We’ll start with the assumption that the laws of physics are sufficiently correct as written. Sufficiently correct means that for purposes of reconciling to our 1st person experience we don’t need to discover any new physics, such as a 5th force, or any new secret particle. The laws of physics may be refined in the future, sure, but for our purposes here we’ll assume we’ve got conceptually everything we need and see where we can go with it. This is particularly important regarding quantum mechanics. Quantum mechanics is one of the most successful, most validated, and most accurate theories ever. But, it is an especially weird subject matter and nothing we can say here can make it intuitive or sensible. The hard truth is, when we look up close at the Universe, things look very different than the reality we are accustomed too. One strange concept, called quantum measurement, connects the observer with the observed in an intimate way. For instance, the angle from which we look at an electron affects the direction it is spinning. Without touching, or otherwise disturbing the electron, the mere direction from which we view it affects the state it will be in! Now, the observer does not have to be a person, it can be any macroscopic device, such as a computer, a camera, photomultiplier tube, something called a Stern-Gerlach apparatus, and much more.
To this day, there is no scientific consensus about how to interpret the strange nature of quantum mechanics. However, very recently results have finally shed light on this problem. This paper, entitled “Understanding quantum measurement from the solution of dynamical models” by A. Allahverdyan et al. (2013) has shown how quantum measurement can be understood. The paper describes some very complex mathematics, yet, a conceptually simple framework and involves some things called quantum dynamical models, statistical mechanics, and “asymptotic cascades to pointer states“. It also describes how quantum measurements occur without any abrupt or time-irreversible
collapse of the wave function. When the coupling between the measuring apparatus and the system is dominant, the system entangles with the measuring apparatus and then cascades to a pointer state registering the measurement, but, when the coupling to the environment dominates, decoherence occurs (when information about the system leaks into the environment). We won’t dwell on the lengthy mathematical details but encourage the interested reader to dive in as this paper is tantamount to a sensible interpretation of quantum mechanics. For our purposes, it will be a sufficient approximation to follow the Copenhagen interpretation and view observation as causing the wave function of the observed to collapse (see Born rule). The wave function will collapse to states dictated by the measurement device (called basis states). This strange point connecting the observer and the observed will turn out to be essential to our attempts to reconcile the 1st and 3rd person perspective, so, below we give a, hopefully, illustrative example.
Suppose we place an electron in a toy box through a door in the top as shown in (figure 4). The electron is spin “up” (spinning in a right-handed way about an axis as depicted by the arrow – the arrow points along the axis of rotation like an axis between the north and south poles of the Earth). Technically, electron spin is a little bit more complex than this description, but, for our purposes, this will suffice to illustrate the concepts involved (the interested reader can dive in deeper to electron spin here). Next, we close the door, and open another door on the front side of the box. Fifty percent of the time the electron will be found pointing toward us (out of the page), and the other fifty percent away from us. There is no way to determine definitively which direction (toward or away) the electron will be spinning, all we can calculate are the probabilities. Whichever side of the box we open, that will determine the axis about which the electron is spinning. If we open a side door, the electron will be found spinning about that axis – toward or away with equal probability. However, if we place the electron in the box through the top, close the door, then re-open the top door, the electron will still be in the same state we left it with near one-hundred-percent probability.
Figure 4: A spinning electron is inserted through the top of a box with spin pointing up (counter-clockwise rotation about the axis). The door is closed. A door on the front is then opened. The electron can only be found pointing toward the observer or away. In this case, it will do so with 50/50% probability. The side of the box that we open determines the axis of the electron – a strange aspect of quantum mechanics that intimately connects the observer with the observed.
