Memory Effect in Spin Chains

1. Memory Effect in Spin Chains 1-Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran 2-Departimento di Fisica, Universita di Camerino, I-62032 Camerino, Italy 3-Computer Science Department, ETH Zurich, CH-8092 Zurich, Switzerland 4-Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK Acknowlegment This poster has been supported by CECSCM Memory less Channel Memory Channel Classical Capacity Assume that in the first transmission, the following state is transferred through the channel We use the following inputs as two shot equiprobable inputs in the memory channel. Spin chains can be used as a channel for short distance quantum communication [1]. The basic idea is to simply place the quantum state by a swap operator at one end of the spin chain which is initially in its ground state, allow it to evolve for a specific amount of time, and then receive it in the receiver register by applying another swap operator. The setup has been shown here. After the first transmission, the state of the channel is After transmission through the channel The Holevo bound for the above equiprobable inputs per each use, as a lower bound for classical capacity, is 1 2 N The effect of the channel when its state is can be specified easily It’s easy to show that the effect of the channel is like an amplitude damping channel. The maximum of Holevo bound over shows that the maximum of C is achieved by separable states. The maximum of Holevo bound is compared with the single shot classical capacity [2] in the following figure Amplitude damping channel Memory channel So the total effect of the is Where the memory evolution is determined by the following Kraus operators The average fidelity over all input states is measure of the quality of the Channel is The results are 1- Separable states achieves the classical capacity 2- Despite that entanglement is not useful, in non optimal time the memory increases the classical capacity. Entanglement Distribution Quantifying the memory Quantum Capacity Coherent information as a lower bound for quantum capacity is In the case of perfect transmission the state of the channel is again reset to the ground state and both of the above evolutions are converged to identity evolution. So we can consider the memory parameter as a distance between the Kraus operators 1 2 N The coherent information when the maximally mixed state is transferred through the chain has been compared with single shot quantum capacity [2] in following figure. where This memory parameter varies from zero for memory less channelto one for full memory Channel. Notice that the memory can help in non optimal time to increase the quantum capacity slightly. Resetting the chain Effect of memory Importance of this model 1- This model is a new model of memory in which the action of the channel is dependent on the state of the previous transmission. So understanding the characteristic of this model is important. 2- This model is more physical than the usual models of memory which are based on the Markovian channels [3] and also it’s easier to implement practically. 3- Studying the capacity of this channel is important because in contrast with the usual memory channels, entanglement is not useful here, however memory can be useful in some cases. Generically, while propagating, the information will also inevitably disperse in the chain and Some information of the state remains in the channel. It is thus assumed that a reset of the spin chain to its ground state is made after each transmission. To reset the chain essentially the system should be interacting with macroscopic apparatus like a zero temperature bath. So the results are: 1- The peaks happens at the same time with the same value in state transferring and entanglement distribution. 2- At non-optimal time memory can improve the quality of state transferring in average. 3- The quality of transmission is dependent on two parameters, one is the memory parameter and the second one is time of evolution. 4- The memory is always destructive for entanglement distribution. Zero temperature bath [1] S. Bose, Phys. Rev. Lett. 91, 207901 (2003). [2] V. Giovannetti and R. Fazio, Phys. Rev. A 71, 032314 (2005). [3] C. Macchiavello, G. M. Palma, Phys. Rev. A 65, 050301 (2002).