Quantum entanglement is a uniquely quantum link between particles that makes their properties inseparable. It underlies the power of many quantum technologies from secure communication to quantum computing, by enabling correlations impossible in classical physics.

Entanglement nevertheless remains poorly understood and is therefore the subject of a lot of research, both in the fields of quantum technologies as well as fundamental physics.

In this context, the idea of separability refers to a composite system that can be written as a simple product (or mixture of products) of the states of its individual parts. This implies there is no entanglement between them and to create entanglement, a global transformation is needed.

A system that remains completely free of entanglement, even after any possible global invertible transformation is applied, is called absolutely separable.  In other words, it can never become entangled under the action of quantum gates.

Necessary and sufficient conditions to ensure separability exist only in the simplest cases or for highly restricted families of states. In fact, entanglement verification and quantification is known to be generically an NP-hard problem.

Recent research published by a team of researchers from Spain and Poland has tackled this problem head-on. By introducing new analytical tools such as linear maps and their inverses, they were able to identify when a quantum state is guaranteed to be absolutely separable.

These tools work in any number of dimensions and allow the authors to pinpoint specific states that are on the border of being absolutely separable or not (mathematically speaking, ones that lie on the boundary of the set). They also show how different criteria for absolute separability, which may not always agree with each other, can be combined and refined using convex geometry optimisation.

Being able to more easily and accurately determine whether a quantum state is absolutely separable will be invaluable in quantum computation and communication.

The team’s results for multipartite systems (systems with more than two parts) also reveal how little we currently understand about the entanglement properties of mixed, noisy states. This knowledge gap suggests that much more research is needed in this area.

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When we interact with everyday objects, we take for granted that physical systems naturally settle into stable, predictable states. A cup of coffee cools down. A playground swing slows down after being pushed.  Quantum systems, however, behave very differently.

These systems can exist in multiple states at once, and their evolution is governed by probabilities rather than certainties. Nevertheless, even these strange systems do eventually relax and settle down, losing information about their earlier state. The speed at which this happens is called the relaxation rate.

Relaxation rates tell us how fast a quantum system forgets its past, how quickly it thermalises, reaches equilibrium, decoheres, or dissipates energy. These rates are important not just for theorists but also for experimentalists, who can measure them directly in the lab.

Recently, researchers discovered that these rates obey a surprisingly universal rule. For a broad class of quantum processes (those described by what physicists call Markovian semigroups) the fastest possible relaxation rate cannot exceed a certain limit. Specifically, it must be no larger than the sum of all relaxation rates divided by the system’s dimension. This constraint, originally a conjecture, was first proven using tools from classical mathematics known as Lyapunov theory.

In a new paper published recently, an international team of researchers provided a new, more direct algebraic proof of this universal bound. There are a number of advantages of the new proof compared to the older one, and it can be generalised more easily, but that’s not all.

The very surprising outcome of their work is that the rule doesn’t require complete positivity. Instead, a weaker condition – two‑positivity is enough. The distinction between these two requirements is crucial.

Essentially, both are measures of how well-behaved a quantum system is, how it is protected from providing nonsensical results. The difference is that two-positivity is slightly less stringent but far more general, and hence very useful for many real-world applications.

The fact that the new proof only requires two-positivity means that it this new universal relaxation rate can actually be applied to a lot more scenarios.

What’s more, even when weakened even further, a slightly softer version of the universal constraint still holds. This shows that the structure behind these bounds is richer and more subtle than previously understood.

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Loss of a limb can significantly impact a person’s independence and quality-of-life, with arm amputations particularly impeding routine daily activities. Prosthetic limbs can restore some of the lost function, but often rely on surface electrodes with low signal quality. A research team at the University of Michigan has now shown that implanted electrodes could provide more accurate and reliable control of hand and wrist prostheses.

Today, most upper-limb prostheses are controlled using surface electrodes placed on the skin to detect electrical activity from underlying muscles. The recorded electromyography (EMG) signals are then used to classify different finger and wrist movements. Under real-world conditions, however, these signals can be impaired by inconsistent electrode positioning, changes in limb volume, exposure to sweat and artefacts from user movements.

