Axioms Of Quantum Mechanics Research Articles

An updated quantum complementarity principle

Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.

Depiction of Hamiltonian PT-symmetry

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian <i>H</i> is invariant neither under parity operator <i>P</i>, nor under time reversal operator <i>T</i>, <i>PTH</i> = <i>H</i>. Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics: real energy levels and unitary time evolution. The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science, Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian. In the paper, we define operator <i>F</i> according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator <i>CPT</i> and operator <i>F</i>. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry broken, and <i>J</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry. If <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, to judge whether the Hamiltonian is PT symmetric. Re<i>F</i> = 1/4||(<i>CPTF</i>+<i>F</i>)||CPT represents the sum of squares of real part of the eigenvalue <i>E<sub>n</sub></i> of Hamiltonian <i>H</i>, Im<i>F</i> = 1/4||(<i>CPTF</i>–<i>F</i>)||CPT is the sum of imaginary part of the eigenvalue <i>E<sub>n</sub></i> of a Hamiltonian <i>H</i>. If Im<i>F</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once Im<i>F</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. Re<i>F</i> = 0 implies that Hamiltonian <i>H</i> is PT-asymmetric, but it is a sufficient condition, not necessary condition. The later is easier to realize in the experiment, but the studying conditions are tighter, and it further requires that <i>CPT</i> <inline-formula><tex-math id="Z-20240108115351">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.png"/></alternatives></inline-formula>(<i>x</i>) = <inline-formula><tex-math id="Z-20240108115401">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.png"/></alternatives></inline-formula>(<i>x</i>). If we only pay attention to whether PT-symmetry is broken, it is simpler to use the latter method. The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.

Unification of the first law of quantum thermodynamics

Underlying the classical thermodynamic principles are analogous microscopic laws, arising from the fundamental axioms of quantum mechanics. These define quantum thermodynamic variables such as quantum work and heat and characterize the possible transformations of open quantum systems. The foremost quantum thermodynamic law is a simple statement concerning the conservation of energy. Nevertheless, there exist ambiguity and disagreement regarding the precise partition of a quantum system’s energy change to work and heat. By treating quantum mechanics as a comprehensive theory, applicable to both the micro and macroscopic domains, and employing dynamical symmetries, we bridge the gaps between five popular thermodynamic approaches to the first law. These include both autonomous and semi-classical formulations, which define work in terms of an ensemble average, as well as the single shot paradigm, where work is defined as a deterministic quantity.

A Quantum Mechanics for interest rate derivatives markets

We present a model-free axiomatic formulation of Option Pricing Theory for interest rate derivatives. In this setting, completely analogous to axiomatic Quantum Mechanics, the role of the wave function is played by the discounted zero-coupon bond price. The theory is linked to term structure models through the Hamiltonian operator, and we show that its associated Schrödinger equation is consistent with the [25] model. We also find the quantum-mechanical equivalent of the standard risk-neutral option pricing formula.In order to take advantage of this quantum framework, we develop a time-dependent perturbation theory and we apply it to obtain an approximate closed-form expression for the price of a call option under the [26] model with a small local volatility perturbation.

A measurable physical theory of hyper-correlations beyond quantum mechanics

A unique characteristic of quantum mechanics is entanglement describing correlations between particles irrespective of their locations. This property, called non-locality, has no classical analogue. Over the past few years, quantum physicists have reached a consensus that we lack a physical theory to account for a class of states whose non-local character exceeds the bounds allowed by quantum mechanics. Motivated by our observation that an extension of the Schrödinger equation with non-linear terms is directly linked to a relaxation of Born’s rule, an axiom of quantum mechanics, we derive a physical theory that accounts for such hyper-correlated states and modifies Born’s rule. We model correlated particles with a generalized probability theory whose dynamics are described with a non-linear version of Schrödinger’s equation and demonstrate how that deviates from the standard formulation of quantum mechanics in experimental probability-prediction. We show also that the violation of the Clauser-Horn-Shimony-Holt inequality, the amount of non-locality, is proportional to the degree of non-linearity, which can be experimentally tested.

