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Optimal pure state cloning and transposition are complementary channels

Vanessa Brzić, Dmitry Grinko, Michał Studziński, Marco Túlio Quintino

Abstract

State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. We first determine the optimal fidelity for transforming $N$ qudits into $K$ copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding $N \to K$ transposition map is the complementary channel of the optimal universal symmetric $N \to N + K$ quantum cloning machine on pure states. We then present an explicit quantum circuit that realises $N \to K$ transposition and $N \to N + K$ cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state $N \to 1$ qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.

Optimal pure state cloning and transposition are complementary channels

Abstract

State cloning and state transposition are fundamental transformations which, despite being desirable, cannot be perfectly realised due to two conceptually distinct constraints of quantum theory: cloning is forbidden by linearity, while transposition is ruled out by complete positivity. In this work, we show that, despite these different constraints, the best physically allowed realisation of both transformations arises from a single physical process described by an isometry, which simultaneously implements their best possible approximations. We first determine the optimal fidelity for transforming qudits into copies of their transposition and show that, for pure input states, it is achieved by an estimation strategy, which is the unique optimal strategy under the worst-case fidelity figure of merit. We further prove that the corresponding transposition map is the complementary channel of the optimal universal symmetric quantum cloning machine on pure states. We then present an explicit quantum circuit that realises transposition and cloning in parallel and analyse its gate efficiency. Finally, we investigate mixed-state qudit transposition and determine its maximal performance in terms of white-noise visibility, yielding the structural physical approximation of transposition in the multicopy regime.
Paper Structure (17 sections, 14 theorems, 159 equations, 1 figure)

This paper contains 17 sections, 14 theorems, 159 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Optimal approximation of $N\! \to\! K$ pure state transposition
  3. An SDP formulation of the problem
  4. Covariance properties of the problem
  5. Proof of optimality
  6. Attainability of the optimal value
  7. Attainability via estimation
  8. A tight upper bound from the dual problem
  9. Uniqueness properties
  10. Single-site output state structure
  11. Optimal $N\! \to\! K$ pure state transposition and $N\to N+K$ pure state cloning are complementary channels
  12. Optimal $N\to 1$ mixed state transposition
  13. Representation theory background
  14. Symmetric group $S_n$
  15. Schur--Weyl duality
  16. ...and 2 more sections

Key Result

Theorem 1

The maximum average fidelity for approximating the transpose map from $N$ input copies to $K$ output copies on pure states is given by where $d_S^N:=\dim(\mathrm{sym}^N(\mathds{C}^d))=\binom{N+d-1}{d-1}$. The optimal performance is attainable by an estimation strategy, which acts on the symmetric subspace as ${\mathcal{T}_\textup{CP}: \mathcal{L}\left(\mathrm{sym}^N\left(\mathds{C}^d\right)\right

Figures (1)

  • Figure 1: A circuit realisation for simultaneous optimal pure state cloning and transposition/conjugation. As detailed in the main text, $U_\text{Sch}$ is the unitary Schur transformation and CG stands for the Clebsch--Gordan transform. The top $N+K$ output wires of the output correspond to cloning (when other bottom wires are traced out), and bottom $K$ wires of the output correspond to transposition/conjugation (when other top wires are traced out).

Theorems & Definitions (26)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Theorem 4
  • proof
  • Theorem 4
  • proof
  • ...and 16 more