Implanted electrodes, tiny contacts that are surgically sutured into muscles, could do a better job. By targeting muscles deeper in the arm, they offer higher signal-to-noise ratios and less susceptibility to daily variations. And although amputation can eliminate many of the muscles that control hand functions, techniques such as regenerative peripheral nerve interface (RPNI) surgery – in which muscle tissue is grafted to nerves in the residual limb – enable electrodes to target missing muscles and record relevant signals for prosthetic control.

Senior author Cynthia Chestek points out that such RPNI grafts are also beneficial for the nerve itself. “They provide a target for nerve endings that prevent the formation of painful neuromas, and that may in turn help reduce phantom limb pain,” she explains “In future, it would also be possible to place electrodes and a wireless transmitter during that same surgery, such that no additional surgeries are required other than the original amputation.”

In their latest work, reported in the Journal of Neural Engineering, Chestek and colleagues investigated whether implanted electrodes could provide stable and high-quality signals for  controlling prosthetic hand and wrist function.

Performance comparisons

The study involved two individuals with forearm amputations and EMG electrodes implanted into RPNIs and muscles in their residual limb. The subjects performed various experiments, during which the team recorded EMG signals from the implanted electrodes plus dry-domed and gelled (used to improve contact with the skin) surface electrodes.

In one experiment, participants were tasked with controlling a virtual hand and wrist in real time by mimicking movements (various grips) on a screen. The researchers used the recorded EMG signals to train linear discriminant analysis classifiers to distinguish the cued grips, training separate classifiers for each electrode type.

They then evaluated the performance of these grip classifiers during a posture classification experiment, in which the subjects actively controlled hand or wrist movements of a virtual hand. Participants achieved faster, more accurate and more reliable control using the implanted electrodes than the surface electrodes.

With participants sitting and keeping their arm still, the implanted electrodes achieved average per-bin accuracies (the percentage of correctly classified time bins) of 82.1% and 91.2% for subjects 1 and 2, respectively. The surface electrodes performed worse, with accuracies of 77.1% and 81.3% for gelled electrodes, and 58.2% and 67.1% for dry-domed electrodes, for subjects 1 and 2, respectively.

The researchers repeated this experiment with the subjects standing and moving their arm to mimic daily activities. Adding movement reduced the classification accuracy in all cases, but affected the implanted electrodes to a far smaller degree. The control success rate (the ability to hold a grip for at least 1 s, within 3 s of seeing a movement cue) also diminished between still and moving conditions, but again, the implanted electrodes experienced smaller decreases.

Overall, the performance of online classifiers using implanted electrodes was only slightly affected by arm movements, while classifiers trained on surface electrodes became unstable. Investigating the reasons underlying this difference revealed that implanted electrodes exhibited higher EMG signal amplitudes, lower cross-correlation between channels, and smaller signal deviations between still and moving conditions.

The Coffee Task

To examine a real-world scenario, subject 1 completed the “Coffee Task”, which involves performing the various grips and movements required to: place a cup into a coffee machine; place a coffee pod into the machine; push the start button; move the filled cup onto a table; and open a sugar packet and pour it into the cup.

The subject performed the task using an iLimb Quantum myoelectric prosthetic hand controlled by either implanted or dry surface electrodes, with and without control of wrist rotation. The participant performed the task faster using implanted electrodes, successfully completing the task on all three attempts. For surface-based control, they reached the maximum time limit of 150 s in two out of three attempts.

Although gelled electrodes are the gold standard for surface EMG, they cannot be used whilst wearing a standard prosthetic socket. “With the Coffee Task, use of the physical prosthetic  hand is needed, so this was only performed with dry-domed surface electrodes and implanted electrodes,” explains first author Dylan Wallace.

The researchers also assessed whether simultaneous wrist and hand control can reduce compensatory body movements (measured using reflective markers on the subject’s torso), compared with hand control alone. Without wrist rotation, the subject had to lean their entire upper body to complete the pouring task. With wrist rotation enabled, this lean was greatly reduced.

This finding emphasizes how wrist control provides significant functional benefit for prosthesis users during daily activities. Chestek notes that in a previous study where participants wore a prosthesis without an active wrist, “almost everything we asked them to do required large body movements”.

“Fortunately, the implantable electrodes provide highly specific and high-amplitude signals, such that we were able to add that wrist movement without losing the ability to classify multiple different grasps,” she explains. “The next step would be to pursue continuous, rather than discrete, movement for all of the individual joints of the hand –  though that will not happen quickly.”

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