# Fundamental limitations in the purifications of tensor networks

## Abstract

We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

- Authors:

- Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching (Germany)
- Department of Computer Science, University College London, London WC1E 6BT, United Kingdom and DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
- Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany)
- Departamento de Análisis Matemático and IMI, Universidad Complutense de Madrid, 28040 Madrid, Spain and ICMAT, C/ Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid (Spain)

- Publication Date:

- OSTI Identifier:
- 22596619

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; MANY-BODY PROBLEM; MATRICES; MIXED STATES; TENSORS

### Citation Formats

```
De las Cuevas, G., Cirac, J. I., Cubitt, T. S., Wolf, M. M., and Pérez-García, D.
```*Fundamental limitations in the purifications of tensor networks*. United States: N. p., 2016.
Web. doi:10.1063/1.4954983.

```
De las Cuevas, G., Cirac, J. I., Cubitt, T. S., Wolf, M. M., & Pérez-García, D.
```*Fundamental limitations in the purifications of tensor networks*. United States. doi:10.1063/1.4954983.

```
De las Cuevas, G., Cirac, J. I., Cubitt, T. S., Wolf, M. M., and Pérez-García, D. Fri .
"Fundamental limitations in the purifications of tensor networks". United States.
doi:10.1063/1.4954983.
```

```
@article{osti_22596619,
```

title = {Fundamental limitations in the purifications of tensor networks},

author = {De las Cuevas, G. and Cirac, J. I. and Cubitt, T. S. and Wolf, M. M. and Pérez-García, D.},

abstractNote = {We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.},

doi = {10.1063/1.4954983},

journal = {Journal of Mathematical Physics},

number = 7,

volume = 57,

place = {United States},

year = {Fri Jul 15 00:00:00 EDT 2016},

month = {Fri Jul 15 00:00:00 EDT 2016}

}

DOI: 10.1063/1.4954983

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