Measurement-device-independent randomness generation with arbitrary quantum states

Measurements of quantum systems can be used to generate classical data that are truly unpredictable for every observer. However, this true randomness needs to be discriminated from randomness due to ignorance or lack of control of the devices. We analyze the randomness gain of a measurement-device-independent setup, consisting of a well-characterized source of quantum states and a completely uncharacterized and untrusted detector. Our framework generalizes previous schemes as arbitrary input states and arbitrary measurements can be analyzed. Our method is used to suggest simple and realistic implementations that yield high randomness generation rates of more than one random bit per qubit for detectors of sufficient quality.

Figure 1

The measurement-device-independent setup for randomness generation (for details see text). The trusted source sends for the input $a\in \{1,...,n_s\}$ a known state $\rho \left(a\right)$ to an untrusted measurement device (black box), which outputs $x$ with $x\in \{1,...,n_o\}$. The outcome randomness ${\mathcal{R}}_X$ is characterized by the observed probability distribution $P_{\mathrm{obs}}(x,a)$, i.e., the probability that the pair $(x,a)$ occurs.

Figure 2

The graphic depicts the relevant systems, states, and measurement to estimate the randomness generation rate, which is conditioned on the outside systems $A$ and $E$. The primed systems are contained in the measurement box and represent internal degrees of freedom ($E^{\prime }$) as well as the incoming state $\rho \left(a\right)$ ($A^{\prime }$). A further internal degree of freedom is given by the unknown measurement $\left\{G\right(x\left)\right\}$, which produces the outcome $x$ in the laboratory. The state $\sigma$ provides correlations between the detector degrees of freedom and Eve's site $E$.

Figure 3

The randomness rate versus the detector quality defined in Eq. (22) in the asymptotic limit and for $q\equiv p_1\to 1$. The solid (dashed) line depicts an extremal POVM with $n_o=4$ ($n_o=2$) outcomes for a tomographically complete set of $n_s=4$ states. The ideal measurement statistics arises from the measurement directions in Eq. (26) ($n_o=4$) and from a ${\sigma }_z$ measurement ($n_o=2$). The lower line corresponds to the case of two nonorthogonal sent states and two outcomes.

Figure 4

The randomness rate as a function of the angle parameter $\alpha$ [see Eq. (28)] between two sent states for several detector qualities $\eta$ and asymmetry $q=\frac{1}{2}$. The statistics of the ideal measurement corresponds to a ${\sigma }_x$ measurement. For $\alpha =0$, the randomness rate is equal to zero.

Figure 5

The randomness rate of a two-state two-outcome setup with the sent states $\left|+\right.〉, \left|0\right.〉$ and different asymmetry parameters $q\equiv p_1$. The ideal measurement statistics corresponds to a ${\sigma }_z$ measurement. The $q\to 1$ line is nonzero for $\eta >\frac{1}{2}$.

Figure 6

The randomness rate per qubit versus the detector quality from Eq. (22) for different setups in the asymptotic limit (order from upper to lower line corresponds to order in legend). The upper two lines correspond to setups with four different sent states per qubit, whereas the lower three lines correspond to two different sent states per qubit. The ideal measurement statistics is given by a ${\sigma }_z^{\otimes m}$ measurement, where $m$ is the number of sent qubits.

Figure 7

The difference of the normalized randomness rates of a one-qubit setup with tomographically complete states, and a two-qubit doubling in the asymptotic limit. The solid line corresponds to a three-outcome POVM, whose statistics is determined by the measurement directions in Eq. (27), and the dashed line to a projective measurement.