Reply to Comment on: ‘Experimental indications of non-classical brain function’ (2022 Journal of Physics Communications 6 105001)

We have recently suggested a proposal to explore non-classicality in the brain, for which we developed an entanglement witness protocol using MRI. The witness protocol intended to find spin interactions which could not be explained by classical interactions, based on intermolecular multiple quantum coherence (iMQC). As for Warren’s comments, we show in more detail that this was indeed the case; our observations were certainly not generated by interactions based on the iMQC model. Further, we discuss some additional details not present in the original paper.

For our study , we were inspired by the fact that in quantum systems local operations and classical communications (LOCC) cannot increase entanglement between quantum systems (Horodecki et al 2009). In NMR, the quantum mechanical description of spin interaction in fluid-like systems only takes LOCC into account (Richter et al 1995). Known as iMQC (intermolecular multiple quantum coherence), it comes as no surprise that the predictions of the iMQC are equivalent to a fully classical derivation considering only magnetic dipole interactions (Jeener 2000), which is named multiple spin echo (MSE) (Deville et al 1979). In the following, we will refer to both as classical.
In our experiments, we were looking for signals which cannot be explained by any classical means. That would be the case if the signal intensity surpasses the maximum of the classical predictions. It has been shown that if the signal exceeds this classical bound, then the system must contain entangled spins (Gärttner et al 2018). We discovered that biological dynamics can create those entanglements under certain circumstances Pérez 2022, Pérez et al 2022). The amount of entanglement depended on conscious awareness , and the complexity of the signal showed a correlation with ageing and short-term memory performance , respectively. These results indicate that entanglement production is a part of cognitive and consciousness-related processes. Currently, it is not clear whether the microscopic interaction between the spins, which is the mediator of the entanglement, is classical or quantum (Hall and Reginatto 2018). For pure spins, some hybrid models can entangle classically. Whether or not this is possible for our findings, where we observed mixed states in thermodynamics, is still an open question. The discovery of entanglement in quantized light, which was coupled with bacteria, has favoured quantum mediation via electrophysiological oscillations (Marletto et al 2018). The presence of heartbeat evoked potentials during our observations suggest a similar mechanism. On the other hand, a recent study has shown that macroscopic biology can mediate entanglement even without any cell activity (Lee et al 2022).
Those findings demonstrate that the mechanisms used by the mediation are beyond biological interest. Instead, the mediator may be of fundamental interest to evolve quantum mechanics from the physic labs into the real world. However, the macroscopic dynamics of the brain (brain functions) are either hybrid quantum systems or quantum systems, which are both non-classical. In both cases, the unknown brain function is nonclassical as claimed in our paper . The entangled spins may then influence many Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. physiological and cognitive processes, which consequently are also non-classical. From here, let us return to our observations.
In 2013, when we first recognized the heartbeat evoked signals, we were fascinated by the fact that the brain can generate signals that cannot be explained by blood flow related mechanisms, movement, or NMR relaxations. We realised, that contrast mechanisms effecting the single quantum coherence (SQC) could be ruled out. Moreover, further investigations indicated that the signals were generated by dipole interactions. Here again, iMQC could be ruled out as contrast mechanism as we will explain below. We concluded that spins in pure states, as studied in SQC and iMQC, may not be the source of our observations. Even entanglement of pure spins seemed unlikely because if pure spins entangled, then those spins would become invisible in the SQC. That means that the net number of spins visible in the SQC signal would decrease. Because we can exclude entangled pure spins, arguments in Braunstein et al (Braunstein et al 1999) aren't relevant for our considerations. However, it doesn't mean that there can't be entanglement. Instead, if pure states are excluded, then the signal must arise from mixed states, which are normally invisible to magnetic resonance imaging (This case was not considered inBraunstein et al (Braunstein et al 1999). However, they can be visible in an MQC design if they become entangled (Gärttner et al 2018). This is, of course, a radical departure from the conventional MRI view. Nevertheless, it is clear that we oppose the idea that the brain is a chemical solution in thermal equilibrium. Living systems perform complex computations, producing, distributing, and saving information, and other tasks. In such a system, the entropy, a measure of information, cannot be maximal to reach a thermal equilibrium. It seems obvious that rules that apply in non-living materials in physics and chemistry cannot apply to living matter. They are not only very different, but they are also not conscious. Consequently, they may handle underlying mechanisms of information in other ways than in the brain. It seems that it is against good scientific practice to transfer knowledge without proof between dissimilar systems. Therefore, we should consider experimental discovery as such, without being biased about what can or cannot be.