A Quantum Mechanics for Interest Rate Derivatives Markets

We present an axiomatic formulation of the Option Pricing Theory for interest rates, completely analogous to axiomatic Quantum Mechanics. The role of the wave function is played in the financial theory by discounted zero-coupon bond prices. The theory is model-free and is linked to term structure models through the Hamiltonian operator. Under the quantum framework, we prove the consistency of the Schrodinger equation with the dynamics of the stochastic string model. We also find an expression for pricing options under this quantum theory, show its consistency with standard option pricing and apply the quantum formalism to Price calls and swaptions.

Approximation of quasi-states on manifolds

Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. They were discovered as a part of an attempt to understand the axioms of quantum mechanics due to von Neumann. A very interesting and fundamental example is given by the so-called median quasi-state on $$S^2$$ . In this paper we present an algorithm which numerically computes it to any specified accuracy. The error estimate of the algorithm crucially relies on metric continuity properties of a map, which constructs quasi-states from probability measures, with respect to appropriate Wasserstein metrics. We close with non-approximation results, particularly for symplectic quasi-states.

PT -symmetric circuit QED

The Hermiticity axiom of quantum mechanics guarantees that the energy spectrum is real and the time evolution is unitary (probability-preserving). Nevertheless, non-Hermitian but $\mathcal{PT}$-symmetric Hamiltonians may also have real eigenvalues. Systems described by such effective $\mathcal {PT}$-symmetric Hamiltonians have been realized in experiments using coupled systems with balanced loss (dissipation) and gain (amplification), and their corresponding classical dynamics has been studied. A $\mathcal {PT}$-symmetric system emerging from a quantum dynamics is highly desirable, in order to understand what $\mathcal {PT}$-symmetry and the powerful mathematical and physical concepts around it will bring to the next generation of quantum technologies. Here, we address this need by proposing and studying a circuit-QED architecture that consists of two coupled resonators and two qubits (each coupled to one resonator). By means of external driving fields on the qubits, we are able to tune gain and losses in the resonators. Starting with the quantum dynamics of this system, we show the emergence of the $\mathcal {PT}$-symmetry via the selection of both driving amplitudes and frequencies. We engineer the system such that a non-number conserving dipole-dipole interaction emerges, introducing an instability at large coupling strengths. The $\mathcal {PT}$-symmetry and its breaking, as well as the predicted instability in this circuit-QED system can be observed in a transmission experiment.

When does a generalized Boolean quasiring become a Boolean ring?

Generalized Boolean quasirings are ring-like structures used as algebraic models in the foundations of axiomatic quantum mechanics. The quantum mechanical system corresponding to such a quasiring turns out to be a classical one if and only if this quasiring is a Boolean ring with unit. We characterize this situation by a single identity.

Towards a Constructive Foundation of Quantum Mechanics

I describe a constructive foundation for Quantum Mechanics, based on the discreteness of the degrees of freedom of quantum objects and on the Principle of Relativity. Taking Einstein's historical construction of Special Relativity as a model, the construction is carried out in close contact with a simple quantum mechanical Gedanken experiment. This leads to the standard axioms of Quantum Mechanics. The quantum mechanical description is identified as a tool that allows describing objects with discrete degrees of freedom in space-time covariant with respect to coordinate transformations. An inherent property of this description is a quantum mechanical interaction mechanism. The construction gives detailed answers to controversial questions, such as the measurement problem, the informational content of the wave function, and the completeness of Quantum Mechanics.

Two-qubit quantum cloning machine and quantum correlation broadcasting

Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: "Is there an optimal quantum cloning machine?" Buzek and Hillery answer to this question and construct their famous B-H quantum cloning machine. The B-H machine clones state of an arbitrary single qubit in optimal manner and hence it is universal. Generalizing this machine for two-qubit system is straightforward, but this procedure does not preserve quantum correlation existing in bipartite state in optimal manner and also, during this procedure, this machine loses its universality and becomes a state-dependent cloning machine. In this paper we propose an optimal universal local quantum state cloner for two qubit systems. Also we present two classes of state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is in general very fragile during broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.

