Storage and retrieval of von Neumann measurements

This work examines the problem of learning an unknown von Neumann measurement of dimension $d$ from a finite number of copies. To obtain a faithful approximation of the given measurement, we are allowed to use it $N$ times. Our main goal is to estimate the asymptotic behavior of the maximum value of the average fidelity function $F_d$ for a general $N\to 1$ learning scheme. We show that $F_d=1-\mathrm{\Theta }\left(\frac{1}{N^2}\right)$ for arbitrary but fixed dimension $d$. In addition to that, we compared various learning schemes for $d=2$. We observed that the learning scheme based on deterministic port-based teleportation is asymptotically optimal but performs poorly for low $N$. In particular, we discovered a parallel learning scheme, which despite its lack of asymptotic optimality, provides a high value of the fidelity for low values of $N$ and uses only two-qubit entangled memory states.

Figure 1

Schematic representations of the setup for learning of von Neumann measurements ${\mathcal{P}}_U$ in the $N\to 1$ scheme.

Figure 2

Schematic representation of DPBT for $N=2$. In this case, the output label $i=2$ of the measurement $\mathcal{Q}$ determines the partial trace of the first system of the remaining quantum state.

Figure 3

Schematic representation of the learning scheme for $N=2$ based on DPBT. In this case, the label $i=2$ of the measurement ${\mathcal{Q}}_0$ indicates that the output of the learning procedure equals $j_i=j_2$.

Figure 4

Schematic representation of the setup, which we use to calculate the upper bound for $F_2$. In this scenario we are given $N$ copies of a unitary channel ${\mathrm{\Phi }}_{\overline{U}}$ in parallel. Our objective is to approximate the von Neumann measurement ${\mathcal{P}}_U$.

Figure 5

Schematic representations of the pretty good learning scheme for $N=3$. In the learning process we obtained three labels: $0,1,0$. As labels “0” are in majority, we reject the label “1” and the associated quantum part.

Figure 6

The average fidelity function (dimensionless) ${\mathcal{F}}_2^{\text{avg}}$ calculated for $N\to 1$ learning scheme: optimal adaptive strategy ${\mathcal{L}}_{\text{Adaptive}}$ (numerical value, blue squares); optimal parallel scheme ${\mathcal{L}}_{\text{Parallel}}$ (numerical value, red crosses); learning scheme based on DPBT ${\mathcal{L}}_{\text{DPBT}}$ (green pentagons); pretty good learning scheme ${\mathcal{L}}_{\text{PGLS}}$ (orange diamonds); learning scheme based on PPBT ${\mathcal{L}}_{\text{PPBT}}$ (gray circles).

Figure 7

The schematic representation of a learning scheme $\mathcal{L}=(\sigma ,{\left\{{\mathcal{C}}_i\right\}}_{i=1}^{N-1},\mathcal{R})$.

Figure 8

Schematic representation of the setup, which we use to upper bound $F_2$. In this scenario we are given $N$ copies of unitary channel ${\mathrm{\Phi }}_{\overline{U}}$ in parallel. Our objective is to approximate the von Neumann measurement ${\mathcal{P}}_U$.

Figure 9

Schematic representations of the right-hand side of Eq. (B5). With probability $1/2$ we prepare one of basis states $|0\rangle$ or $|1\rangle$ and calculate the probability that we obtain output $i$. Equation (B5) is then the cumulative probability that provided the state $|i\rangle \langle i|$ we measure $i$. The learning scheme $\mathcal{L}$ is given as $\mathcal{L}=(\sigma ,{\left\{{\mathcal{C}}_i\right\}}_{i=1}^N,\mathcal{R})$ and the storage $\mathcal{S}$ (marked with a dashed line) is defined as a composition of an initial memory state $\sigma$ and processing channels ${\left\{{\mathcal{C}}_i\right\}}_{i=1}^N$.