# Poisson states in quantum information

This is a Perspective on "Poisson Quantum Information" by Mankei Tsang, published in Quantum 5, 527 (2021).

By Cosmo Lupo (Department of Physics & Astronomy, University of Sheffield).

A Poisson process is a mathematical model used to describe random events occurring in time or space. It has a wide range of applications, including natural sciences, operation research, and telecommunication.

Physicists are familiar with Poisson processes arising from particle decay and detection. For example, consider counting detection events during a time interval of duration $T$. If $\Delta p$ is the probability that a particle is detected within a small time $\Delta t = T/M$, then the probability of observing $n$ particles is given by the binomial distribution,

P(n) = \binom{M}{n} \Delta p^n (1-\Delta p)^{M-n} \, .

The Poisson limit is defined as $M \rightarrow \infty$ while $M \Delta p = T \Delta p/\Delta t$ is kept constant. In this limit, we obtain the Poisson distribution

P(n) \rightarrow e^{-N} \frac{N^n}{n!} \, ,

where $N = T \Delta p / \Delta t$ is the expected number of events during time $T$.

As a matter of fact, Poisson processes can be defined in a completely abstract way. The events do not necessarily need to be points in time or in the physical space. In general, a Poisson process is defined by assigning a probability density on an abstract space, and events to points in said space, as long as they are statistically independent.

In an article recently published in $Quantum$ [1], Mankei Tsang considered a Poisson process where the points are vectors in a Hilbert space $H_1$.
To account for when the event does not happen, this space needs to be extended via direct sum into $H_0 \oplus H_1$, where $H_0 = \{ |0\rangle \}$ is a $vacuum$ space that can be assumed to be one-dimensional without loss of generality, whereas the space $H_1$ may have arbitrary dimensions, depending on the physical degrees of freedom. The quantum state

\tau = (1-\epsilon) |0\rangle \langle 0| + \epsilon | \psi \rangle \langle \psi |

describes a situation where the event $| \psi \rangle$ happens with probability $\epsilon$.

Consider an i.i.d. source that generates $M$ copies of such a state, $\tau^{\otimes M}$.
We may want to compute the probability of observing $n$ events, which is the expectation value, on the $\tau^{\otimes M}$, of the operator $O = \sum_\pi \pi(\Pi_1^{\otimes n})$, where $\Pi_1$ is the projector onto $H_1$ and the sum is over the ${ M \choose n}$ permutations. In the Poisson limit of $M \rightarrow \infty$, with $N = M \epsilon$ constant, this probability is again given by the Poisson distribution. More generally, we may not only count the events, but also observe the degrees of freedom in the $\textit{single-particle}$ space $H_1$. Given a single-particle measurement with POVM elements $E_j>0$, such that $\sum_j E_j = \Pi_1$, the probability of obtaining $n_j$ times the outcome $j$ is as well expressed, in the Poisson limit, by the Poisson distribution.

The state $\tau^{\otimes M}$, in the Poisson limit, may be called a $\textit{Poisson state}$.
Mankei Tsang showed that the Poisson state is succinctly described by the so-called $\textit{intensity operator}$

\Gamma := N \tau_1 \, .

Note that this is a single-particle operator, being defined in $H_1$, yet it describes the statistics in the limit of $M \rightarrow \infty$. Quantum information functionals, as the fidelity, the relative entropy, the Fisher information matrix, etc., can be written in a simple way in terms of the intensity operator. These properties lift to Poisson channels, which are defined as those channels that map Poisson states to Poisson states.

The work of Mankei Tsang is motivated by quantum imaging in the regime of weak signals, where detecting a photon is a rare event [2]. More generally, following a seminal idea by the same author and collaborators [3], a number of protocols of quantum metrology and hypothesis testing have been recently proposed that exploit weak signals. By establishing a theoretical framework for analysing rare events in quantum information theory, this work has the ambition of kicking off the field of $\textit{Poisson quantum information}$. I am excited by the possibility that these formal concepts may be as fruitful as Gaussian quantum information in providing new insights and applications. I warmly invite to read the paper, which is pleasant and thought-provoking.

### ► References

[1] M. Tsang, Quantum 5, 527 (2021).
https:/​/​doi.org/​10.22331/​q-2021-08-19-527

[2] M. Tsang, Contemporary Physics 60, 279 (2019).
https:/​/​doi.org/​10.1080/​00107514.2020.1736375

[3] M. Tsang, R. Nair, and X.-M. Lu, Physical Review X 6, 031033 (2016).
https:/​/​doi.org/​10.1103/​PhysRevX.6.031033

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