## Theory of Quantum State Generation and Discrimination

Staff members: | Dr. Ulrike Herzog |

## 1. Theory of biphoton generation in a single-resonant OPO far below threshold

In parametric down-conversion, a pump photon is split into two photons with lower frequency. The resulting two-photon state, consisting of a signal photon and an idler photon , is called a biphoton. Upon post-selection on the idler photons, the process of spontaneous parametric down-conversion provides a source of heralded single photons. For an efficient atom-photon coupling, as required for several recently proposed quantum networks, the photon bandwidth has to be by orders of magnitude smaller than the bandwidth of the photons emitted in spontaneous parametric down-conversion. In order to reduce the photon bandwidth, parametric down-conversion in a double-resonant optical parametric oscillator (OPO) far below threshold with subsequent spectral filtering by means of an external cavity has been applied [J. S. Neergaard-Nielsen *et. al.*, Opt. Exp. **15**, 7940 (2007)]. Continuous biphoton generation in a single-resonant OPO far below threshold has been demonstrated in our group using a setup where only the signal mode experiences resonance enhancement in the cavity while the orthogonally polarized idler mode is nonresonant due to deflection by an intra-cavity polarizing beam splitter [M. Scholz *et al.*, Appl. Phys. Lett. **91**, 191104 (2007)]. Additional spectral filtering can produce narrow-band photons.

Different from the approaches used for the theoretical description of the double-resonant OPO [Y. J. Lu and Z. Y. Ou, Phys. Rev. A **62**, 033804 (2000)], free-field quantization of the idler field is inevitable for a theoretical treatment in the single-resonant case. In our theoretical investigations [1] we derive expressions for the biphoton production rate and the biphoton wave function, and we determine both the spectral properties of the emitted radiation and the second-order signal-idler cross-correlation function for a single-resonant OPO far below threshold.

[1] U. Herzog, M. Scholz, and O. Benson, *Theory of biphoton generation in a single-resonant optical parametric oscillator far below threshold*, Phys. Rev. A **77**, 023826 (2008).

## 2. Theory of quantum state discrimination and identification

Quantum information is encoded in the state of a quantum system, and therefore different states have to be distinguished in order to read out the information. Hence quantum state discrimination [1] is a fundamental problem for many tasks in quantum communication, cryptography and computing. In the standard problem, a quantum system is prepared, with given prior probability, in a definite state out of a set of given states which are known, and the actual state of the system has to be determined. When the possible states are non-orthogonal, it is impossible to devise a measurement that can distinguish between them perfectly. Therefore measurement strategies have been developed that are optimized with respect to various criteria. Several of these strategies have been experimentally realized by means of linear optical measurements using weak laser light to represent single-photon states. (R. Clarke *et al.*, Phys. Rev. A **63**, 040305(R ), 64, 012303 (2001), Mohseni *et al.*, Phys. Rev. Lett. **93**, 200403 (2004)] .

The earliest and simplest of the optimization criteria is the requirement that the overall probability of getting a wrong result be as small as possible, with inconclusive results being forbidden and all states being individually distinguished. In the corresponding measurement strategy, called **minimum-error discrimination**, based on the outcome of the measurement a guess is made as to what the state of the quantum system was.

Another basic strategy requires that, whenever a definite outcome is returned after the state-distinguishing measurement, the result should be error-free, i.e. unambiguous. This can be achieved at the expense of allowing for a non-zero probability of inconclusive outcomes, where the measurement fails to give a definite answer. When the overall probability of failure is minimized, **optimum unambiguous discrimination** is achieved.

A recent application of the tools developed for optimum state discrimination is the problem of **quantum state identification**, where the system is prepared in a set of pure states that are unknown but that are encoded in a set of reference copies. Despite of the complete lack of information about the involved states themselves, the identification of the state of the system with one of the reference states is possible due to symmetry properties that are intrinsically quantum mechanical. The reference states can be considered as program states in a programmable quantum state discriminator.

The theoretical work performed at the Humboldt-University in the field of quantum state discrimination and identification has been mostly done in collaboration with Janos Bergou from the Hunter College of the City University of New York. It refers to both minimum-error discrimination and optimum unambiguous discrimination. In the latter case, particular attention is given to the discrimination between sets of pure states, or between mixed states, respectively. Moreover, several aspects of quantum state identification have been treated [9].

[1] for an overview, see J. A. Bergou, U. Herzog, and M. Hillery, *Discrimination of Quantum States*, Review article in Lect. Notes Phys. **649**: Quantum State Estimation, M. Paris and J. Rehacek eds., Springer, Berlin (2004)

[2] U. Herzog, *Discrimination between Non-orthogonal Two-photon Polarization states*, Fortschr. Phys. **49**, 981 (2001)

[3] U. Herzog and J. A. Bergou, *Minimum-error discrimination between subsets of linearly dependent quantum states*, Phys. Rev. A **65**, 050305(R) (2002)

[4] J. A. Bergou, U. Herzog, and M. Hillery, *Quantum Filtering and Discrimination between Sets of Boolean Functions*, Phys. Rev. Lett. **90**, 257901 (2003)

[5] U. Herzog, *Minimum-error discrimination between a pure and a mixed two-qubit state*, J. Opt. B 6, **24**, (2004)

[6] U. Herzog and J. A. Bergou, *Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination*, Phys. Rev. A **70**, 022302 (2004)

[7] J. A. Bergou, U. Herzog, and M. Hillery, *Optimal unambiguous filtering of a quantum state: An instance in mixed state discrimination*, Phys. Rev. A **71**, 042314 (2005)

[8] U. Herzog and J. A. Bergou, *Optimum unambiguous discrimination between two mixed quantum states*, Phys. Rev. A **71**, 050301(R) (2005)

[9] J. A. Bergou, V. Buzek, E. Feldman, U. Herzog, and M. Hillery, *Programmable quantum state discriminators with simple programs*, Phys. Rev. A **73**, 062334 (2006)

[10] U. Herzog, *Optimum unambiguous discrimination of two mixed quantum states and application to a class of similar states*, Phys. Rev. A **75**, 052309 (2007)