On Quantum Entropy

Von Neumann entropy [3] captures the randomness associated with not-knowing precisely the quantum state, but does not capture the randomness associated with the observables. Thus, it requires the existence of classical statistics elements (mixed states) in order not to vanish. Wehrl entropy [4] is based on Husimi’s [5] quasiprobability distribution, rooted in projecting states to an overcomplete basis representation of coherent states. These quasiprobability distributions are not relativistic invariant. Note that no two coherent states are orthogonal to each other. Therefore, the Kolmogorov third axiom for a probability distribution, requiring that elementary events be mutually exclusive, is not satisfied. Consequently some probability properties, such as the monotonicity of probabilities and the complement rule, are not satisfied by Husimi’s quasiprobability distribution. These limitations prevent Wehrl entropy from correctly counting the random values of the observables. For example, a quantum state where the projection in position space is a Dirac delta function at $x_0$ produces a non-zero value distribution for all possible position coordinates x in the classical phase space coordinate $(x,p)$, where p is the momentum coordinate. Clearly, this is not the description of the random position x in QM. Indeed Wehrl [6] referred to his proposed entropy as a classical entropy for the classical phase space.

The entropic uncertainty principle (EUP) [7,8,9,10] is an extension of the standard uncertainty principle [11,12] for a conjugate pair of observables, where the product of the variances of each observable is replaced by the sum of the entropies for each observable. It is a static statement. We view the sum of these two entropies as a single entropy in phase space, which evolves over time, and the EUP as a lower bound for our proposed entropy for pure states.

Works on quantum thermalization [13,14,15,16] and their references, suggest consideration of a quantum system as a bipartite set of environment states and a subsystem of interest and application of the von Neumann procedure of tracing out the density matrix for the environment states. Then, these works establish a relation between the von Neumann reduced density matrix of a subsystem of interest and the classical entropy. We argue that a complete quantification of randomness of the system, including the randomness of the observables, will lead to a more accurate understanding of the role of entropy in physics.

Our starting points are pure states, where the DOFs associated with position and spin are precisely specified. We investigate the inherent quantum randomness associated with the observables. This randomness is fully captured by two conjugate pairs of observables satisfying the uncertainty principle [11,12], one associated with space and momentum, and the other associated with the internal spin state. The extension of the uncertainty principle to the entropic uncertainty principle (EUP) [7,8,9,10] suggests that the two entropy components associated with position and momentum play a role in physics. We propose a definition of entropy associated with a quantum pure state, and refer to it as coordinate-entropy. Furthermore, we extend the coordinate-entropy to mix states.

We study the coordinate-entropy’s time evolution for some physical systems, including a coherent state evolution through a potential free Dirac equation, and the hydrogen atom in an excited state transitioning to the ground state with a photon emission. In these scenarios the entropy increases. We also study a collision of two spinless particles. As they come close with each other, due to the superposition of the position wave functions and conservation laws, only the annihilation of these particles and the creation of new particles can prevent the entropy from decreasing. In the process of this analysis, we propose a property of the time evolution of the entropy associated with a potential free Dirac equation.

We then hypothesize an entropy law, universally applicable to particle physics, stating that in a closed physical system the entropy never decreases. The motivation is that the inverse of the amount of randomness, information, cannot be gained in a closed system. Such a law implies irreversibility of time for all physical scenarios where entropy does not stay constant. We complete the paper examining the consequences of such a law in physics.

We associate with a state $|\psi \rangle$ and its density operator $\rho =|\psi \rangle \langle \psi |$ their projection onto the QM eigenstates $|x\rangle$ and $|p\rangle$ of the operators $\widehat{x}$ and $\widehat{p}$, respectively. Either one projection, $\psi \left(x\right)=\langle x|\psi \rangle$ or $\varphi \left(p\right)=\langle p|\psi \rangle$, is sufficient to recover the other one via a Fourier transform. However, to account for the randomness of the observables, both $|\psi \rangle$ and $|\varphi \rangle$, are needed. The quantum coordinate phase space is defined by projecting the density operator to obtain the probability densities ${\rho }_{\mathrm{x}}\left(x\right)=\langle x|\rho |x\rangle ={\left|\psi \left(x\right)\right|}^2$ and ${\rho }_p\left(p\right)=\langle p|\rho |p\rangle ={\left|\tilde{\varphi }\left(p\right)\right|}^2$.

