QUANTUM STATE RECONSTRUCTION VIA
THE JAYNES MAXENT PRINCIPLE

MaxEnt principle and observation levels

The state of a quantum system can always be described by a statistical density operator $\hat \rho$. Depending on the system preparation, the density operator represents either a pure quantum state (complete system preparation) or a statistical mixture of pure states (incomplete preparation). The degree of deviation of a statistical mixture from the pure state can be best described by the uncertainty measure $\eta [\hat \rho ]$

$$\eta [\hat \rho ]=-k_B\,{\rm Tr}(\hat \rho \ln \hat \rho ) \eqno(1)$$

where $k_B$ is the Boltzmann constant. The uncertainty measure (entropy) $\eta [\hat \rho ]$ is equal to zero for pure states and $\eta [\hat \rho ]>0$ for statistical mixtures. For $N$-dimensional Hilbert space of the system the uncertainty measure $\eta [\hat \rho ]$ takes its maximum value $k_B\ln N$ when $\hat \rho =\hat1/N$. In this case all pure states in the mixture appear with the same probability equal to $1/N$. It can be shown with the help of the Liouville equation that in the case of an isolated system the uncertainty measure is a constant of motion, i.e., $d\eta(t)/dt=0$.

MaxEnt principle

There are situations when instead of the density operator $\hat \rho$, expectation values $G_\nu$ of a set ${\cal O}$ of operators $\hat G_\nu$ $(\nu = 1,\ldots , n)$ are given. The set of linearly independent operators is referred to as the observation level ${\cal O}$. The operators $\hat{G}_{\nu}$ which belong to a given observation level do not commutate necessarily. A large number of density operators which fulfill the conditions

$${\rm Tr} \,\hat \rho _{\{\hat G\}} = 1 \eqno(2a)$$

$${\rm Tr}\,(\hat \rho _{\{\hat G\}}\hat G_\nu )=G_\nu \,, \,\,\,\,\nu = 1,2,...,n ; \eqno(2b)$$

can be found for a given set of expectation values $G_\nu = \langle \hat {G_\nu}\rangle$. Each of these density operators $\hat \rho _{\{\hat G\}}$ can posses a different value of the uncertainty measure $\eta [\hat \rho _{\{\hat G\}}]$. If we wish to use only the expectation values $G_\nu$ of the chosen observation level for determining the density operator, we must select a particular density operator $\hat \rho _{\{\hat G\}} = \hat \sigma _{\{\hat G\}}$ in an unbiased manner. According to the Jaynes principle of the Maximum Entropy this density operator $\hat \sigma _{\{\hat G\}}$ must be the one which has the largest uncertainty measure (entropy) $\eta [\hat \sigma _{\{\hat G\}}]$ and simultaneously fulfills constraints (2). The variation determining the maximum of $\eta [\hat \sigma _{\{\hat G\}}]$ under the conditions (2) leads to a generalized canonical density operator

$${\hat \sigma _{\{\hat G\}}={1 \over {Z_{\{\hat G\}}}} \exp \,(-\sum\limits_\nu {\lambda _\nu \hat G_\nu })} \eqno(4)$$

$${Z_{\{\hat G\}}(\lambda _1,...,\lambda _n)= {\rm Tr}[\exp (-\sum\limits_\nu {\lambda _\nu \hat G_\nu })]} \eqno(5)$$

where $\lambda_n$ are the Lagrange multipliers and $Z_{\{\hat G\}}(\lambda_1,\ldots \lambda_n)$ is the generalized partition function. By using the derivatives of the partition function we obtain the expectation values $G_{\nu}$ as

$$G_\nu ={\rm Tr}(\hat \sigma _{\{\hat G\}}\hat G_\nu )= -{\pa... ...nu }} \ln Z_{\{\hat G\}}(\lambda _1,...,\lambda _n) . \eqno(6)$$

The Lagrange multipliers can be expressed as functions of the expectation values $\lambda _\nu =\lambda _\nu (G_1,...,G_n)$. The maximum uncertainty measure regarding an observation level ${\cal O}_{\{ \hat G\}}$ will be referred to as the entropy $S_{\{\hat G\}}$

$$S_{\{\hat G\}}\equiv \eta [\hat \sigma _{\{\hat G\}}] =-k_B{... ... \sigma _{\{\hat G\}} \ln \hat \sigma _{\{\hat G\}}). \eqno(7)$$

This means that to different observation levels different entropies are related expressing thus our knowledge of an unknown pure state (on a given observation level). Zero entropy means complete knowledge (reconstruction) of the state.

