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# Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

## Abstract

The trace distance between two quantum states, ρ and σ, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of ρ–σ. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either ρ or σ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Here, our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1565902

- Alternate Identifier(s):
- OSTI ID: 1547973

- Report Number(s):
- LA-UR-19-22724

Journal ID: ISSN 2469-9926; PLRAAN

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review A

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 2; Journal ID: ISSN 2469-9926

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Information Science; Mathematics

### Citation Formats

```
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, and Cincio, Lukasz. Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.100.022103.
```

```
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, & Cincio, Lukasz. Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states. United States. doi:10.1103/PhysRevA.100.022103.
```

```
Coles, Patrick Joseph, Cerezo, Marco Vinicio Sebastain, and Cincio, Lukasz. Tue .
"Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states". United States. doi:10.1103/PhysRevA.100.022103.
```

```
@article{osti_1565902,
```

title = {Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states},

author = {Coles, Patrick Joseph and Cerezo, Marco Vinicio Sebastain and Cincio, Lukasz},

abstractNote = {The trace distance between two quantum states, ρ and σ, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of ρ–σ. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either ρ or σ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Here, our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.},

doi = {10.1103/PhysRevA.100.022103},

journal = {Physical Review A},

number = 2,

volume = 100,

place = {United States},

year = {2019},

month = {8}

}

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