Steering-based randomness certification with squeezed states and homodyne measurements

High-quality randomness, certified to be unpredictable by eavesdroppers, is key to secure information processing. Quantum mechanics enables randomness certification with minimal trust in the devices used, by exploiting quantum nonlocality. However, such full device independence is challenging to implement. We present a scheme for quantum randomness certification based on quantum steering. The protocol is one-sided device independent, providing high security, but requires only states and measurements that are simple to realize on quantum optics platforms—squeezed vacuum states and homodyne detection. This ease of implementation is demonstrated experimentally and implies that gigahertz random bit rates should be attainable with current technology. Furthermore, our scheme is immune to the detection loophole and represents the closest to full device independence that can be achieved using purely Gaussian states and measurements.

Figure 1

Setup for steering test and randomness certification using Gaussian states and measurements, consisting of a source of squeezed, entangled states and two parties, Alice and Bob, who perform homodyne measurements. The source emits either a two-mode squeezed vacuum or a single-mode squeezed vacuum split on a balanced beam splitter. Alice measures one of two conjugate quadratures, according to a binary input $x$, and applies periodic binnings into $o_A$ outcomes $a$. Bob either performs full state tomography or measures $m_B$ different quadratures, according to input $y$, and applies a nonperiodic binning into $o_B$ outcomes $b$. For certifying steering and randomness, Alice's device and the source state are untrusted, while Bob's device is assumed to be well characterized. Randomness is extracted from Alice's measurement outcome, for one of her inputs.

Figure 2

Optimal min-entropy vs transmission efficiency, for the two-mode squeezed (TMS) and the split single-mode squeezed (SMS) vacuum source with and without noise (0.01 shot-noise units). $H_{\min }$ is maximized over Alice's binning period in the range $T_q\in [2,10]$. In all cases, Alice has a choice of $m_A=2$ observables, $q$ and $p$, binned to $o_A=8$ outcomes. Bob performs tomography (tomo.; black curve) or measures in $m_B=2,4,6$ different directions equally spaced between $q$ and $p$, binned to $o_B=16$ outcomes. For TMS the squeezing is $-4$ dB, and for SMS it is $-6$ dB.

Figure 3

Results obtained from experimental data of Ref. [42]. Min-entropy vs Alice's number of outputs. The optimal values of $H_{\min }$ from the idealized theoretical model (black dots) and the lower bounds on $H_{\min }$ from the experimental data (pink circles) are pictured. Alice and Bob both measure $q$ and $p$, and randomness is extracted from Alice's $q$ measurement. $H_{\min }$ is maximized over Alice's binning period in the range $T_q\in [2,10]$. Bob has $o_B=16$ outcomes.

Figure 4

Orthogonality parameter $\epsilon$ as a function of the squeezing parameter $s$ and the cutoff in photon number $m$.

Figure 5

Bound on the min-entropy $H_{\min }$ as a function of the cutoff $m$ in the Fock basis for different squeezing parameters. Alice's measurement parameters are $o_A=8$ and $T_q=3$.

Figure 6

Lower bound on the min-entropy vs Fock-basis cutoff, for several values of Alice's outputs $o_A$. The measurement settings at each value of $o_A$ correspond to the maximum values of $H_{m}in$ in Fig. 3 of the main text.