A time evolution of a density function $\rho$ according to a Hermitian Hamiltonian H is described by ${\rho }_t={\mathrm{e}}^{-\mathrm{i}\frac{H}{\hslash }t}\rho {\mathrm{e}}^{\mathrm{i}\frac{H}{\hslash }t}$ and so the evolution of a state in the quantum coordinate phase space is given by the pair of probability densities ${\rho }_{\mathrm{x}}(x,t)=\langle x|{\rho }_t|x\rangle ={\left|\psi (x,t)\right|}^2$ and ${\rho }_p(p,t)=\langle p|{\rho }_t|p\rangle ={\left|\tilde{\varphi }(p,t)\right|}^2$.

Our formulation of the coordinate-entropy is motivated by previous work including Boltzmann and Gibbs [1], Shanon [17], Jaynes [18], von Neumann entropy, Wehrl entropy, and it amounts to the sum of the two entropies in the EUP. More precisely, let the entropy associated only with the spatial coordinates be the differential entropy ${\mathrm{S}}_{\mathrm{x}}=-\int {\rho }_{\mathrm{x}}(x,t)ln{\rho }_{\mathrm{x}}(x,t){\phantom{(}\mathrm{d}}^{3}x$. Let $k=\frac{1}{\hslash }p$ be the spatial frequency (Fourier conjugate of x), ${\rho }_k(k,t)={\hslash }^3{\rho }_p(p,t)$ the associated probability density, and ${\mathrm{S}}_{\mathrm{k}}=-\int {\rho }_{\mathrm{k}}(k,t)ln{\rho }_{\mathrm{k}}(k,t){\phantom{(}\mathrm{d}}^{3}k$. Then we define the entropy associated with the quantum coordinate phase space distributions as where $\rho (x,k,t)={\rho }_{\mathrm{x}}(x,t){\rho }_k(k,t)$. The entropy is dimensionless and thus, invariant under changes of the units of measurements.

A natural extension of this entropy to an N-particle QM system is where ${\rho }_{\mathrm{x}}(x_1,\cdots ,x_N,t)={\left|\psi (x_1,\cdots ,x_N,t)\right|}^2$ and ${\rho }_{\mathrm{k}}(k_1,\cdots ,k_N,t)={\left|\varphi (k_1,\cdots ,k_N,t)\right|}^2$ are defined in QM via the projection of the state ${|{\psi }_t\rangle }^N$ of N particles (the product of N Hilbert spaces) onto $\langle x_1|\cdots \langle x_N|$ and $\langle k_1|\cdots \langle k_N|$ coordinate systems.

Fields in QFT are described by the operators $\Psi (x,t)$, where $(x,t)$ is the space-time, and their spatial Frequency transform $\Phi (k,t)$. They are written in terms of operators that create and the annihilate particles at a position and time $(x,t)$. A representation for a system of particles is based on Fock states with occupation number $|n_{q_1},n_{q_2},\cdots ,n_{q_i},\cdots n_{q_K}\rangle$, where $n_{q_i}$ is the number of particles in a QM state $|q_i\rangle$. The number of particles in a Fock state is then $N={\sum }_{i=1}^K{n}_{q_i}$, and a QFT state is described in a Fock space as $|\mathrm{state}\rangle ={\sum }_m{\alpha }_m|n_{q_1},n_{q_2},\cdots ,n_{q_i},\cdots \rangle$, where m is an index over the configurations of a Fock state, ${\alpha }_m\in \mathbb{C}$, and $1={\sum }_m{\left|{\alpha }_m\right|}^2$. The QFT phase space state associated with an initial $|\mathrm{state}\rangle$ and quantum fields $\Psi (x,t)$ and $\Phi (k,t)$ is then given by $\left(\langle \mathrm{state}|{\rho }_{\mathrm{x}}^{\mathrm{QFT}}(x,t)|\mathrm{state}\rangle ,\langle \mathrm{state}|{\rho }_{\mathrm{k}}^{\mathrm{QFT}}(k,t)|\mathrm{state}\rangle \right)$, where the probability density operators for the spatial coordinates are

The QFT coordinate-entropy is then described by the same formulae (1), but for QFT ${\rho }_{\mathrm{x}}(x,t)=\langle \mathrm{state}|{\rho }_{\mathrm{x}}^{\mathrm{QFT}}(x,t)|\mathrm{state}\rangle$ and ${\rho }_{\mathrm{k}}(k,t)=\langle \mathrm{state}|{\rho }_{\mathrm{x}}^{\mathrm{QFT}}(k,t)|\mathrm{state}\rangle$.

