# A know-nothing test of entropy

June 1, 2015

Friday we had a new paper out on arXiv, here: https://arxiv.org/abs/1505.07802 . This one is about device-independent tests of entropy.

Generally, to learn about the physical properties of something, such as temperature or mass, we use a measurement device that we already know well. We want our device to be well calibrated and characterised such that we can have confidence in the information it gives about whatever we are studying. But interestingly, there are some properties which can be inferred essentially without knowing anything about the devices used in the experiment.

One example of such a property is dimension, meaning the number of degrees of freedom that a physical system has. We can perform an experiment where a particle is prepared by one device and then sent to another which measures it. For various combinations of settings for the preparation and measurement devices, we collect data about how likely different measurement outcomes are. Without knowing anything more about how the devices function or about what the particle is, just by studying this data we can infer the number of degrees of freedom that the particle must possess.

Another example is quantum entanglement. In a similar experiment with two devices, from the data describing combinations of inputs and outputs one can infer that the devices share an entangled quantum state. Again without knowing anything about what is actually going on inside them.

In this paper we identify another quantity which can be tested in such a device-independent way, namely entropy. In a prepare-and-measure experiment, the entropy measures the number of degrees of freedom which are exploited by the particle on average. That is, while the dimension tells us how many degrees of freedom the particle must have, the entropy tells us to which degree it must exploit them to reproduce the data. These two things can be very different – as we show in the paper, it is possible to have experiments that require arbitrarily high dimension, while the entropy remains very small. We also show that quantum particles do better than classical ones when it comes to entropy – for the same dimension, when the particles are quantum, the same task can be realised with lower entropy than for classical particles.

From a fundamental point of view, it is interesting to ask what quantities can be assessed in a device-independent manner. So it’s nice that we now know entropy is one of them, as well as some of its basic characteristics. From a more practical point of view, device-independent tests of both dimension and entanglement have promising applications for quantum cryptography and random number generation. So it might very well be that device-independent entropy tests will also be useful for quantum information processing.