Two recent theoretical advances with potential impact on quantum technology Open Access

Any practical quantum device must interact with the classical world of our direct experience in some way. For instance, a quantum sensor needs some kind of display, and a quantum computer should presumably interface with a classical computer to achieve its full potential. A Hamiltonian coupling between quantum systems and their classical environment is expected to be the most effective method for transferring “results” gained by quantum sensors or computers. However, a detailed understanding of dissipative couplings between quantum and classical systems is also important, if only to minimize undesirable dissipative interactions.

The good news of the first part of this paper is that there exists a general thermodynamic framework for coupling quantum systems to their classical environments, which are assumed to evolve according to the laws of reversible and irreversible dynamics. This framework has been established by quantizing a geometric formulation of classical irreversible thermodynamics, offering a significant extension of the theory of open quantum systems.^1,2

The development of new and improved quantum devices would undoubtedly benefit from a more intuitive understanding of quantum mechanics. In particular, entanglement, which embodies the holistic nature of quantum mechanics, is counterintuitive but crucial. In the standard approach, it leads to several well-known paradoxes. When a quantum system is divided into two subsystems, such as two particles or groups of particles, entanglement is the phenomenon that each subsystem cannot be described independently of the state of the other subsystem, even when the subsystems are separated by a large distance.

After 100 years of quantum mechanics, rather than adhering to dogmatism and relying solely on its mathematical framework, we should demand a convincing interpretation. Should we not consider the so-called measurement problem of quantum mechanics a potential obstacle for the advancement of quantum technology?^3–6 

At least Feynman was amusingly irritated when he began contemplating what quantum computers could be good for,⁷ “we always have had a great deal of difficulty in understanding the worldview that quantum mechanics represents. At least I do, because I am an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it … you know how it always is. every new idea, it takes a generation or two until it becomes obvious that there's no real problem. It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore, I suspect there's no real problem, but I'm not sure there's no real problem.”

The good news in the second part of this paper is that there exists a new interpretation of quantum mechanics, which offers a fresh perspective on entanglement and eliminates the paradoxes that arise in the standard approach to quantum mechanics.

The emergence of irreversibility has been intensely investigated and heavily debated for ∼150 years. Boltzmann’s transport equation for rarefied gases⁸ marks the first milestone in this development. It is an irreversible equation for the single-particle probability density in position and momentum space based on the collision laws obtained from the Hamiltonian dynamics of gas particles. It also implies an evolution equation for entropy. By the end of the 19th century, Boltzmann had clearly understood the probabilistic nature of the second law of thermodynamics, recognizing that violations would never be observed for macroscopic systems but could become noticeable in very small systems.

Fluctuation–dissipation relations^9–13 and projection-operator methods^14–18 played a significant role in advancing the field of irreversible thermodynamics. In the 1960s, the foundational principles for formulating linear irreversible thermodynamics were well-established and documented in a classical textbook.¹⁹ 

An elegant geometric formulation of classical nonequilibrium thermodynamics, initiated by Grmela^20,21 in 1984, has led to the so-called GENERIC framework (general equation for the nonequilibrium reversible–irreversible coupling):^22,23 Reversible dynamics is generated by energy via a Poisson bracket, whereas irreversible dynamics is generated by entropy via a dissipative bracket.

Understanding the emergence of irreversibility on the quantum level is expected to be considerably more challenging, certainly not easier than for classical systems. We aim to avoid limitations imposed by perturbation theory, simplistic models like reservoirs composed of harmonic oscillators, or approximations of unclear validity.

Lindblad formulated a quantum master equation for the density matrix of a dissipative quantum system based on the assumptions of linearity and complete positivity.²⁴ Grabert has used the projection-operator method to derive a quantum master equation that is nonlinear in the density matrix.^18,25 Both equations address the dissipative coupling of a quantum system to a bath. In many applications, the classical environment should not be limited to a heat bath. Ideally, the coupling of a quantum system to an arbitrary classical nonequilibrium system would be desirable.

