The operator-sum-difference representation of a quantum noise channel. (English) Zbl 1317.81050
Summary: When a model for quantum noise is exactly solvable, a Kraus (or operator-sum) representation can be derived from the spectral decomposition of the Choi matrix for the channel. More generally, a Kraus representation can be obtained from any positive-sum (or ensemble) decomposition of the matrix. Here we extend this idea to any Hermitian-sum decomposition. This yields what we call the “operator-sum-difference” (OSD) representation, in which the channel can be represented as the sum and difference of “subchannels.” As one application, the subchannels can be chosen to be analytically diagonalizable, even if the parent channel is not (on account of the Abel-Galois irreducibility theorem), though in this case the number of the OSD representation operators may exceed the channel rank. Our procedure is applicable to general Hermitian (completely positive or non-completely positive) maps and can be extended to the more general, linear maps. As an illustration of the application, we derive an OSD representation for a two-qubit amplitude-damping channel.
MSC:
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
| 94A40 | Channel models (including quantum) in information and communication theory |
| 46L07 | Operator spaces and completely bounded maps |
References:
| [1] | Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002) · Zbl 1053.81001 |
| [2] | Weiss, U.: Quantum Dissipative Systems. World Scientific, Singapore (2008) · Zbl 1166.81005 · doi:10.1142/6738 |
| [3] | Banerjee, S., Ghosh, R.: Dynamics of decoherence without dissipation in a squeezed thermal bath. J. Phys. A: Math. Theor. 40, 13735 (2007) · Zbl 1129.81013 · doi:10.1088/1751-8113/40/45/014 |
| [4] | Srikanth, R., Banerjee, S.: Squeezed generalized amplitude damping channel. Phys. Rev. A 77, 012318 (2008) · doi:10.1103/PhysRevA.77.012318 |
| [5] | Omkar, S., Srikanth, R., Banerjee, S.: Dissipative and non-dissipative single-qubit channels: dynamics and geometry. Quant. Info. Proc. 12, 3725 (2013) · Zbl 1303.81039 · doi:10.1007/s11128-013-0628-3 |
| [6] | Turchette, Q.A., Myatt, C.J., King, B.E., Sackett, C.A., et al.: Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs. Phys. Rev. A 62, 053807 (2000) · doi:10.1103/PhysRevA.62.053807 |
| [7] | Brune, M., Hagley, E., Dreyer, J., Maitre, X., et al.: Observing the progressive decoherence of the meter in a quantum measurement. Phys. Rev. Lett. 77, 4887 (1996) · doi:10.1103/PhysRevLett.77.4887 |
| [8] | Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015 |
| [9] | Kraus, K.: States, Effects and Operations. Springer, Berlin (1983) · Zbl 0545.46049 |
| [10] | Choi, M.D.: Positive linear maps on c*-algebras. Can. J. Math. 24, 520 (1972) · Zbl 0235.46090 · doi:10.4153/CJM-1972-044-5 |
| [11] | Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285-290 (1975) · Zbl 0327.15018 · doi:10.1016/0024-3795(75)90075-0 |
| [12] | Jordan, T.F., Shaji, A., Sudarshan, E.C.G.: Dynamics of initially entangled open quantum systems. Phys. Rev. A 70, 052110 (2004) · Zbl 1227.82041 · doi:10.1103/PhysRevA.70.052110 |
| [13] | Leung, D.W.: Choi’s proof and quantum process tomography. J. Math. Phys. 44, 528-33 (2003) · Zbl 1061.81004 · doi:10.1063/1.1518554 |
| [14] | Ficek, Z., Tanaś, R.: Entangled states and collective nonclassical effects in two-atom systems. Phys. Rep. 372, 369 (2002) · Zbl 0999.81009 · doi:10.1016/S0370-1573(02)00368-X |
| [15] | Artin, E.: Galois Theory. Dover, (2004) · Zbl 1227.82041 |
| [16] | Banerjee, S., Ravishankar, V., Srikanth, R.: Entanglement dynamics in two-qubit open system interacting with a squeezed thermal bath via quantum nondemolition interaction. Euro. Phys. J. D. 56, 277 (2010) · Zbl 1186.81014 · doi:10.1140/epjd/e2009-00286-2 |
| [17] | Banerjee, S., Ravishankar, V., Srikanth, R.: Dynamics of entanglement in two-qubit open system interacting with a squeezed thermal bath via dissipative interaction. Ann. Phys. 325, 816 (2010) · Zbl 1186.81014 · doi:10.1016/j.aop.2010.01.003 |
| [18] | Shabani, A., Lidar, D.A.: Maps for general open quantum systems and a theory of linear quantum error correction. Phys. Rev. A 80, 012309 (2009) · doi:10.1103/PhysRevA.80.012309 |
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