Channel Capacities of an Exactly Solvable Spin-Star System Nigum Arshed, 1 A. H. Toor, 1 and Daniel A. Lidar 2 $^{1}$Department of Physics, Quaid-i-Azam University Islamabad 45320, Pakistan $^{2}$Departments of Physics, Chemistry, and Electrical Engineering, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles 90089, USA We calculate the entanglement-assisted and unassisted channel capacities of an exactly solvable spin star system, which models the quantum dephasing channel. The capacities for this non-Markovian model exhibit a strong dependence on the coupling strengths of the bath spins with the system, the bath temperature, and the number of bath spins. For equal couplings and bath frequencies, the channel becomes periodically noiseless. PACS numbers: 03.67.-a,03.67.Hk,89.70.Kn,75.10.Jm One of the fundamental tasks of quantum information the- ory is to determine the information transmission capacities of quantum channels [1, 2]. The maximum amount of informa- tion that can be reliably transmitted over a channel, per chan- nel use is known as its capacity [3]. Classical channels can be uniquely characterized by their capacity [4]. The situation in the quantum realm is significantly more involved, with vari- ous capacities required to characterize a quantum channel [5]. Studies of quantum channel capacities can be broadly di- vided into those considering memoryless quantum channels, for which the output at a given time depends only upon the corresponding input and not upon any previous inputs [6– 22], and quantum memory channels, where successive uses of the channel modify its properties and description [23–30]. Another important distinction is between channels generated by Markovian vs . non-Markovian environments or baths. Markovian channels describe memoryless baths, while for non-Markovian channels bath memory plays a role [31, 32]. Many quantum optical [33] and nuclear magnetic resonance systems [34] are accurately described by Markovian chan- nels, but the Markovian limit is always an approximation [35]. Non-Markovian effects are especially important in con- densed matter systems, such as coupled electron or nuclear spins [36]. The master equations describing the dynamics of non-Markovian systems are often (though not always [37]) complicated integro-differential equations which are rarely exactly solvable [31]. A channel can be memoryless yet non- Markovian. This situation arises when successive uses do not modify the channel, but a proper description of each use of the channel requires a non-Markovian treatment accounting for bath memory effects. In this work we investigate how non- Markovian effects modify channel capacities by studying an exactly solvable model of a non-Markovian memoryless chan- nel: the Ising spin-star system [38]. One reason to consider spin systems is that they are good candidates for the physical realization of quantum computa- tion and communication, in part due to their relatively long re- laxation and decoherence times [39–43]. Spin chains have at- tracted much recent interest as quantum communication chan- nels [44]. Capacities of a spin chain with ferromagnetic Heisenberg interactions were calculated by studying the qubit amplitude damping channel [21], and its successive use with- out resetting (quantum memory channel) was investigated for quantum and classical communication [30]. Different flavors of the spin-star system, with both diago- nal and non-diagonal coupling, have been used to study top- ics such as entanglement distribution [45], the dynamics of entanglement of two central spins [46], and analytically solv- able models of decoherence [38, 47, 48]. However, spin-star systems are so far unexplored for quantum transmission of information, and this is our goal in the present paper. The system qubit in our communication model is represented by a spin located at the center of the star. It interacts with all non-central spins, comprising the environment, via Ising cou- plings. This provides a non-Markovian quantum dephasing channel whose dynamics can be solved exactly [38]. We allow arbitrary couplings between the system and environment spins and unlike the spin chain channels studied in Refs. [21, 30], obtain analytical expressions for the capacities of this model. We do not consider the quantum memory channel setting of successive channel uses, wherein a new spin is repeatedly in- troduced into the same channel [28]. Rather, we consider the parallel use setting of a memoryless channel [21], where n messages (classical or quantum) are simultaneously transmit- ted over n identical spin-star systems. Thus, in our treatment, non-Markovian memory effects are entirely associated with the non-Markovian dynamics of each spin-star system. The organization of the paper is as follows. In Sec. II we give a brief review of quantum channels and their capacities. In Sec. III we describe the model of a quantum dephasing channel obtained by coupling a system spin via Ising inter- actions to a spin bath, and review its exact solution in the Kraus representation. In Sec. IV we present our communi- cation model, calculate its capacities and study some limiting cases. Finally, in Sec. V we discuss the results and present our conclusions. Appendix A contains a technical calculation. Formally, a quantum channel E is a completely positive and trace preserving map (CPTP) of a quantum system from an initial system state ρ$_{S}$ to the final state E ( ρ$_{S}$ ) [2, 31]. Quan- tum channels arise by joint unitary evolution U of the system I. INTRODUCTION arXiv:1001.1785v1 [quant-ph] 12 Jan 2010 II. QUANTUM CHANNEL CAPACITIES