A systematic framework for quantum systems in contact with finite quantum heat reservoirs has been established in a pioneering paper.²⁶ This approach elaborates the meaning of entropy production and sheds light on the emergence of irreversibility in the limit of large heat reservoirs. More recent developments are summarized in a broad collection of over 40 papers²⁷ and in a recent review article,²⁸ both of which emphasize fluctuation theorems and information-theoretic aspects while also addressing experimental achievements and practical applications.

The geometric structure of the GENERIC framework offers the opportunity to extend Dirac’s approach to quantization from Hamiltonian to dissipative systems. Instead of deriving suitable master equations for dissipative quantum systems emerging from the reversible equations of quantum mechanics, a quantization procedure, in the spirit of replacing Poisson brackets by commutators, is applied to dissipative classical systems. As illustrated in Fig. 1, this idea eliminates the most challenging task of explaining the emergence of irreversibility at the quantum level.

As the von Neumann entropy is readily available as a generator of irreversible dynamics for quantum systems described by density matrices, one only needs to find a quantization rule for the dissipative bracket, analogous to Dirac’s replacement of Poisson brackets with commutators.

This idea has been pursued in Ref. 29 and further formalized and generalized in Ref. 30. The argumentation and notation in those papers are very abstract because the emphasis is on the deep structural features of the procedure. Here, we offer a much simpler reformulation suitable for practical applications.

The variables chosen to describe a quantum subsystem and its classical environment are summarized in Table I. In addition, energy and entropy as the generators of reversible and irreversible dynamics, respectively, are listed in this table.

The proper arena for quantum mechanics is provided by separable complete Hilbert spaces, which are complex vector spaces equipped with inner products.^31,32 Observables are self-adjoint linear operators on a Hilbert space $H$⁠. Here, we focus on the evolution of the density matrix ρ, also known as the statistical operator on $H$⁠. The density matrix characterizes the state of our quantum subsystem, and its time evolution determines the evolution of the averages $Aρ=tr(ρA)$ of all quantum observables A. This perspective corresponds to the Schrödinger picture, which we use throughout this letter.

The discrete, continuous, or mixed set of variables x for the classical environment forms a manifold $M$⁠. Observables are functions or functionals on the manifold $M$⁠. For notational simplicity, we assume a discrete set of variables labeled by an index j or k. The modifications required for continuous sets of variables, in particular the proper generalization of matrices and partial derivatives, are explained in detail in Sec. 2 2 and Appendix C of Ref. 33.

Note that we allow the Hamiltonian H(x) to depend on the variables of the environment, thereby introducing a reversible coupling of the quantum system and its environment. The appearance of classical external fields in the Hamiltonian of a quantum system is quite common, for example, a static magnetic field in the Schrödinger equation for discussing Larmor precession or the electromagnetic four-vector potential in the Pauli equation.

The first term describes reversible dynamics of the environment generated by the energy. The energy gradient is multiplied by the antisymmetric Poisson matrix L(x), which is given by the symplectic matrix transformed to non-canonical coordinates. The Poisson bracket of two observables is obtained by multiplying the Poisson matrix from both sides with the gradients of the two arguments of the bracket.

The second term in Eq. (7) describes irreversible dynamics generated by the entropy gradient. The friction matrix M(x) is assumed to be positive-semidefinite so that irreversible dynamics essentially follows the entropy gradient.

The remaining term represents the dissipative coupling between the quantum system and its environment. It is constructed such that the change in the energy of the quantum system, as obtained from Eq. (6) for A = H(x), is compensated by the change in energy of the classical environment, as obtained by multiplying the evolution Eq. (7) with ∂E^tot(x)/∂x.

Note that the reversible evolution is governed by the commutator with the Hamiltonian, whereas the dissipative contribution possesses a double commutator structure involving coupling and free energy operators. The classical analog of a quantum master equation is a diffusion of the Fokker–Planck equation for the evolution of a probability density, incorporating first and second derivatives for the reversible and irreversible contributions, respectively.