The framework used, QM or QFT, will define which probability density operator is being employed and to which kind of state is being applied.

It is not possible to know simultaneously the spin of a particle in all three dimensional directions, and this uncertainty, or randomness, was exploited in the Stern–Gerlach experiment [19] to demonstrate the quantum nature of the spin. We explore elsewhere [2] the entropy associated with the quantum spin phase space.

In classical physics canonical transformations are studied for phase spaces mapping $(x,k,t)\mapsto (x^{\prime },k^{\prime },t)$ that preserves the form of Hamilton’s equations. We adopt the QFT description of canonical transformations, where the phase space coordinates $(x,k)$ are conjugate variables and not operators.

A special case of canonical transformations of the coordinates $x\mapsto x^{\prime }$, is known as point transformation. Attempts by [21] to extend it to a quantum mechanics point transformation, where the conjugate variables become conjugate operators, are interesting. However, a large set of classical point transformations cannot yield a quantum point transformation, including the transformation from Cartesian coordinates to a spherical coordinate system [22]. The case of translations is possible, and studied by [23] as quantum reference frames. So now we adopt the QM representation, but it is not difficult to adapt it to a QFT representation. When a quantum reference frame is translated by $x_0$ along x, the state $|{\psi }_t\rangle$ in the position representation becomes $\psi (x-x_0,t)=\langle x-x_0|{\psi }_t\rangle =\langle x|{\widehat{T}}_P(-x_0)|{\psi }_t\rangle$, where ${\widehat{T}}_P(-x_0)={\mathrm{e}}^{\mathrm{i}{x}_0\frac{1}{\hslash }\widehat{P}}$, and where $\widehat{P}$ is the momentum operator conjugate to $\widehat{X}$. When the reference frame is translated by $p_0$ along p, the state $|{\psi }_t\rangle$ in the momentum representation becomes $\tilde{\varphi }(p-p_0,t)=\langle p-p_0|{\psi }_t\rangle =\langle p|{\widehat{T}}_X(-p_0)|{\psi }_t\rangle$, where ${\widehat{T}}_X(-p_0)={\mathrm{e}}^{\mathrm{i}\frac{1}{\hslash }{p}_0\widehat{X}}$, and where $\widehat{X}$ is the position operator conjugate to $\widehat{P}$.

We will be focusing on fermions, and thus on the Dirac spinors equation, though the results apply to bosons as well. We return to the QFT represenation where the Dirac Hamiltonian is

A QFT solution $\Psi (x,t)$ satisfies $[{\mathcal{H}}^{\mathcal{D}},\Psi (x,t)]=-\mathrm{i}\hslash \frac{\Psi (x,t)}{t}$ and the C, P, and T symmetries provide new solutions from $\Psi (x,t)$. As usual, ${\Psi }^{\mathrm{C}}(x,t)=C{\overline{\Psi }}^{\mathsf{T}}(x,t)$, ${\Psi }^{\mathrm{P}}(-x,t)=P\Psi (-x,t)$, ${\Psi }^{\mathrm{T}}(x,-t)=T{\Psi }^{*}(x,-t)$, and ${\psi }^{\mathrm{CPT}}(-x,-t)=CPT{\overline{\psi }}^{\mathsf{T}}(-x,-t)$. For completeness, we briefly review the three operations, Charge Conjugation, Parity Change, and Time Reversal.