It is important to notice that we have never claimed to have measured an iMQC effect (It is, nevertheless, confusing that we have been referring to the sequence as a MQC. This, however, originates from the Gärttner et al. paper (Gärttner et al 2018), which refers to the entanglement witness as an upper bound of the signal that could be generated by a MQC sequence design.) We instead claimed that we have used a sequence, which could be used to measure zero spin echoes (ZSE) or intermolecular zero quantum coherence (iZCQ) by its design (the ZQC sequence design consists of two RF pulses with embedded gradients (called the asymmetric gradients in (Kerskens and Pérez 2022)), which represents the simplest set-up for ZQC.). However, although the sequence is capable of measuring iZQC in principle, it is not sensitive enough to do so. Consequently, if this sequence detects dipole interactions, then the assumptions made in ZSE or iZQC are not fulfilled. Therefore, a lengthy discussion about iMQC is unnecessary for the discord surrounding our findings.
It is, however, important to discuss Warren's opposition to our witness protocol. He claims that a iZQC sequence could detect signals up to 41% (we only detected signals up to 15%). His claim is based on the following equation (Rizi et al 2000) where J 1 is the first-order Bessel function; M 0 is the initial magnetization before the first RF pulse; s z 3 1 2 ] in which ŝ is the unit vector along the gradient direction, and zˆis the unit vector of the static magnetic field; τ d is the dipolar demagnetizing time (≈310 msec for pure water at 3T). For small x = 1 (J 1 (x) ≈ x/2), the signal is proportional to M 0 2 ( ) . The maximum signal of 41% is reached at TE = 806 ms. The equation ignores relaxation, which is present in the brain tissue and which we have detected experimentally.
In our paper, we considered the possible signal intensity very generously in order to include any eventualities. However, considering our sequence parameters, we used a TE = 5 ms, the maximum signal could reach 0.4% of the initial magnetization. If relaxation is included, then the reduction will be down to 0.13%, which is the best-case scenario without any saturation. The initial magnetization is proportional to the iZQC signal S M iZQC 0 2 µ ( ) for short TE. In ultra-fast MRI sequences, M 0 is reduced by more than a magnitude by saturation, which means S iZQC < 0.01%. This is far below the detection limit and many magnitudes below the signals we detected. It is now clear that we have not measured iZQC. We could, however, reproduce the signal dependencies of the parameters α and Δ s in the above equation, including T2 relaxation. Furthermore, we found that the gradient direction represented in Δ s influenced the signal as expected. However, the correlation distance, as predicted for ZQC, played no role (We varied the magnet field gradient moments as mentioned in the methods chapter of (Kerskens and Pérez 2022) from 32.5 ms mT m −1 to around 80 ms mT m −1 ).
A classical mechanism also fails if we consider the following gedankenexperiment. Instead of an image train resulting in a highly saturated magnetization with the signal intensity S 0 , we consider a single ZQC experiment during the time window where we observe the ZQC signals S ZQC . Then, the S ZQC should be at least by a magnitude higher than in the saturated case, which means for our observation S ZQC > S 0 . With S ZQC < 0.42 · S 0 , we get 42 > 100, which clearly shows that the classical relation S M ZQC 0 2 µ ( ) is in contradiction with our findings.
In conclusion, the flow of our paper  is logical, and the experimental details are clearly documented. In our study, we have measured an effect that cannot be explained by classical or basic quantum mechanical models based on decades of published work. Since the signals observed in  were above the classical bound, we interpreted them as being an entanglement witness (Gärttner et al 2018). Finding the entanglement witness was the methodical aim of the paper. It allows us to have a glimpse at the foundation of consciousness, which otherwise is difficult to do. This is of course only the cause, if our assumptions about the maximal classical signal are right, which are based on models derived for chemical solutions but which also have been used for many years in biological systems. The question to ask is; could biology create a classical mechanism that could match our observations? This is where constructive criticism could be interesting.
In summary, we proposed an entanglement witness, which is based on the idea that entangled spins can create higher signal intensities than classically interacting spins in a MQC MR sequence. Hence, the iMQC model, which considers only classical interactions, is a good approximation to estimate the maximal achievable classical signal. It can then be used as a classical threshold, above which non-classicality comes into the game. We found signal intensities, which were at least a thousand times higher than this threshold. This and further considerations suggest that the signals cannot be described by a classical iMQC model. Hence, we suggested a non-classical mechanism.

Data availability statement
No new data were created or analysed in this study.