quantum information theory Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...

, the channel-state duality refers to the correspondence between

quantum channel In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information i ...

s and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from ''A'' to C^''n''×''n'', where ''A'' is a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...

and C^''n''×''n'' denotes the ''n''×''n'' complex entries, and positive linear functionals (

state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...

s) on the tensor product :

\mathbb^ \otimes A.

Details

Let ''H''₁ and ''H''₂ be (finite-dimensional) Hilbert spaces. The family of linear operators acting on ''H_i'' will be denoted by ''L''(''H_i''). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in ''L''(''H_i'') respectively. A

, in the Schrödinger picture, is a completely positive (CP for short), trace-preserving linear map :

\Phi : L(H_1) \rightarrow L(H_2)

that takes a state of system 1 to a state of system 2. Next, we describe the dual state corresponding to Φ. Let ''E_{i j}'' denote the matrix unit whose ''ij''-th entry is 1 and zero elsewhere. The (operator) matrix :

\rho_ = (\Phi(E_))_ \in L(H_1) \otimes L(H_2)

is called the ''Choi matrix'' of Φ. By

Choi's theorem on completely positive maps In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belav ...

, Φ is CP if and only if ''ρ''_Φ is positive (semidefinite). One can view ''ρ''_Φ as a density matrix, and therefore the state dual to Φ. The duality between channels and states refers to the map :

\Phi \rightarrow \rho_,

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

Choi–Jamiołkowski isomorphism In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by completely positive map In mathematics a positive map is a map between C*-algebras that ...

Applications

This isomorphism is used to show that the "Prepare and Measure"

Quantum Key Distribution Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then b ...

(QKD) protocols, such as the

BB84 BB84 is a quantum key distribution scheme developed by Charles Bennett and Gilles Brassard in 1984. It is the first quantum cryptography protocol. The protocol is provably secure, relying on two conditions: (1) the quantum property that informat ...

protocol devised by C. H. Bennett and G. Brassard are equivalent to the " Entanglement-Based" QKD protocols, introduced by A. K. Ekert. More details on this can be found e.g. in the book Quantum Information Theory by M. Wilde.M. Wilde, "Quantum Information Theory" - Cambridge University Press 2nd ed. (2017), §22.4.1, pag. 613

References

{{DEFAULTSORT:Channel-State Duality Quantum information theory