Quantum operation

Builds on Hilbert space Trace class operator Completely positive map Positive operator Density matrix

Definition

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be complex Hilbert spaces, and let $\mathcal{T}(\mathcal{H})$ denote the trace class operators on $\mathcal{H}$ . A quantum operation from $A$ to $B$ is a linear map $\mathcal{E} \colon \mathcal{T}(\mathcal{H}_A) \to \mathcal{T}(\mathcal{H}_B)$ such that:

$\mathcal{E}$ is completely positive: for every $n \ge 1$ , the map $\mathcal{E} \otimes \id_n$ on $\mathcal{T}(\mathcal{H}_A \otimes \C^n)$ sends positive operators to positive operators;
$\mathcal{E}$ does not increase the trace: $\tr \mathcal{E}(\rho) \le \tr \rho$ for every positive $\rho \in \mathcal{T}(\mathcal{H}_A)$ .

For a density operator $\rho$ with $\tr \mathcal{E}(\rho) > 0$ , the number $\tr \mathcal{E}(\rho)$ is the probability that the process occurs, and the output state is $\mathcal{E}(\rho)/\tr \mathcal{E}(\rho)$ .

Remarks

Example: for a projection $P$ on $\mathcal{H}_A = \mathcal{H}_B$ , the map $\mathcal{E}(\rho) = P\rho P$ is a quantum operation. It is completely positive since $(P \otimes I)X(P \otimes I) \ge 0$ for $X \ge 0$ , and $\tr(P\rho P) = \tr(P\rho) \le \tr \rho$ for $\rho \ge 0$ . Non-example: the transpose map $\rho \mapsto \rho^T$ on $\C^{2 \times 2}$ preserves positivity and the trace, but clause 1 fails for $n = 2$ . Indeed, applied to $\psi\psi^*$ with $\psi = e_0 \otimes e_0 + e_1 \otimes e_1$ , it gives the swap operator, which has the eigenvalue $-1$ . In finite dimensions every quantum operation has the Kraus form $\mathcal{E}(\rho) = \sum_k K_k \rho K_k^*$ with $\sum_k K_k^* K_k \le I$ .

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Tag	`B9N3V8Q4Z2`
URI	`m2://B9N3V8Q4Z2`
Kind	definition
Writing	W0 stub
MSC	81P47, 81P16, 46L60
Defines	Quantum operation
Uses	Hilbert space, Trace class operator, Completely positive map, Positive operator, Density matrix
Changed	3 November 2026

Quantum operation

Definition

Remarks

Reading path

Further reading

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