Extension and reduction of the observation level

If an observation level ${\cal O}_{\{\hat{G}\}}\equiv \hat G_1, \ldots , \hat G_n$ is extended by including further operators $\hat M_1, \ldots , \hat M_l$, then additional expectation values $M_1 = \langle \hat M_1\rangle , \ldots , M_l = \langle \hat M_l\rangle$ can only increase amount of available information about the state of the system. This procedure is called the extension of the observation level (from ${\cal O}_{\{\hat{G}\}}$ to ${\cal O}_{\{\hat{G}, \hat{M}\}}$) and is associated with a decrease of the entropy. More precisely, the entropy $S_{\{\hat G,\hat M\}}$ of the extended observation level ${\cal O}_{\{\hat G, \hat M\}}$ can be only smaller or equal to the entropy $S_{\{\hat G\}}$ of the original observation level ${\cal O}_{\{ \hat G\}}$,

$$S_{\{\hat G,\hat M\}}\le S_{\{\hat G\}}$$

We can also consider a reduction of the observation level if we decrease number of independent observables which are measured, e.g., ${\cal O}_{\{ \hat{G}, \hat{M}\}}\rightarrow {\cal O}_{\{ \hat{G}\}}$ (here $\hat{G}_{\nu}$ and $\hat{M}_{\mu}$ are independent). This reduction is accompanied with an increase of the entropy due to the decrease of the information available about the system.

Examples of observation levels

Complete observation level

${\cal O}_{0}\equiv\{ (\hat{a}^{\dagger})^k\hat{a}^l;~\forall k,l\}$
The set of operators $\vert n \rangle \langle m \vert$ (for all $n$ and $m$) are referred to as complete observation level.

Thermal observation level

${\cal O}_{\rm th}\equiv\{ \hat{a}^{\dagger}\hat{a}\}$
The total reduction of the complete observation level ${\cal O}_{0}$ results in a thermal observation level ${\cal O}_{\rm th}$ characterized just by one observable, the photon number operator $\hat{n}$, i.e., quantum-mechanical states of light on this observation level are characterized only by their mean photon number $\bar{n}\equiv\langle\hat{n} \rangle$.

Phase-sensitive observation levels:

 
${\cal O}_{1}\equiv\{\hat{a}^{\dagger}\hat{a},\hat{a}^{\dagger},\hat{a}\}$
We can extent the thermal observation level if in addition to the observable $\hat{n}$ we consider also the measurement of mean values of the operators $\hat{a}$ and $\hat{a}^{\dagger}$ (that is, we consider a measurement of the observables $\hat{q}$ and $\hat{p}$).
${\cal O}_{ 2}\equiv\{ \hat{a}^{\dagger}\hat{a},(\hat{a}^{\dagger})^2,\hat{a}^2, \hat{a}^{\dagger},\hat{a}\}$
On this observation level not only the mean photon number $\bar{n}$ and mean values of $\hat{q}$ and $\hat{p}$ are known, but also the variances $\langle \left(\Delta \hat{q}\right)^{2}\rangle$ and $\langle \left(\Delta \hat{p}\right)^{2}\rangle$ are measured.

Phase-insensitive observation levels:

 
${\cal O}_{\rm A}\equiv \{\hat{P}_n=\vert n \rangle\langle n\vert;~~\forall n\}$
${\cal O}_{\rm B}\equiv \{\hat{n}, \hat{P}_{2n}=\vert 2n \rangle\langle 2n\vert;~~\forall n\}$
${\cal O}_{\rm C}\equiv \{\hat{n}, \hat{P}_{2n+1}=\vert 2n +1\rangle\langle 2n+1\vert;~~\forall n\}$
${\cal O}_{\rm D}\equiv \{\hat{n}, \hat{P}_{N}=\vert N\rangle\langle N\vert\}$; namely ${\cal O}_{\rm D1}$ for $N=0$ and ${\cal O}_{\rm D2}$ for $N=\bar{n}$

Fig. 1. The reconstructed Wigner functions of the coherent state $\vert\alpha \rangle$ with $\bar{n}=2$. We consider the observation levels as indicated in the figure.

Fig. 2. The reconstructed Wigner functions of the even coherent state ${{\cal N}\over 2} (\vert\alpha\rangle + \vert-\alpha\rangle)$ with $\bar{n}=2$.