Do we really need Quantum Mechanics?

Author: Katie Handford, Girls in Quantum

Do we really need quantum mechanics? If I’m being entirely honest, when this question was first posed to me at the beginning of the year, my initial reaction, as someone who had only the week before accepted a PhD position in quantum information, was, I really hope so. Upon further consideration however, it struck me that for those of us who use quantum mechanics on a daily basis, we never really consider much more than, it works! As the use of quantum applications in technology, academia and industry increases, it is worth knowing whether we are neglecting alternative formalisms for these systems, simply blinded by the appealing aspects of a quantum theory.

Beginning from the introduction of discrete energy levels to explain the blackbody problem, moving through the introduction of light and atomic quantisation and the discovery of wave-particle duality, all culminating in Heisenberg’s development of matrix mechanics, closely followed by Schrödinger’s theory of wave mechanics; quantum mechanics has been contributed to by many individuals and has been consistently at the forefront of scientific advancement for the last century.

The development of quantum mechanics did not come without its criticisms, as it contradicts our own experience of the universe. Classically, an object has physical properties regardless of whether it is being observed. If I have a purple book and I then place it under my desk, so it is no longer being observed, we aren’t going to argue that the book is still purple, despite the fact we are no longer looking at it. However, in quantum mechanics, objects do not possess definite physical properties, therefore by placing my book under my desk, out of sight, it is no longer known to be purple and can be considered to take any colour. In quantum mechanics, the value of any physical property is a consequence of measurement being performed, whereas classically, physical properties have pre-fixed values, simply revealed by the act of measurement.

It is these ideas, contradicting our own experience of the universe, that led to reservations regarding quantum theory. One of the most vocal sceptics was Einstein who argued that any theory lacking definite values of physical properties could not be a “complete” theory capable of explaining all elements of reality [1].

Einstein’s argument can be visualised using the following example. Consider a neutral pion. It is observed that, upon decay, the pion emits an electron and a positron moving in opposite directions. As a neutral pion is a zero-spin particle and electrons and positrons are spin half-particles, these must exist in an entangled state with opposite spin values. As this is a quantum system, the electron and positron do not have definite spin values until they are measured.

It is seen that if you measure the electron and find it has a value of spin up, this induces an instantaneous collapse of the wavefunction and forces the positron to have a value of spin down. This means, by measuring one of the particles, the value of the other can be known with absolute certainty.

As these two particles are separated by a large physical distance and they do not have pre-determined spin values, in order for the measurement result of the electron to dictate the positron value, a signal must travel between the two, communicating the electron’s measurement outcome so the positron value can be fixed, ensuring the total spin of the pion can be conserved. For this measurement of the electron to instantaneously impact the positron would require a signal to travel between the two, faster than the speed of light.

Einstein argued that, as the theory of special relativity prohibits such a communication, the fact that the measurement value of my positron can be predicted with certainty from the outcome with my electron, then “elements of reality” must exist that gives pre-fixed values for this quantity. John Bell proposed a thought experiment designed to test Einstein’s argument, by completing quantum mechanics through the addition of these elements of reality, in the form of hidden variables [2]. The general process for the thought experiment is as follows. A central user, Charlie, distributes two particles, one to Alice and one to Bob, who are separated by a physical distance. Alice and Bob both randomly select between one of two properties of the particle to measure. The key assumption is the requirement for local realism. Realism ensures that the properties selected to be measured have pre-fixed values. Assuming locality means that both Alice and Bob perform their measurements simultaneously, therefore the measurement performed by Alice cannot impact the measurement undertaken by Bob, and vice versa.

The measurement outcomes found by Alice and Bob respectively are reliant on both the local settings of their measurement devices and a hidden variable, which encodes all the physical properties of the particle. It is this inclusion of hidden variables that introduces the “elements of reality” Einstein suggested were required to explain the “spooky action at a distance”, allowing measurement of space-like separated particles to seemingly have instantaneous impacts.

By considering the average values for all possible combinations of measurements performed and all the possible probabilities of achieving any outcome, an upper limit on the average of all possible measurement outcomes is determined. This upper limit is known as the Bell inequality. This limit must be obeyed by all classical correlations and provides a strict criterion for the limitations of a classical description. If it could be demonstrated that all the statistical predictions of quantum mechanics obey this bound, then Einstein’s argument that quantum mechanics should be completed by including hidden variables would be confirmed.

In the context of our question, if it could be proven that all the predictions of quantum mechanics respect this limit, then it would demonstrate that all the results achieved by quantum mechanics could be reproduced entirely classically. This would mean that our use of quantum mechanics would be a matter of simplicity rather than necessity as nothing prohibits a classical approach.

