Geometrical picture of photocounting measurements

We revisit the representation of generalized quantum observables by establishing a geometric picture in terms of their positive operator-valued measures (POVMs). This leads to a clear geometric interpretation of Born's rule by introducing the concept of contravariant operator-valued measures. Our approach is applied to the theory of array detectors, which is a challenging task as the finite dimensionality of the POVM substantially restricts the available information about quantum states. Our geometric technique allows for a direct estimation of expectation values of different observables, which are typically not accessible with such detection schemes. In addition, we also demonstrate the applicability of our method to quantum-state reconstruction with unbalanced homodyne detection.

Figure 1

An array detector consists of a set of $N$ on-off detectors. An incident signal is split into multiple beams with equal intensities, represented by a unitary $U\left(N\right)$, which are individually measured with on-off detectors. Dashed lines indicate the $N-1$ vacuum inputs.

Figure 2

Schematic presentation of the observable $\widehat{B}$ in the POVM basis $\{{\widehat{\mathrm{\Pi }}}_0,{\widehat{\mathrm{\Pi }}}_1\}$. The vector representation of the observable $\widehat{B}$ is the sum of the vector ${\widehat{B}}_c={\sum }_{n=0}^1{B}^n{\widehat{\mathrm{\Pi }}}_n$ and the orthogonal completion $\widehat{R}$, according to Eq. (4).

Figure 3

Graphical presentation of POVM and COVM bases, Eqs. (16) and (17), respectively, for $\eta =0.5$. (a) The observable $\widehat{B}=\left|0\right.〉\left.〈0\right|+2\left|1\right.〉\left.〈1\right|$ and (b) the density operator $\widehat{\varrho }=0.9\left|0\right.〉\left.〈0\right|+0.1\left|1\right.〉\left.〈1\right|$, with the corresponding covariant and contravariant coordinates, are shown. The coordinate ${\rho }^1$ has a clear negative value.

Figure 4

The HS mismatch of the projectors $\left|m\right.〉\left.〈m\right|$ for different numbers $m$ in the case of $N=10$.

Figure 5

The HS mismatch $\parallel \widehat{R}{\parallel }_{\mathrm{HS}}$ of the operators $\widehat{B}=exp(-t\widehat{n})$ (dashed line) and $\widehat{B}=\text{:}exp(-t\widehat{n})\text{:}$ (solid line) vs the parameter $t$ in the case of $N=10$ on-off detectors.

Figure 6

The HS mismatches $\parallel \widehat{R}{\parallel }_{\mathrm{HS}}$ (black bars) and moments (gray bars) in logarithmic scale for a photon-number state $\left|5\right.〉$ in the case of $N=20$ as a function of the order of moments $m$: (a) the truncated moments $\langle {\widehat{n}}^{\left[m\right]}\rangle$ [see Eq. (38)] and (b) the truncated normal-ordered moments $\langle \text{:}{\widehat{n}}^{\left[m\right]}\text{:}\rangle$ [see Eq. (39)].

Figure 7

Scheme for unbalanced homodyne detection [10, 11]. The signal is combined with a local oscillator on a beam splitter with a transmission coefficient close to one. The output is then sent to an array detector (see also Refs. [48, 49]).

Figure 8

The HS mismatch for the operator $\widehat{P}(\alpha ;s)$ (solid line) and for its truncations up to $M=7$ (dashed line) and $M=2$ (dot-dashed line) as functions of the parameter $s$.

Figure 9

Theoretical (dashed lines) and reconstructed from the simulated data (solid lines) Cahill-Glauber phase-space distributions with (a) $s=0$ [Wigner function $W\left(\alpha \right)$] and (b) $s=0.2$ [distribution $P(\alpha ;0.2)$] for a squeezed vacuum for (i) $\mathrm{Im}\alpha =1$ and (ii) $\mathrm{Im}\alpha =0$. Further details of the simulation are given in the text.