Charge Conjugation transforms particles $\Psi (x,t)$ into antiparticles ${\overline{\Psi }}^{\mathsf{T}}(x,t)={\left({\Psi }^{†}{\gamma }^0\right)}^{\mathsf{T}}(x,t)$. As $C{\gamma }^{\mu }{C}^{-1}=-{\gamma }^{\mu \mathsf{T}}$, ${\Psi }^{\mathrm{C}}(x,t)$ is also a solution for the same Hamiltonian. In the standard representation, $C=\mathrm{i}{\gamma }^2{\gamma }^0$ up to a phase. Parity Change $P={\gamma }^0$, up to a sign, effects the transformation $x\mapsto -x$. Time Reversal effects $t\mapsto -t$ and is carried by the operator $\mathcal{T}=T\widehat{K}$, where $\widehat{K}$ applies conjugation. In the standard representation $T=\mathrm{i}{\gamma }^1{\gamma }^3$, up to a phase. For simplicity of notation we will drop for the QFT superscript of the probability density operator in the theorem that now follows.

Note that in QFT, one scales the operator $\Phi (k,t)$ by $\sqrt{2{\omega }_{\mathrm{k}}}$, that is, one scales the creation and the annihilation operators ${\alpha }^{†}\left(k\right)=\sqrt{{\omega }_{\mathrm{k}}}{\mathbf{a}}^{†}\left(k\right)$ and $\alpha \left(k\right)=\sqrt{\omega }\mathbf{a}\left(k\right)$. In this way, the density operator ${\Phi }^{†}(k,t)\Phi (k,t)$ scales with ${\omega }_{\mathrm{k}}$ and becomes a relativistic scalar. Also, with such a scaling, the infinitesimal probability of finding a particle with momentum $p=\hslash k$ in the original reference frame is invariant under the Lorentz transformation, though it would be found with momentum $p^{\prime }=\hslash k^{\prime }$.

We introduce the concept of a QCurve to specify a curve (or path) in a Hilbert space parametrized by time. In QM a QCurve is represented by a triple $\left(|{\psi }_0\rangle ,U\left(t\right),\delta t\right)$ where $|{\psi }_0\rangle$ is the initial state, $U\left(t\right)={\mathrm{e}}^{-\mathrm{i}\frac{H}{\hslash }t}$ is the evolution operator, and $[0,\delta t]$ is the time interval of the evolution. Of course, one may also represent the initial state by a triple $\left({\rho }_0,U\left(t\right),\delta t\right)$, where ${\rho }_0=|{\psi }_0\rangle \langle {\psi }_0|$ is the density matrix. Alternatively, we can represent the initial state in the quantum coordinate phase space by $(\langle x|{\rho }_0|x\rangle ,\langle k|{\rho }_0|k\rangle )$ or $(\langle x|{\psi }_0\rangle ,\langle k|{\psi }_0\rangle )$. In QFT the unitary evolution may be represented by the initial condition $\left({\Psi }_0=\Psi (x,0),{\Phi }_0=\Phi (k,0)\right)$ or by the initial phase space state $\left({\rho }_{\mathrm{x}}(x,0)=\langle \mathrm{state}|{\rho }_{\mathrm{x}}^{\mathrm{QFT}}(x,0)|\mathrm{state}\rangle ,{\rho }_{\mathrm{k}}(k,0)=\langle \mathrm{state}|{\rho }_{\mathrm{k}}^{\mathrm{QFT}}(k,0)|\mathrm{state}\rangle \right)$.

We will use any of these representations to describe a QCurve as more convenient for manipulations for the problem at hand.

Consider stationary states $|{\psi }_t\rangle =|{\psi }_E\rangle {\mathrm{e}}^{-\mathrm{i}\omega t}$ with $\omega =E/\hslash$, where E is an energy eigenvalue of the Hamiltonian, and $|{\psi }_E\rangle$ is the time-independent eigenstate of the Hamiltonian associated with E.

Dirac’s free-particle Hamiltonian in QM [26] is

It can be diagonalized in the spatial Fourier domain $|k\rangle$ basis to obtain where $\omega \left(k\right)$ is the frequency component of the Hamiltonian. We focus on the positive energy solutions and so the group velocity becomes

In (9) we will use the Taylor expansion of (5) up to the second order, thus requiring the Hessian $H\left(k\right)$, with the entries for the positive energy solution. The three (positive) eigenvalues of $H\left(k\right)$ are where $\mu \left(\right|\left|k\right|\left|\right)=\hslash \left|\right|k\left|\right|/c$ is the kinetic energy in mass units. The Hessian is positive definite for positive energy, and so it gives a measure of the dispersion of the wave.

