Device-independent witness of arbitrary-dimensional quantum systems employing binary-outcome measurements

Device-independent dimension witnesses (DWs) are a remarkable way to test the dimension of a quantum system in a prepare-and-measure scenario imposing minimal assumptions on the internal features of the devices. However, as the dimension increases, the major obstacle in the realization of DWs arises due to the requirement of many-outcome quantum measurements. In this article we propose a variant of a widely studied communication task (random access code) and take its average payoff as the DW. The presented DW applies to arbitrarily large quantum systems employing only binary-outcome measurements.

Figure 1

Scheme of the $d\mathrm{RAC}$. Alice gets the input $a_0,\cdots ,a_{n-1}$ and sends a message $m$ to Bob. Besides the message, Bob also receives the input $y\in \{0,\cdots ,n-1\}$. His task is to give the output $b$, which obeys the relation $b=a_y$.

Figure 2

Scheme of the binary RAC. Alice gets the input $a_0,\cdots ,a_{n-1}$ and sends the message $m$ to Bob. Besides the message Bob receives two inputs $y\in \{0,1,\cdots ,n-1\}$ and $k\in \{0,1,\cdots ,d-1\}$. His task is to guess whether $a_y=k$ or not. His answer is encoded in $G$, which is 0 when his guess is yes and 1 when it is no.

Figure 3

Scheme of the binary RAC. Bob is split into two parts: $B_I$ (initial Bob) and $B_F$ (final Bob).

Figure 4

Dependence of $H$ on $T$ for $d=8$, $x=1,2,3,4$, and $p_{\text{crit}}=0.14$. We note that the entropy for the strategy with $x=1$ is not always the largest in the established ranges of $T$. According to the numerical procedure, this is an example in which, at step 3c, ${\mathrm{\Delta }}_i\le 0$ and our algorithm skips from step 3c to step 3c. Vertical lines indicate the limits of the ranges $[T_0,T_1^{x=i}]$.

Figure 5

Dependence of $H$ on $T$ for $d=8$, $x=1,2,3,4$, and $p_{\text{crit}}=0.18495$. The largest entropy is obtained with exactly one strategy for which $x=1$. According to our procedure, this is an example in which, at step 6, ${\mathrm{\Delta }}_i>0$ for every $i\in \{2,3,\cdots ,\lceil \frac{d}{2}\rceil \}$ and our algorithm returns $p_{\text{crit}}$. Vertical lines indicate the limits of the ranges $[T_0,T_1^{x=i}]$.

Figure 6

Numerical calculation of values of the minimal $p_{\text{crit}}$ and the corresponding maximal $T_{\text{yes}}$ as a function of dimension with accuracy ${\varepsilon }_{p_{\text{crit}}}={10}^{-5}$.