Our toy box is a metaphor for a spinning electron in a magnetic field. In the laboratory, electrons may only have spin +1/2 
Another strange aspect of quantum mechanics is entanglement. We can entangle electrons so that they become one
system of electrons. In such an entangled system, we can know all there is to know about the system without knowing anything about the individual state of a particle within the system (see a more detail description here). The system of electrons has literally become one
thing. Moreover, entanglement is the only phenomenon like this in all of physics, that is, it is the only means by which particles may become one system of particles larger than themselves. Furthermore, there is no theoretical limit on how many particles may be entangled together or how complex the system may be. Now, there are many ways to entangle electrons, but one way is to bring them close together so that they “measure” each other (so their magnetic coupling to one another is stronger than the coupling to the surrounding environment). When one electron measures the other, it will either find it spinning the same direction or opposite – just like any measurement must. The electron is not big enough to force a single outcome like we see when we open the toy box, so the two electrons will sit in a quantum superposition of states: they with both be spinning in the same direction and spinning opposite at the same time (we refer the reader back to this paper for a detailed mathematical description of what “big enough” is). They will remain entangled in a superposition until stronger external interactions, like with the environment or macroscopic observers, cause decoherence or specifically measure the system. Generally, in the laboratory, this happens very fast, like femtoseconds (one millionth of one billionth of a second) and is the reason most scientists are skeptical of quantum mechanics playing anything but a trivial role in biological systems. This paper, “The Importance of Quantum Decoherence in Brain Processes” by Max Tegmark (1999) shows, for example, that whole neurons are way too big to exist in a superposition of states for biologically relevant timescales due to decoherence. Still, there is abundant evidence that quantum superpositions and entanglement exist in biological systems on smaller scales. A comprehensive list can be found in the nascent field of quantum biology for which the reader can find an excellent introduction in this book: “Life on the Edge: The Coming Age of Quantum Biology” by J. Al-Khalili and J. McFadden (2014). The best studied example of quantum effects in biology occurs in the process of photosynthesis in the FMO complex of green sulfur bacteria where quantum states are observed to persist for as long as picoseconds (one trillionth of a second) and which allows plants to convert light from the sun into chemical energy in a perfect, fast, 100% efficient process.
Figure 5: Quantum biology – photosynthesis. Diagram of the FMO complex. Light excites electrons in an antenna. The quantum exciton then transfers through various proteins in the FMO complex to the reaction center to further photosynthesis. by By OMM93 – Own work, CC BY-SA 4.0 via Wikipedia
Decoherence rates are highly dependent on the surrounding environment and dynamics of the system. For example, if it is really, really cold (~ absolute zero), and isolated, then entanglement can theoretically last on the order of seconds or longer. The dynamics are critical too – like if something keeps cyclically pushing particles together – entanglement can, again, in theory, last much longer. If the cyclicality happens faster than decoherence, entanglement may, in theory, be sustained indefinitely. Interestingly, biomolecules in organisms are really buzzing, vibrating at frequencies ranging from nanoseconds to femtoseconds. Comparing these times to the picosecond decoherence times observed in photosynthesis and a plausible means by which quantum effects may persist in biological systems becomes apparent. The interested reader can find a mathematical description of this type of dynamic entanglement in these papers:
“Persistent Dynamic Entanglement from Classical Motion: How Bio-Molecular Machines can Generate Nontrivial Quantum States” by G. G. Guerreschi, J. Cai, S. Popescu, and H.J. Briegel (2012)
“Dynamic entanglement in oscillating molecules and potential biological implications” by J. Cai, S. Popescu, and H.J. Briegel (2010)
“Generation and propagation of entanglement in driven coupled-qubit systems” by J. Li and G.S. Paraoanu (2010)
“Steady-state entanglement in open and noisy quantum systems” by L. Hartmann, W. Dür, and H.J. Briegel (2005).
They describe a theory by which entanglement can be sustained in noisy, warm environments in which no static entanglement can survive. In other words, the way physicists try to build quantum computers today by completely shielding the qubits from the outside world probably won’t work in biological systems, but this isn’t the only way to go about it. This is an important point and is critical for the reconciliation we propose in this essay. We should note that this theory of dynamic entanglement has not been experimentally verified, but no one, so far, has done the experiments to look for it. For the rest of this essay, we will assume that this theory pans out experimentally and entanglement will be found to be sustainable in biological systems.
Now, suppose we entangle two electrons so that their spins are pointing parallel (see here for more about how this is accomplished in practice). We can place them in two separate boxes as shown in (figure 6). When we open any other door of the box, even if the boxes are at opposite ends of the galaxy, the electrons will be found to be spinning parallel to each other. The measurement of one instantaneously affects the state of the other no matter how far away it is. This is the property of quantum mechanics that Einstein labeled “spooky action at a distance”. Nonetheless, experiment after experiment has supported this strange property (see “Physicists address loophole in tests of Bell’s inequality using 600-year-old starlight“).
Figure 6: Two entangled electrons placed into two boxes through the top, separated by a galaxy, then opened on the side (or any side for that matter), will always be found spinning in the same direction -it could be either left or right, but A and B will always point the same way. Picture of Milky Way Galaxy here.
