Fundamental limitations in the purifications of tensor networks

Journal Article · Fri Jul 15 00:00:00 EDT 2016 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4954983· OSTI ID:22596619

De las Cuevas, G.; Cirac, J. I. ^[1]; Cubitt, T. S. ^[2]; Wolf, M. M. ^[3]; Pérez-García, D. ^[4]

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1. Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching (Germany)
2. 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)
3. Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany)
4. Departamento de Análisis Matemático and IMI, Universidad Complutense de Madrid, 28040 Madrid, Spain and ICMAT, C/ Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid (Spain)

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.

• OSTI ID: 22596619
• Journal Information: Journal of Mathematical Physics, Vol. 57, Issue 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
MANY-BODY PROBLEM
MATRICES
MIXED STATES
TENSORS