An Evolutionary Formalism for Weak Quantum Measurements

Unitary evolution and projective measurement are fundamental axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.

An Evolutionary Formalism for Weak Quantum Measurements

Unitary evolution and projective measurement are fundamental axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.

Energy quantisation and time parameterisation

We show that if space is compact, then trajectories cannot be defined in the framework of quantum Hamilton--Jacobi equation. The starting point is the simple observation that when the energy is quantized it is not possible to make variations with respect to the energy, and the time parameterisation t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. It should be stressed that this follows directly form the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunneling and the energy quantization theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similarly to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.

What Is a Quantum-Mechanical “Weak Value” the Value of?

A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes (e.g., the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.

Classical and quantum-mechanical axioms with the higher time derivative formalism

A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibration) reference frames with the random initial conditions? One of the most generalized ways of descriptions (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher temporal derivatives of the coordinates in the Lagrange function. Such formalism describing physical objects in the infinite dimensions space does not contradict to the quantum mechanics and infinite dimensions Hilbert space.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

Basic quantum mechanics for three Dirac equations in a curved spacetime

We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, iff the field of Dirac matrices $\gamma ^\mu $ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the $\gamma ^\mu $ matrices. It similarly restricts the choice of the $\gamma ^\mu $ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermiticity condition depends on the choice of the $\gamma ^\mu $ matrices.

On second quantization on noncommutative spaces with twisted symmetries

By the application of the general twist-induced ⋆-deformation procedure we translate second quantization of a system of bosons/fermions on a symmetric spacetime into a noncommutative language. The procedure deforms, in a coordinated way, the spacetime algebra and its symmetries, the wave-mechanical description of a system of n bosons/fermions, the algebra of creation and annihilation operators and also the commutation relations of the latter with functions of spacetime; our key requirement is the mode-decomposition independence of the quantum field. In a minimalistic view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind. In a non-conservative one, we obtain a closed, covariant framework for quantum field theory (QFT) on the corresponding noncommutative spacetime consistent with quantum mechanical axioms and Bose–Fermi statistics. One distinguishing feature is that the field commutation relations remain of the type ‘field (anti)commutator=a distribution’. We illustrate the results by choosing as examples interacting non-relativistic and free relativistic QFT on Moyal space(time)s.

Observation of parity–time symmetry in optics

A photonic system that shows behaviour similar to that of a violation of parity–time symmetry provides a convenient test bed to explore this and related phenomena. It could also lead to a new class of optical materials with exotic properties that exploit non-reciprocal light flow. One of the fundamental axioms of quantum mechanics is associated with the Hermiticity of physical observables1. In the case of the Hamiltonian operator, this requirement not only implies real eigenenergies but also guarantees probability conservation. Interestingly, a wide class of non-Hermitian Hamiltonians can still show entirely real spectra. Among these are Hamiltonians respecting parity–time (P T) symmetry2,3,4,5,6,7. Even though the Hermiticity of quantum observables was never in doubt, such concepts have motivated discussions on several fronts in physics, including quantum field theories8, non-Hermitian Anderson models9 and open quantum systems10,11, to mention a few. Although the impact of P T symmetry in these fields is still debated, it has been recently realized that optics can provide a fertile ground where P T-related notions can be implemented and experimentally investigated12,13,14,15. In this letter we report the first observation of the behaviour of a P T optical coupled system that judiciously involves a complex index potential. We observe both spontaneous P T symmetry breaking and power oscillations violating left–right symmetry. Our results may pave the way towards a new class of P T-synthetic materials with intriguing and unexpected properties that rely on non-reciprocal light propagation and tailored transverse energy flow.