The master Eq. (9) is our fundamental equation for open quantum systems. As a consequence of the definition (1), the second term in Eq. (9) is generally nonlinear in ρ. This nonlinearity of the irreversible contribution is caused by the noncommutativity of quantum observables and implies that, in general, our master equation cannot be of the popular Lindblad form (see, for example, Sec. 3 7 of Ref. 1 for a discussion of nonlinear quantum master equations). The specific conditions under which the GENERIC quantum master equation is linear in ρ are discussed in the  Appendix.

Equations (7) and (9) describe the evolution of the state variables for the classical environment and the quantum system introduced in Table I. They represent the dissipative coupling of a quantum system to a general classical nonequilibrium system as its environment, achieved through Dirac-style quantization of the GENERIC framework. An additional reversible coupling is included in the dependence of the Hamiltonian H(x) of the quantum system on the classical variables x.

The first term in Eq. (10) describes the entropy production in the classical environment. Note that only the symmetric part of the friction matrix M(x) contributes to entropy production. An antisymmetric contribution to M(x) would describe irreversible processes without entropy production. Historically, this possibility has been introduced by Casimir.^19,35 Recent examples of irreversible processes without entropy production include slip phenomena and the energy cascade in turbulence.³⁶ 

The second term in Eq. (10) describes the entropy production associated with the dissipative coupling of the quantum system and its classical environment. Note that both contributions to the entropy production (10) are always non-negative. In a thermodynamic setting, this property is more relevant than the complete positivity assumed in Lindblad’s approach.

The first derivation of a nonlinear quantum master equation of the type (9) for a quantum system coupled to a heat bath was achieved by using the projection-operator method.^18,25 The same type of nonlinear equation has been recovered from the quantization of GENERIC and illustrated for the examples of a two-level system and a damped harmonic oscillator.³⁴ The zero-temperature limit of the thermodynamic quantum master equation has been discussed in Ref. 37. In addition, heat transport in quantum spin chains has been discussed with this master equation.³⁸ 

It has been shown that the nonlinear quantum master Eq. (9) leads to a biexponential decay, a realistic susceptibility profile, and ultralong coherence of a qubit, which is not limited by the energy relaxation time because complete positivity is not imposed.³⁹ It has been recognized that the thermodynamic quantum master equation, which is generally nonlinear, may be of the linear Davies–Lindblad type if the coupling operators Q_α are eigenoperators of the Hamiltonian of the quantum subsystem and the associated coupling matrices $Kuα(x)$ are chosen suitably (see  Appendix).³⁰ 

The powerful tool of stochastic unravelings¹ of quantum master equations for dissipative quantum systems in terms of stochastic jump processes in Hilbert space has been applied to nonlinear equations of the type (9). The nonlinearity can be produced by mean-field interactions in the stochastic jump process.⁴⁰ One- and two-process unravelings have been developed and tested in Ref. 41.

The dissipative coupling of a quantum system to a time-evolving environment has been explored in Ref. 42. Practical applications include vibrational relaxations in liquids,⁴³ where slower rotational and translational modes can be treated by classical thermodynamics and hydrodynamics, or the Marcus theory of electron transfer in molecular systems,⁴⁴ where the dielectric environment can be treated by classical thermodynamics and electrodynamics. Further applications include spin-selective radical-ion-pair reactions relevant to photochemistry and photosynthesis,⁴⁵ quantum dots exchanging energy with two heat baths,⁴⁶ the coupling of quantum systems to classical opto-electronic systems for the modeling of laser devices,⁴⁶ semi-classical drift-diffusion-reaction models for the transport of charge carriers in opto-electronic devices,⁴⁷ and coupled spin dynamics for a sensitivity enhancement of magnetic resonance imaging and spectroscopy.⁴⁸ 

As a final application, we mention that quantum master Eq. (9) provides the foundations of dissipative quantum field theory [cf. Eq. (1.45) of Ref. 49]. In this approach, dissipative smearing regularizes quantum field theory at short distances. Some ontological implications of dissipative quantum field theory have been discussed in Ref. 50. The unraveling of the quantum master equation leads to a new simulation technique for quantum field theory, where the simulation time corresponds to real time. In this context, it has been realized that it is natural to treat interactions as stochastic jumps.

