Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 1

A quantum channel is a linear completely positive trace-preserving map. Its Kraus representation isKraus operators are not unique: two representations of the same channel are related by an isometry on the Kraus index, and by a unitary when both lists are minimal and equally long. A differentiable Markovian family of infinitesimal channels gives the Lindblad equation

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Strict trace-distance contraction means that for some $\eta <1$,for all states. Equivalently, the induced trace norm on nonzero traceless Hermitian operators is strictly below one. This immediately gives a unique fixed state by the contraction mapping theorem.

The standard algebraic condition is that the channel be a primitive quantum channel: some finite productsspan the full matrix algebra, equivalently some power of the channel maps every nonzero positive operator to a positive-definite one. Spectrally, eigenvalue $1$ is then simple and every other eigenvalue has modulus less than one. This condition is necessary and sufficient for convergence to a unique full-rank fixed point. Irreducibility alone gives uniqueness but may leave periodic peripheral eigenvalues.

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Let ${\rho }_{\ast }=|{\psi }^{-}⟩⟨{\psi }^{-}|$ and chooseFor any orthonormal two-qubit basis $\{|\mu ⟩{\}}_{\mu =1}^4$, a Kraus representation isThe completeness relation holds and $\mathcal{E}({\rho }_{\ast })={\rho }_{\ast }$. Differences of states are traceless, soThe channel is strictly contractive with coefficient $1/2$. On operator space, ${\rho }_{\ast }$ spans the eigenvalue-one direction and every traceless operator has eigenvalue $1/2$, hence

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The Stinespring dilation isometry isApplying it successively to fresh environment systems givesContracting the final system with a boundary vector turns every amplitude into a boundary contraction of the matrices $K_{\alpha }$. This is a matrix product state with bond dimension at most the system dimension.

For an injective MPS, the fundamental gauge freedom istogether with the inverse transformation of boundary vectors; an overall phase is also immaterial. Its channel converges to a unique fixed state exactly when eigenvalue one is simple and there are no other peripheral eigenvalues, equivalently when it is a primitive quantum channel. A unique fixed state without convergence only requires the eigenvalue-one eigenspace itself to be one-dimensional.

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