Once two electrons are entangled we can perform quantum operations on them to put the system into a superposition of four different states at once: (1) “A” up and “B” down, (2) “A” down and “B” up, (3) “A” up and “B” up, and (4) “A” down and “B” down. Entanglement is not limited to just electrons – photons (light), nuclei, molecules, many forms of quasiparticles including macroscopic collections of billions of particles can be entangled (see, for example, SQUIDs, phonons or solitons). To the extent these configurations have binary states (electrons: spin up or spin down, photons: right-handed or left-handed polarization, etc.) they may represent quantum bits or qubits like in a quantum computer (although analog quantum computing is just as plausible as a binary system). In the case of electrons, spin-up could represent a value of, say, “0” and spin-down a value of, say, “1”, to perform powerful quantum computations. The power of such computations is derived from the quantum computer’s ability to be in superposition of many states at once, equal to 
However, to take advantage of all these superpositions, we must find a clever way to make them interfere with each other. We can’t look at the result until the very end of the computation and when we do the superposition will collapse. If we can get the states to interfere in the right way we can get the system to be in a superposition that is concentrated, with near 100% probability, in a state that corresponds to a solution to our problem. This is the case with a special algorithm known as Shor’s algorithm that can solve certain NP problems in polynomial time. NP problems are those that require exponential time on a classical computer to solve (see more on P vs NP here, and here). Shor’s algorithm uses something called the quantum Fourier transform to achieve this speed-up and is used to factor large integers. This is an important problem in cryptography and is the “go to” technique that encrypts substantially all the traffic over the internet. For example, factoring a 500-digit integer will take longer than the age of the Universe on a classical computer, but less than two seconds on a quantum computer – hat tip to John Preskill for the stats, see his great introductory video lecture on quantum computing here (2016). To perform such computations, it is thought to require about 1,000 qubits. Other examples of NP problems include the infamous traveling salesman problem. It is an open problem whether all NP problems can be solved in polynomial time on a quantum computer. The vast majority of physicists and computer scientists think this is unlikely, however.
Figure 7: Quantum subroutine in Shor’s algorithm By Bender2k14 – Own work. Created in LaTeX using Q-circuit CC BY-SA 4.0 via Wikipedia
It is time to note, though, that there is a strange, unproven, powerful possibility lurking in quantum mechanics. All the quantum computers that have been designed to date utilize something called the Linear Schrödinger Equation (LSE) – an ubiquitous form in quantum mechanics. That means that the qubits are contained by forces external to the qubit system itself, like an external magnetic field. A much rarer, and controversial, situation in quantum physics involves the Non-Linear Schrödinger Equation (NLSE) and this occurs only when the quantum state wave functions interact with themselves. NLSE systems may potentially have a profound impact on quantum computation because they can theoretically solve NP-complete problems in polynomial time (only if the time evolution of the Schrödinger equation turns out to be nonlinear), and this means they can solve all NP problems in polynomial time. The interested reader can dive in here: “Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for NP-Complete and #P Problems” by D. Abrams, S. Lloyd (1998), and in chapter 5 here “NP-Complete Problems and Physical Reality” by S. Aaronson (2005). So far, such a system has never been implemented, nor even a theoretical design proven though the idea continues to be debated. Here is the latest on the subject: “Nonlinear Optical Quantum-Computing Scheme Makes a Comeback” by D. Brod and J. Combes (2016).
The NLSE does appear in the study of a number of special physical systems, , for example, in Bose-Einstein condensates (BEC) (see this paper on building a quantum transistor using the NLSE in a BEC), in fiber optic systems, and also, especially interestingly, in biology: like the Davydov Alpha-Helix Soliton which forms a complex quasiparticle that transports energy up and down the chain of protein molecules, and in Fröhlich condensates which have recently been observed experimentally in biomolecules upon exposure to THz radiation (see Lundholm, et al. 2015). The interested reader can dive into Weinberg’s original paper (1989) on the NLS Equations here for more, and further enhancements of the theory here and here that address some difficulties. Also, see here for the development of the NLSE using Ricati equations.
To get some hands-on experience with basic quantum computers, IBM has a 5-qubit machine, albeit strictly implementing a linear Schrödinger equation, online right now that is freely available to everyone. Go to www.ibmexperience.com to learn more.
Figure 8: Examples of solutions to Non-Linear Schrödinger Equations. Absolute value of the complex
envelope of exact analytical breather solutions of the Nonlinear Schrödinger (NLS) equation in nondimensional form. (A) The Akhmediev breather; (B) the Peregrine breather; (C) the Kuznetsov–Ma breather. From: Miguel Onorato, Davide Proment, Günther Clauss and Marco Klein (2013) “Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test”. PLoS One 8(2): e54629, doi: 10.1371/journal.pone.0054629, PMC 3566097 via Wikipedia
