Random purification channel made simple (2511.23451v1)

Abstract: The recently introduced random purification channel, which converts $n$ i.i.d. copies of any mixed quantum state into a uniform convex combination of $n$ i.i.d. copies of its purifications, has proved to be an extremely useful tool in quantum learning theory. Here we give a remarkably simple construction of this channel, making its known properties -- and several new ones -- immediately transparent. In particular, we show that the channel also purifies non-i.i.d. states: it transforms any permutationally symmetric state into a uniform convex combination of permutationally symmetric purifications, each differing only by a tensor-product unitary acting on the purifying system. We then apply the channel to give a one-line proof of (a stronger version of) the recently established Uhlmann's theorem for quantum divergences.

• The paper introduces a simplified construction of the random purification channel, clarifying its structure for both i.i.d. and symmetric inputs.
• The work proves a one-line derivation of the Uhlmann theorem for quantum divergences, broadening its applicability to various divergence measures.
• The channel leverages Haar randomness and symmetry to uniformly generate purifications, facilitating practical advancements in quantum tomography and learning.

A Simple Construction and Extended Properties of the Random Purification Channel

Introduction and Context

The random purification channel, first introduced in recent works by Tang, Wright, and Zhandry, is a quantum channel that takes $n$ i.i.d. copies of a mixed state, $\rho_A$, and outputs $n$ i.i.d. copies of a uniformly random purification of $\rho_A$ (Tang et al., 8 Oct 2025). This channel has significant relevance in quantum learning theory, particularly in reducing mixed state tomography to pure state tomography (Pelecanos et al., 19 Nov 2025). Prior constructions of this channel leveraged intricate machinery from representation theory and Schur–Weyl duality. The present work provides a radically simplified, explicit construction of the channel, elucidating both its structure and a variety of new properties (2511.23451).

Explicit Construction of the Random Purification Channel

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be isomorphic Hilbert spaces. The channel,

$$\Lambda^{(n)}(\cdot) \coloneqq \sqrt{R_n} \big( \cdot \otimes 1_{B^n} \big) \sqrt{R_n},$$

is constructed using a positive semi-definite operator $R_n$ defined as the expectation over Haar-random unitaries $U_B$ acting on $\mathcal{H}_B$:

$$R_n \coloneqq \mathbb{E}_{U_B} \Big[ (1_{A^n} \otimes U_B^{\otimes n}) \, \Gamma_{AB}^{\otimes n} \, (1_{A^n} \otimes (U_B^\dagger)^{\otimes n}) \Big],$$

where $\Gamma_{AB}$ is the unnormalized maximally entangled state $\sum_i |i\rangle_A \otimes |i\rangle_B$.

This channel is completely positive and trace-preserving, and as proved in the paper, precisely coincides with the formulation of the random purification channel of (Tang et al., 8 Oct 2025, Pelecanos et al., 19 Nov 2025). The explicit construction makes immediate several of its salient properties and further generalizes its applicability.

Structural and Operational Characterization

A core property is that for any permutationally symmetric state $\rho_{A^n}$,

$$\Lambda^{(n)}(\rho_{A^n}) = \mathbb{E}_{U_B}\Big[ (1_A \otimes U_B^{\otimes n}) \, (\psi_\rho)_{A^nB^n}\, (1_A \otimes U_B^{\dagger\otimes n}) \Big],$$

where $(\psi_\rho)_{A^nB^n}$ is a fixed (permutationally symmetric) purification of $\rho_{A^n}$. For $\rho_{A^n} = \rho_A^{\otimes n}$, this corresponds to $n$ copies of a Haar-random purification of $\rho_A$.

A key result is the extension to inputs that are not strictly i.i.d.: for arbitrary permutationally symmetric inputs, the channel outputs uniform convex combinations of permutationally symmetric purifications differing only by a tensor-product unitary on the purifying subsystem. This property was not previously established.

Applications to Quantum Divergences

The channel admits new applications outside its original context. The most prominent is a one-line proof of the Uhlmann theorem for quantum divergences, substantially strengthening the result in [Mazzola et al., IEEE Trans. Inf. Theory 71, 7039 (2025)]. For any divergence $\mathbb{D}$ satisfying the data processing inequality and a weak quasi-concavity property (which includes the Umegaki relative entropy, sandwiched and measured Rényi divergences, and their variants), and for any extension $\rho_{AB}$ of $\rho_A$, the following holds:

$$\mathbb{D}^\infty(\rho_A \Vert \sigma_A) = \mathbb{D}^\infty(\rho_{AB} \Vert \mathcal{C}^{\sigma_A}_{AB}),$$

where $\mathcal{C}^{\sigma_A}_{AB}$ denotes the set of all extensions of $\sigma_A$ to $AB$. The optimizing sequence of extensions is given explicitly by applying the channel $\Lambda_{\mathrm{purify}}^{(n)}$ followed by an arbitrary channel acting on the purifying system.

This result applies not only to Umegaki relative entropy but, crucially, to additivity and regularized versions of all divergences satisfying the weak quasi-concavity property, a much broader class than previously considered.

Strong Statements and Immediate Corollaries

• Generalization beyond i.i.d. inputs: The channel acts on all permutationally symmetric states, mapping them to mixtures of symmetric purifications, not only for i.i.d. or product inputs but in full generality.
• Uhlmann's theorem for divergences: The identity holds for all additive divergences in the regularized sense, with the universal sequence of optimizers given by the output of the random purification channel.
• Efficiency of the construction: The new explicit construction avoids the technical machinery of Schur–Weyl duality, simplifying both theoretical analysis and practical implementation.

In addition, the use of Haar measure invariance and symmetry arguments proves that the channel and its output are independent of the specific purification chosen, as long as it purifies the input.

Implications and Theoretical Significance

The random purification channel, as now explicitly constructed, provides a conceptual bridge connecting state symmetrization, purification, and group representation averaging in quantum information. Its application to proofs of equality for regularized divergences under extensions is immediate and modular, as demonstrated by the compact and elementary derivation of the Uhlmann theorem for divergences. This generalizes the relationship between the distinguishability of states and their purifications in asymptotic analysis.

From a practical perspective, the new construction suggests explicit protocols for quantum learning, tomography, and Shannon-theoretic tasks where purification or symmetrization is advantageous. Furthermore, the ability to handle arbitrary symmetric inputs paves the way for applications in tasks involving de Finetti-type symmetrizations and analysis of symmetric quantum protocols.

Theoretically, the generalization to broad classes of divergences opens avenues for the axiomatic treatment of quantum resource theories and further reveals the intricate structure of state extension problems and their asymptotics.

Future extensions may target operational interpretations in quantum channel simulation, generalized resource manipulation, and potentially extend to infinite-dimensional settings or general probabilistic theories. The random purification paradigm, as a result of this work, may find algorithmic or complexity-theoretic applications in both classical and quantum learning settings.

Conclusion

This work significantly clarifies the structure and operational properties of the random purification channel by providing a simplified explicit construction and establishing its action on a broad class of inputs. The extended results on quantum divergences, notably a strengthened, universally applicable version of the Uhlmann theorem, demonstrate the deep, structural interrelations between purification, symmetry, and quantum information measures. The techniques and results will serve as foundational tools for continued investigation into quantum learning, hypothesis testing, resource theory, and the mathematics of symmetric operations in quantum information theory (2511.23451).