Channel-state duality

Summary

In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from A to Cn×n, where A is a C*-algebra and Cn×n denotes the n×n complex entries, and positive linear functionals (states) on the tensor product

Details

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Let H1 and H2 be (finite-dimensional) Hilbert spaces. The family of linear operators acting on Hi will be denoted by L(Hi). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in L(Hi) respectively. A quantum channel, in the Schrödinger picture, is a completely positive (CP for short), trace-preserving linear map

 

that takes a state of system 1 to a state of system 2. Next, we describe the dual state corresponding to Φ.

Let Ei j denote the matrix unit whose ij-th entry is 1 and zero elsewhere. The (operator) matrix

 

is called the Choi matrix of Φ. By Choi's theorem on completely positive maps, Φ is CP if and only if ρΦ is positive (semidefinite). One can view ρΦ as a density matrix, and therefore the state dual to Φ.

The duality between channels and states refers to the map

 

a linear bijection. This map is also called Jamiołkowski isomorphism or Choi–Jamiołkowski isomorphism.

Applications

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This isomorphism is used to show that the "Prepare and Measure" Quantum Key Distribution (QKD) protocols, such as the BB84 protocol devised by C. H. Bennett and G. Brassard[1] are equivalent to the "Entanglement-Based" QKD protocols, introduced by A. K. Ekert.[2] More details on this can be found e.g. in the book Quantum Information Theory by M. Wilde.[3]

References

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  1. ^ C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing”, Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, 175 (1984)
  2. ^ Ekert, Artur K. (1991-08-05). "Quantum cryptography based on Bell's theorem". Physical Review Letters. 67 (6). American Physical Society (APS): 661–663. Bibcode:1991PhRvL..67..661E. doi:10.1103/physrevlett.67.661. ISSN 0031-9007. PMID 10044956.
  3. ^ M. Wilde, "Quantum Information Theory" - Cambridge University Press 2nd ed. (2017), §22.4.1, pag. 613