Efficient simulations based on unravelings, in which the interactions of reversible quantum systems are treated as stochastic jumps, have been developed and tested in Refs. 51 and 52. This reformulation naturally leads to a new interpretation of quantum mechanics,⁵³ which is discussed in Sec. III.

It is desirable for an interpretation of quantum mechanics to be based on quantum field theory. For example, in the hydrogen atom, the electron and the proton do not really interact through the static, classical Coulomb potential appearing in the Schrödinger equation, but rather through the exchange of photons.

At any given time, a hydrogen atom consists of an electron, a proton, and a number of photons mediating electromagnetic interactions between the charged particles. As the proton is no longer considered a fundamental particle, one might prefer to say that a hydrogen atom consists of an electron, three quarks, and a number of photons and gluons mediating electromagnetic and strong interactions. In any case, the hydrogen atom has a well-defined content of fundamental particles at any given time. These remarks should clarify that the usual quantum mechanical treatment of the hydrogen atom is a semi-classical approximation involving static classical interactions at a distance. An illustrative toy version of a quantum field theoretical calculation illustrates how bound states can be treated on a more fundamental level.⁵⁴ The comparison to quantum mechanics is based on the Fourier transformation of the wave functions from position to momentum space.

In contrast to wave functions or density matrices, which describe the properties of an ensemble of hydrogen atoms, stochastic unravelings of density matrices allow us to describe individual atoms. We thus gain a new perspective on both quantum mechanics and quantum technology. The idea that any quantum system has a well-defined particle content suggests that, at any given time, the system can be described by a multiple of a Fock space base vector.^49,55,56

Inspired by the discussion of the hydrogen atom, our goal is to reformulate the equations of reversible quantum mechanics in terms of stochastic jump processes, where interactions are treated as discrete collision events. Therefore, we need a splitting of the full Hamiltonian into free and interacting contributions, H = H^free + H^int. We further assume that there exists a distinguished basis of orthonormal eigenstates |m⟩ of the free Hamiltonian H^free, which are labeled by the natural number m. The corresponding eigenvalues of H^free are given by E_m. Finally, we assume that the strict superselection rule of quantum field theory is inherited by quantum mechanics, meaning the state of the quantum system at any time t is described by a complex multiple of some base vector |m_t⟩. No superpositions between different base vectors are allowed.

The strict superselection rule, which states that linear combinations of different base vectors do not correspond to physical states, reduces the enormous number of possible stochastic jump processes $ϕt$ and $ψt$⁠. It naturally guides us to a construction of piecewise continuous trajectories with interspersed jumps among basis vectors for the two independent, identically distributed stochastic processes. The following unique stochastic jump process has been constructed in Ref. 53.

If the system between the times t′ and t is represented by a multiple of the base vector $m$⁠, the complex prefactor oscillates in time and leads to an overall phase shift given by −E_m(t − t′)/ℏ.

The stochastic bra-ket formulation of quantum mechanics offers a new interpretation of quantum mechanics. It may be considered an alternative to the currently favored interpretations: Bohmian mechanics,^57–61 the GRW approach,^62–64 and the many-worlds interpretation.^65,66

Note that a constant shift of the Hamiltonian H^int does not affect the von Neumann equation, but as it shifts the matrix elements ⟨m|H^int|m⟩, it affects the jump processes. Therefore, to obtain a unique unraveling, it is important to choose the zero of the interaction energy based on physical arguments.

If there is no natural choice for the zero of energy, one might choose the average energy to be zero. A disadvantage of this choice is that the Hamiltonian depends on the energy of the initial state. An advantage is that steady pure states are described by a time-independent $E(ϕt)$⁠.

In Ref. 53, the usefulness of Eq. (20) is demonstrated in the context of the Einstein–Podolsky–Rosen experiment.^67–72 Entanglement arises from the averaging over independent individual states on the bra and ket sides. The wavelike behavior of quantum particles results from an interplay between the bra and ket vectors, as illustrated by the double-slit experiment.⁵³ 

We have shown how a theory of quantum dissipation can be developed by quantizing the geometry-based GENERIC framework of nonequilibrium thermodynamics. Thermodynamics is invaluable because it provides a sound language for science and engineering. According to Einstein’s famous appraisal of thermodynamics,⁷³ “It is the only physical theory of universal content, which I am convinced that within the framework of applicability of its basic concepts will never be overthrown (for the special attention of those who are skeptics on principle).”

