbed unitaries

16 Channel entropy The channel entropy seems the most suitable estimator of the channel noise Recall that the Choi Jamiolkowski state is the NxN matrix of elements Lxy/N Its von Neumann entropy, i.e. the channel entropy, is given by: > that is the entropy of environmental spin chain ground state after it has evolved through a random application of the perturbed unitaries The computation of the channel entropy for large n is impractical BUT it is possible to give upper and lower bounds that can be reliably computed

17 Averaged channel fidelity Define the fidelity between input state +> and output Choi Jamiolkowski state (input output fidelity for a maximally correlated state):

18 Averaged channel fidelity Define the fidelity between input state +> and output Choi Jamiolkowski state (input output fidelity for a maximally correlated state): This gives an UPPER BOUND for the channel entropy, through the quantum Fano inequality ( binary entropy function) Therefore, for forgetful channels we also have a bound for Q:

19 Averaged channel fidelity Define the fidelity between input state +> and output Choi Jamiolkowski state (input output fidelity for a maximally correlated state): It is directly related to the average transmission fidelity of the map averages taken with respect to the uniform Haar measure

20 Averaged channel fidelity Define the fidelity between input state +> and output Choi Jamiolkowski state (input output fidelity for a maximally correlated state): The fidelity is numerically computed by averaging over Nav randomly chosen couples (x,y) of initial conditions: For the analyzed sizes, fidelity evaluated with come within an error of order

21 Averaged channel fidelity Gaussian decay in time (fix n) Short t: perturbation theory Long t: system integrability Non analytic behavior at criticality

22 Averaged channel fidelity Gaussian decay in time (fix n) exponential decay with the size n Short t: perturbation theory Long t: system integrability Non analytic behavior at criticality

23 Averaged channel fidelity The linear dependence of in n is partly compromised > evidence of correlations in the associated channel model at criticality: enhancement of the channel response to a perturbation of the environment > loss of predictability in the channel behavior with critical environments

24 Averaged channel purity In the same way, define the purity of the Choi Jamiolkowski state (output purity for a maximally correlated state): This gives a LOWER BOUND for the channel entropy ( Rényi entropy)

25 Averaged channel purity In the same way, define the purity of the Choi Jamiolkowski state (output purity for a maximally correlated state): This gives a LOWER BOUND for the channel entropy ( Rényi entropy) Analogously for the fidelity, it is directly related to the average output purity of the map: The purity is computed by approximating the summation with a random sampling:

26 Average output purity in the short time limit <1: oscillations in time; asymptotically tends to average constant value 1: the drop is enhanced (stronger system env. correlations) 1: higher values of purity (in the limit watchdog effect)

27 Generalized model An extra number m of spins between two consecutive qubits is introduced, such to modulate memory effects

28 Generalized model: fidelity Differences for various m are tiny, even though sensitivity increases at criticality Out of criticality only local properties of the chain are important At criticality distance of the spins coupled to the qubits also matters

29 Generalized model: fidelity

30 Conclusions Connect the study of capacities for quantum memory channels to the analysis of many body systems Bounds for the channel entropy by means of the averaged channel fidelity and purity are strongly influenced by spin correlations inside the environment > at criticality: enhancement of the channel response to a perturbation of the environment How to determine the capacity of this class of channels? ( see also M. Plenio & S. Virmani, 2007 ) Generalizations, other integrable / nonintegrable environments?