# Quantum Tomography: From Markovianity to Non-Markovianity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Basic Markovian Tomography Framework

- (1)
- Experiment preparation. Prepare a set of experiments, where each experiment consists of the quantum state, circuit, and measurement.
- (2)
- Data collection. Execute the prepared experiments and record the measurement samples.
- (3)
- Tomography reconstruction. Reconstruct the tomographic target by performing a post-process algorithm based on the collected data.

#### 3.1. Basic QST

#### 3.2. Basic QPT

#### 3.3. Basic GST

## 4. Basic Non-Markovian Tomography Framework

- (P1)
- Linearity. ${\mathcal{T}}_{0:k}\left(a\mathbf{A}+b\mathbf{B}\right)={\mathcal{T}}_{0:k}\left(a\mathbf{A}\right)+{\mathcal{T}}_{0:k}\left(b\mathbf{B}\right)$, for any $a.b\in \mathbb{R}$.
- (P2)
- Complete positivity. ${\mathcal{T}}_{0:k}\otimes {\mathcal{I}}_{anc}\left({\mathbf{A}}_{S,anc}\right)\ge 0$, where ${\mathcal{I}}_{anc}$ is the identity process on the ancilla, for any instruments ${\mathbf{A}}_{S,anc}$ act on the system and the ancilla.
- (P3)
- Containment. ${\mathcal{T}}_{i:j}$ is contained in ${\mathcal{T}}_{l:k}$, where $l\le i\le j\le k$.

#### 4.1. Basic PTT

#### 4.2. Basic IST

## 5. Maximum Likelihood Estimation-Based Methods

**State:**A quantum state $\widehat{\rho}$ is constrained to be completely positive, meaning its density matrix must be positive semi-definite with a unit trace,

**Process:**A quantum process is constrained to be CPTP. The CP constraints require the Choi state of $\widehat{\mathcal{G}}$ to be completely positive,

**Measurement:**A quantum measurement $\widehat{\mathcal{M}}$ and its complementary are constrained by a complete positive that

**Process tensor:**The process tensor $\widehat{\mathcal{T}}$ should be constrained by CP and causality. The CP constraints require the Choi state of the process tensor to be positive semi-definite,

## 6. Further Developments

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Examples

**Example**

**A1**

**(CP**

**violation)**.

## Appendix B. Likelihood Function

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**Figure 1.**Schematic view of quantum tomography. For an unknown quantum device, which can be a quantum state preparation circuit, quantum process, etc., one can obtain the explicit representation with some SPAM circuits.

**Figure 2.**Model of operational open quantum process. Components marked by green blocks are inaccessible to the experimenter, while instruments marked by red blocks are the only accessible components.

**Figure 3.**Workflow of MLE-based quantum tomography. Components are specified in the Markovian settings for facilitating understanding, although the workflow is general for both Markovian and non-Markovian situations.

**Figure 4.**Utilizing the black box to represent the whole space of the tomographic target, the space of physical implementable results is represented by the blue region. The green dot is the ideal tomographic target to be characterized. Due to the probabilistic sampling, the sampled target may lie in the red region. Assuming the sampled target to be ∗ out of physical implementable space, the MLE result lies on the boundary of the implementable region, which can be represented by ×.

Terminology Type | Abbreviation | Explanation |
---|---|---|

Tomography | (L)QST | (Linear inverse) quantum state tomography |

(L)QPT | (Linear inverse) quantum process tomography | |

(L)GST | (Linear inverse) gate set tomography | |

PPT | Process tensor tomography | |

IST | Instrument set tomography | |

SPAM | State preparation and measurement | |

Mathematical | CP | Completely positive |

TP | Trace-preserving | |

PTM | Pauli transfer matrix | |

CJI | Choi–Jamiolkowski isomorphism | |

MLE | Maximum likelihood estimation | |

Noise-related | SE | System–environment |

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**MDPI and ACS Style**

Luan, T.; Li, Z.; Zheng, C.; Kuang, X.; Yu, X.; Zhang, Z.
Quantum Tomography: From Markovianity to Non-Markovianity. *Symmetry* **2024**, *16*, 180.
https://doi.org/10.3390/sym16020180

**AMA Style**

Luan T, Li Z, Zheng C, Kuang X, Yu X, Zhang Z.
Quantum Tomography: From Markovianity to Non-Markovianity. *Symmetry*. 2024; 16(2):180.
https://doi.org/10.3390/sym16020180

**Chicago/Turabian Style**

Luan, Tian, Zetong Li, Congcong Zheng, Xueheng Kuang, Xutao Yu, and Zaichen Zhang.
2024. "Quantum Tomography: From Markovianity to Non-Markovianity" *Symmetry* 16, no. 2: 180.
https://doi.org/10.3390/sym16020180