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Random purification channel made simple

Filippo Girardi, Francesco Anna Mele, Ludovico Lami

TL;DR

The paper addresses simplifying the random purification channel and broadening its applicability beyond iid inputs. It provides a transparent, elementary construction using R_n and Haar randomness, avoiding heavy representation-theoretic machinery. It shows the channel preserves purifications for permutationally symmetric inputs and yields a concise, universal proof of Uhlmann's theorem for a broad class of divergences. These results streamline quantum Shannon-theory arguments and extend tools for quantum learning and state discrimination.

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.

Random purification channel made simple

TL;DR

The paper addresses simplifying the random purification channel and broadening its applicability beyond iid inputs. It provides a transparent, elementary construction using R_n and Haar randomness, avoiding heavy representation-theoretic machinery. It shows the channel preserves purifications for permutationally symmetric inputs and yields a concise, universal proof of Uhlmann's theorem for a broad class of divergences. These results streamline quantum Shannon-theory arguments and extend tools for quantum learning and state discrimination.

Abstract

The recently introduced random purification channel, which converts i.i.d. copies of any mixed quantum state into a uniform convex combination of 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.

Paper Structure

This paper contains 8 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. A simple construction of the random purification channel
  3. Uhlmann's theorem for divergences (in one line)
  4. Divergences and weak quasi-concavity
  5. Ulhmann's theorem for divergences
  6. Some proofs
  7. Proof of Proposition \ref{['prop:weak']}
  8. Proof of Lemma \ref{['lem:quasi_conc']}

Key Result

Lemma 1

Let $\mathcal{H}_A$ be an Hilbert space. For any $n\geq 1$, there exists a quantum channel $\Lambda_{\rm purify}^{(n)}:\mathcal{L}(\mathcal{H}_A^{\otimes n})\to \mathcal{L}((\mathcal{H}_A\otimes\mathcal{H}_B)^{\otimes n})$, where $\mathcal{H}_B$ is isomorphic to $\mathcal{H}_A$, such that, for any a where the expectation is taken over Haar-random unitaries $U_B$ on $\mathcal{H}_B$, $(\psi_\rho)_{A

Theorems & Definitions (28)

  • Lemma 1: (Random purification channel)
  • Lemma 2
  • proof
  • Theorem 3: (A simple construction of the random purification channel)
  • proof
  • Definition 4: (Divergence)
  • Definition 5: (Weak quasi-concavity)
  • Remark 6: (Quasi-concavity implies weak quasi-concavity)
  • Definition 7: (Sandwiched Rényi divergences newRenyiWilde2014)
  • Definition 8: (Measured Rényi divergences newRenyiWilde2014)
  • ...and 18 more