The geometric structure on which GENERIC is based should, whenever possible, be preserved in developing numerical methods for solving practical problems. For reversible equations, symplectic integrators that preserve the underlying Hamiltonian structure are known to be powerful numerical tools.^74,75 For classical dissipative systems, promising initial steps have been taken to reproduce the correct behavior of energy and entropy and to preserve the underlying bracket structure.^76–80 For dissipative quantum systems, the development of structure-preserving methods will be even more challenging, particularly when stochastic simulations are included.

The general reversible and irreversible coupling of quantum systems to classical environments is clearly a cornerstone of quantum technology. It is not only a key tool for simplifying or solving problems of practical importance, but it also offers a framework for discussing the measurement problem.

Even closer to the foundations of quantum mechanics is the stochastic bra-ket interpretation described in the second part of this paper. By eliminating the famous paradoxes from quantum mechanics through the application of a strict superselection rule, we may gain deeper intuition about the quantum world. The usual distinction between the classical and quantum worlds is not fundamental but rather a declaration of our lack of understanding and intuition, even after 100 years of quantum mechanics.

According to the bra-ket interpretation, two stochastic jump processes are required to describe an individual quantum system. According to Eq. (13), there are two sides or aspects of quantum variables: a bra and a ket side. The standard equations of quantum mechanics arise after averaging over ensembles of individual systems. The stochastic bra-ket interpretation provides a new quantum reality with a novel implementation of entanglement.

Realism is nice to have for engineers, as well as for down-to-earth scientists and philosophers. In the words often attributed to Max Planck, “When you change the way you look at things, the things you look at change.” An alternative interpretation of quantum mechanics invites new ways of thinking and new questions to be asked, such as: Is a piecewise linear trajectory of a free particle between collisions a valid concept? How close in space and time must the bra and ket trajectories be to contribute to a local measurement? Do particles cease to exist if their bra and ket trajectories move so far apart that they cannot be detected by any local measurement? Should the corresponding loss of particles be compensated by the simultaneous creation of bra and ket vectors in all possible momentum states? Is there a mechanism for bra and ket vectors to stay spatially close, say by favoring momenta in properly selected directions during collisions or by an attractive bra-ket interaction? Can the bra and ket processes describing individual quantum systems be manipulated separately, say by magnetic fields?

A promising tool for addressing these questions is generalized versions of double-slit experiments,^81,82 where the electrons of the bra and ket processes always pass through specific slits, but the bra and ket versions of the electrons might pass through different slits.⁵³ By varying slit widths and analyzing high-precision intensity profiles, one could investigate whether interfering spherical waves arise uniformly along slits (according to the Huygens–Fresnel principle) or only by interactions at the edges. With configurations involving more than two slits, possibly arranged in multiple layers, one could try to find out whether the bra and ket electrons pass through well-defined slits or sequences of slits. While time-resolved experiments would provide valuable insights into path lengths, they likely remain beyond current technological capabilities.

The possibility of describing individual quantum systems, rather than ensembles, opens up opportunities for novel applications, although—or perhaps precisely because—these individual systems are subject to the intrinsic randomness of quantum mechanics. This might be relevant in the context of electronic or optical devices that exhibit shot noise. For quantum computers, it might be possible to simulate random variables rather than probability densities. The interplay between the bra and ket aspects of the world may also be key to understanding the transition from particle-wave duality to classical behavior.⁸³ 

Advances in the foundations of quantum theory may be valuable for making progress in quantum technology. However, this is by no means a one-way street. The development of quantum devices, conversely, helps us to develop experience with and eventually intuition for quantum mechanics. The challenge of solving urgent practical problems may provide a stronger driving force for progress than the intellectual desire to understand what holds the world together at its core. Curiosity-driven research sometimes leads to curiosities and aberrations. In any case, quantum physics must become classical!

The author has no conflicts to disclose.

Hans Christian Öttinger: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.