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Structure and Classification of Matrix Product Quantum Channels

Giorgio Stucchi, J. Ignacio Cirac, Rahul Trivedi, Georgios Styliaris

Abstract

We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.

Structure and Classification of Matrix Product Quantum Channels

Abstract

We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.
Paper Structure (13 sections, 23 theorems, 106 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 106 equations, 1 figure.

Table of Contents

  1. Quantum Channels Basics
  2. Tensor-Network Formulation of Quantum Channels
  3. Matrix Product Quantum Channels (MPQC)
  4. Local Purification (LP) of Quantum Channels
  5. Inequivalence of the LP and MPQC Classes
  6. Local Purification and Homogeneity
  7. Structure of Homogeneous MPI
  8. Classification for Locally Purifiable Channels
  9. Beyond Homogeneous Channels
  10. Structure of scaled Homogeneous Matrix Product Isometries (sMPI)
  11. Physical implementation of scaled Matrix Product Isometries
  12. sMPIs as MPUs with GHZ input
  13. Locality and Causality Preservation under Matrix Product Isometries

Key Result

Theorem 1

Any hMPI can be written as a depth-two brick wall quantum circuit of isometric gates $u,v$, satisfying $u^\dagger u =\mathbbm{1}_{d_{\mathrm{in}}^2}$ and $v^\dagger v =\mathbbm{1}_{\ell r}$, after blocking at most $D^4$ times: The isometries $u:\mathbb{C}^{d_{\mathrm{in}}^2}\mapsto \mathbb{C}^{\ell r}$ and $v:\mathbb{C}^{\ell r}\mapsto \mathbb{C}^{\chi^2d_{\mathrm{out}}^2}$ satisfy $d_{\mathrm{in

Figures (1)

  • Figure 1: We analyze the structure, classification, and physical implementation of matrix product quantum channels (MPQCs) composed of a repeated tensor $A$. These act on matrix product density operators (MPDOs), preserving their matrix-product form.

Theorems & Definitions (52)

  • Definition 1: Local purification
  • Theorem 1
  • Definition 2: Equivalence
  • Theorem 2
  • Example 1
  • Theorem 3
  • Theorem 4
  • Definition 1: MPQC class
  • Definition 2: LP class
  • Proposition 1
  • ...and 42 more