# Single-photon quantum key distribution in the presence of loss

## Abstract

We investigate two-way and one-way single-photon quantum key distribution (QKD) protocols in the presence of loss introduced by the quantum channel. Our analysis is based on a simple precondition for secure QKD in each case. In particular, the legitimate users need to prove that there exists no separable state (in the case of two-way QKD), or that there exists no quantum state having a symmetric extension (one-way QKD), that is compatible with the available measurements results. We show that both criteria can be formulated as a convex optimization problem known as a semidefinite program, which can be efficiently solved. Moreover, we prove that the solution to the dual optimization corresponds to the evaluation of an optimal witness operator that belongs to the minimal verification set of them for the given two-way (or one-way) QKD protocol. A positive expectation value of this optimal witness operator states that no secret key can be distilled from the available measurements results. We apply such analysis to several well-known single-photon QKD protocols under losses.

- Authors:

- Center for Quantum Information and Quantum Control, Department of Physics and Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G4 (Canada)
- (Canada)
- Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 (Canada)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 20982505

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052336; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DISTRIBUTION; EVALUATION; EXPECTATION VALUE; INFORMATION THEORY; LOSSES; MATHEMATICAL SOLUTIONS; OPTIMIZATION; PHOTONS; QUANTUM CRYPTOGRAPHY; QUANTUM MECHANICS; SECURITY; VERIFICATION

### Citation Formats

```
Curty, Marcos, Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1, Moroder, Tobias, and Quantum Information Theory Group, Institut fuer Theoretische Physik I, and Max-Planck Research Group, Institute of Optics, Information and Photonics, Universitaet Erlangen-Nuernberg, D-91058 Erlangen.
```*Single-photon quantum key distribution in the presence of loss*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.052336.

```
Curty, Marcos, Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1, Moroder, Tobias, & Quantum Information Theory Group, Institut fuer Theoretische Physik I, and Max-Planck Research Group, Institute of Optics, Information and Photonics, Universitaet Erlangen-Nuernberg, D-91058 Erlangen.
```*Single-photon quantum key distribution in the presence of loss*. United States. doi:10.1103/PHYSREVA.75.052336.

```
Curty, Marcos, Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1, Moroder, Tobias, and Quantum Information Theory Group, Institut fuer Theoretische Physik I, and Max-Planck Research Group, Institute of Optics, Information and Photonics, Universitaet Erlangen-Nuernberg, D-91058 Erlangen. Tue .
"Single-photon quantum key distribution in the presence of loss". United States.
doi:10.1103/PHYSREVA.75.052336.
```

```
@article{osti_20982505,
```

title = {Single-photon quantum key distribution in the presence of loss},

author = {Curty, Marcos and Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 and Moroder, Tobias and Quantum Information Theory Group, Institut fuer Theoretische Physik I, and Max-Planck Research Group, Institute of Optics, Information and Photonics, Universitaet Erlangen-Nuernberg, D-91058 Erlangen},

abstractNote = {We investigate two-way and one-way single-photon quantum key distribution (QKD) protocols in the presence of loss introduced by the quantum channel. Our analysis is based on a simple precondition for secure QKD in each case. In particular, the legitimate users need to prove that there exists no separable state (in the case of two-way QKD), or that there exists no quantum state having a symmetric extension (one-way QKD), that is compatible with the available measurements results. We show that both criteria can be formulated as a convex optimization problem known as a semidefinite program, which can be efficiently solved. Moreover, we prove that the solution to the dual optimization corresponds to the evaluation of an optimal witness operator that belongs to the minimal verification set of them for the given two-way (or one-way) QKD protocol. A positive expectation value of this optimal witness operator states that no secret key can be distilled from the available measurements results. We apply such analysis to several well-known single-photon QKD protocols under losses.},

doi = {10.1103/PHYSREVA.75.052336},

journal = {Physical Review. A},

number = 5,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}