Theoretical physicists completely determine the statistics of quantum entanglement

For the first time, theoretical physicists from the Institute of Theoretical Physics (IPhT) in Paris-Saclay have completely determined the statistics that can be generated by a system using quantum entanglement. This achievement paves the way for exhaustive test procedures for quantum devices.

The study is published in the journal Nature Physics.

After the advent of transistors, lasers and atomic clocks, the entanglement of quantum objects—as varied as photons, electrons and superconducting circuits—is at the heart of a second quantum revolution, with quantum communication and quantum computing in sight.

What’s involved? Two objects prepared together in a quantum state—two horizontally or vertically polarized photons, for example—retain the memory of their common origin, even if they are moved far apart from each other. When the quantum state of the two entangled objects is measured—their polarization, in the proposed example—a distinct correlation is observed between the measurement results.

Measurement obeying quantum statistics

What does this correlation depend on? First, the degree of entanglement between the two objects may vary, depending on the nature of the source of the entangled quantum objects—in the example, horizontally polarized photons may be produced more frequently than vertically polarized ones. Then, a choice of measurement must be made—such as selecting a direction in which to measure the polarization—which may impact its result.

In order to generate meaningful quantum correlations, it is indeed essential that each object may be measured using a minimum of two distinct measurements, each offering at least two potential outcomes.

In the simplest experiment revealing the extent of quantum entanglement, five parameters can thus affect the measurement statistics: the degree of entanglement between the objects and the two directions in which both apparatuses perform their measurements. Generally speaking, however, quantum physics allows for intricate systems with numerous degrees of freedom, leading to a wide variety of correlations.

Extracting knowledge from a black box

Quantum correlations have remarkable characteristics, notably their ability to pass a Bell test. When this happens, the results of a quantum experiment are “non-local” in the sense that they cannot be explained in terms of local hidden variable models, which capture our intuitive understanding of correlations. The experimental demonstration of this striking property was celebrated by the physics Nobel prize awarded in 2022 to Alain Aspect, John F. Clauser and Anton Zeilinger. But quantum correlations have more than one trick up their sleeve.

It turns out that physical attributes can often be estimated directly from the statistics obtained upon measurement of an entangled quantum state. For instance, observed correlations can certify that the observed measurement results are random. Importantly, this conclusion is attainable from the measurement results alone, without any assumption on the behavior of the quantum devices at hand, considered as “black-boxes.” Ultimately, some quantum statistics have the property of fully identifying the physical model describing the entangled objects.

Unveiling the extent of quantum correlations

This stunning property, referred to as “self-testing,” plays a crucial role in device-independent quantum information protocols. Because these protocols do not rely on any assumption regarding the proper functioning of the source and measurement apparatuses, they offer unparalleled reliability. So far, several self-testing results have been obtained. For example, it is known that all qubit states can be self-tested, although all possible self-tests are not known yet. Indeed, only self-tests corresponding to maximally entangled states of two qubits have been fully characterized.

IPhT theoretical physicists Victor Barizien and Jean-Daniel Bancal have now demonstrated that it is also possible to describe exactly and completely the statistics obtained when measuring partially entangled objects.

“The idea, which is cute but hard to explain, was to describe the statistics from partially entangled states using what we understand of maximally entangled ones. We found a mathematical transformation that allows for a fruitful physical interpretation,” state the researchers.

In turn, identifying all correlations that can self-test partially entangled two-qubit states provided a complete description of the quantum statistics.

Considerable scope, both fundamental and applied

Having a complete knowledge of the quantum statistics achievable when entanglement is involved has wide consequences. On one hand, it identifies limits of quantum theory itself. In doing so, it bounds the extent of experimental results one can expect to observe provided that nature abides by the rules of quantum physics. On the other hand, it offers exceptionally effective test procedures, applicable to all types of entangled objects and measurements, and therefore to many different types of systems.

In particular, the security of devices using quantum entanglement can be enhanced by tests based on the results of observations made at each instant, rather than on the physical properties of the apparatuses, which are likely to evolve over time. More generally, the way is open to new protocols for quantum testing, communications, cryptography and computation.