Now, we can consider a two qubit maximally entangled state and apply this system to the experimental procedure described above. In this case, the first qubit is sent to Alice, and the second qubit is sent to Bob, where measurements are performed using the Pauli matrix operators [3]. The average values of the possible measurement combinations can be calculated using the quantum expectation values and when these are applied to the Bell inequality, derived in the classical experiment, it is shown that the value obtained exceeds the upper bound. The Bell inequality is violated. Violation of the Bell inequality shows that the two qubit state exhibits quantum correlations that cannot be achieved classically, therefore demonstrating the need for a theory beyond a classical approach to account for these properties.

Not all quantum states violate the Bell inequality. It is often quantum interference that is responsible for this. This is present most notably in pure entangled states, such as the maximally entangled two qubit state, used above. As violation of the inequality, is reliant on the entangled nature of the state, Bell’s work also acts as evidence for the verification of entanglement as a truly quantum property.

Following on from Bell’s work, it is interesting to consider whether there are other quantum phenomena where a classical model can be conclusively excluded. An example of a property for which this is possible is quantum contextuality. Quantum theory challenges the idea of prefixed property values prior to measurement by showing that the outcome of measuring these properties is dependent on the set of compatible measurements undertaken at the same time; the context. Quantum contextuality refers to the inability to produce a context-independent classical model capable of reproducing all the predictions of quantum mechanics. This can be demonstrated simply using the Peres-Mermin square and is widely considered the clearest indicator of true “quantumness” [4].

The Heisenberg uncertainty relation is arguably one of the most well-known features of quantum mechanics, providing a lower limit on the accuracy of simultaneous measurement of conjugate variables, such as position and momentum.

This intrinsic property of quantum systems is a consequence of non-commutation and is often considered the fundamental hallmark of quantum theory, attributed with the unique departure from a deterministic, classical description of reality. In contrast to the Bell inequality, which is a classical result, the uncertainty relation is a quantum result by which all quantum systems must oblige and any violation of this bound is proof of classicality.

It can in fact be demonstrated that a classical analogue of the uncertainty principle can be derived both experimentally and theoretically. As in classical theory the momentum and position of a particle can be simultaneously known exactly, any classical implementation is an epistemic restriction, resulting from the limitation on the observer’s knowledge.

As the Heisenberg uncertainty principle is often considered responsible for the classical-quantum divide, it is of interest to consider whether it is simply this relation that is responsible for the “quantum” properties characteristic of the theory. If this is the case it would imply that it is in fact the uncertainty relation that has prompted the need for a non-classical theory. As classical analogues can be both theoretically applied and experimentally produced, if all of quantum mechanics could be re-formalised in this way, the use of quantum theory to describe statistical systems may be a convenient choice rather than a necessity.

To investigate this, the ψ-epistemic research program [5] was developed by Terry Rudolph and his team where it was suggested that a quantum state is not a direct representation of reality, but rather a representation of an observer’s incomplete knowledge of a classical system. Using Liouville mechanics, epistemic restriction was applied such that the limitations on the knowledge of the system are equivalent to that allowed under the Heisenberg uncertainty relation. By applying these appropriate restrictions, this classically-rooted theory can demonstrate operational equivalence to Gaussian quantum mechanics.

This framework can also be used to demonstrate the reproduction of principles andproperties that had been previously considered to be fundamentally “quantum” phenomena. Some examples included the non-commutation of measurements which can be expressed as using the commutativity of the Poisson bracket, wave function collapse simplifies to a Bayesian update, resulting from an improvement in knowledge as a result of measurement, followed by a random disturbance, to ensure the restriction is still obeyed and the no-cloning theorem can be demonstrated using classical fidelities and the data processing inequality [5]. This inequality prohibits transformations resulting in a decrease in fidelity, which would be required for cloning to occur.

Whilst great success was found in the reproduction of “quantum” properties, this work also reveals several properties for which this restriction is not enough. Both Bell

inequality violations and quantum contextuality cannot be reproduced. This further solidifies what we observed previously, that these properties are representations of true “quantumness” and do warrant a non-classical description.

So, returning to our original question, do we really need quantum mechanics? Yes, we do. Those of us who have chosen to build our careers in quantum theory can breathe a sigh of relief, we all get to keep our jobs. Properties like Bell inequality violations and the existence of contextuality conclusively exclude a classical description, therefore proving the necessity of quantum mechanics. However, as we have shown, some properties that are typically considered characteristic of quantum theory do in fact exist entirely classically, therefore greater caution must be taken when considering whether a property truly warrants the “quantum” label.

References

[1] A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of

Physical Reality Be Considered Complete?” Phys. Rev., vol. 47, pp. 777 – 780, 10 May 1935

[2] J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika, vol. 1, pp. 195 – 200, 3 Nov. 1964

[3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition. Cam bridge University Press, 2010.

[4] C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-Å. Larsson, “Kochen-Specker contextuality,” Reviews of Modern Physics, vol. 94, no. 4, Dec. 2022, issn: 1539–0756.

[5] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A, vol. 86, p. 012103, 1 Jul. 2012