Husimi Q representation

The Husimi-Q distribution is a mathematical tool used in physics. The Q-representation is a quasi-probability distribution commonly used to represent the quantum state of light. It is used in the field of quantum optics and particularly for tomographic purposes.

Definition and properties

The Husimi-Q distribution (called Q-function in the context of quantum optics) is the simplest distribution of quasi-probability in phase space. It is defined as

: $Q(alpha)=frac\{1\}\{pi\}langlealpha|\; ho|alpha\; angle,$

which can be interpreted as $pi$ times the probability distribution for the state $\text{'}\; ho\text{'}$ to be found in the state $|alpha\; angle\; langle\; alpha\; |$. It can be used to produce a pictorial representation of the state $\text{'}\; ho\text{'}$ which sometimes make it easier to grasp its properties.

The Q-distribution is normalized to unity,

: $int\; Q(alpha),dalpha^2\; =\; 1$

and is non-negative definite and bounded:

: $0\; leq\; Q(alpha)\; leq\; frac\{1\}\{pi\}.$

References

* Ulf Leonhard, "Measuring the Quantum State of Light", Cambridge Studies in Modern Optics (1997).