We now consider initial solutions that are localized in space, ${\psi }_{{\mathrm{k}}_0}(x-x_0)={\psi }_0(x-x_0){\mathrm{e}}^{\mathrm{i}{k}_0·x}$, where $x_0$ is the mean value of x. Assume that the variance, $\int {\phantom{(}\mathrm{d}}^{3}x{(x-x_0)}^2{\rho }_{\mathrm{x}}\left(x\right)$, is finite, where ${\rho }_{\mathrm{x}}\left(x\right)={\left|{\psi }_0\left(x\right)\right|}^2$. In a Cartesian representation, we can write the initial state in the spatial frequency domain as ${\varphi }_{{\mathrm{x}}_0}(k-k_0)={\varphi }_0(k-k_0){\mathrm{e}}^{-\mathrm{i}(k-k_0)·x_0}$, where ${\varphi }_0\left(k\right)$ is the Fourier transform of ${\psi }_0\left(x\right)$, and so the variance of ${\rho }_{\mathrm{k}}\left(k\right)={\left|{\varphi }_{{\mathrm{x}}_0}(k-k_0)\right|}^2$ is also finite, with the mean in the spatial frequency center $k_0$.

The time evolution of ${\psi }_{{\mathrm{k}}_0}(x-x_0)$ according a Hamiltonian with a dispersion relation $\omega \left(k\right)$, and written via the inverse Fourier transform, is

As ${\varphi }_{{\mathrm{x}}_0}(k-k_0)$ fades away exponentially from $k=k_0$, we expand (5) in a Taylor series and approximate it by where $v_p\left(k_0\right)$, $v_g\left(k_0\right)$, and $H\left(k_0\right)$ are the phase velocity $\omega \left(k_0\right){\widehat{k}}_0/|k_0|$, the group velocity (6), and the Hessian (7) of the dispersion relation $\omega \left(k\right)$, respectively. Then after inserting (9) into (8), we obtain the quantum dispersion transform where $x_{k_0}^t=x_0+v_g\left(k_0\right)t$, ${\Phi }_{x_{k_0}^t}(k-k_0)={\varphi }_0(k-k_0){\mathrm{e}}^{-\mathrm{i}(k-k_0)·x_{k_0}^t}$, with Fourier transform ${\psi }_{{\mathrm{k}}_0}(x-x_{k_0}^t)$; ∗ denotes a convolution, $Z_r$ and $Z_k$ normalize the amplitudes, and $\mathcal{N}$ is a normal distribution. Consequently, ${\psi }_{{\mathrm{k}}_0}(x-x_{k_0}^t,t)$ is the spatial Fourier transform of ${\Phi }_{x_{k_0}^t}(k-k_0,t)$.

The probability densities associated with the probability amplitudes in (10) are

The time invariance of the density ${\rho }_{\mathrm{k}}^{\mathrm{S}}(k,t)$, and therefore of ${\mathrm{S}}_{\mathrm{k}}$, reflects the conservation law of momentum for free particles.

Coherent states, represented by state $|\alpha \rangle$, are eigenstates of the annihilator operator. The 1D quantum phase space of observables $(x,p)$ can be constructed by the unitary operator $U(x_0,p_0)={\mathrm{e}}^{\frac{\mathrm{i}}{\hslash }(x_{0}X-p_{0}P)}$ applied to zero-state $|x=0,p=0\rangle$, that is, they can be constructed as $|\alpha \rangle =|x_0,p_0\rangle ={\mathrm{e}}^{\frac{\mathrm{i}}{\hslash }(x_{0}X-p_{0}P)}|0,0\rangle$, where $\alpha =x_0+\mathrm{i}{p}_0$. Projecting the state to position space yields ${\psi }_{\alpha }\left(x\right)=\langle x|\alpha \rangle ={\mathrm{e}}^{-\frac{p_0^2}{2}}{\mathrm{e}}^{-\frac{1}{2}{\left(x-\sqrt{2}\alpha \right)}^2}/{\mathsf{\pi }}^{\frac{1}{4}}$, where $\alpha =(x_0+\mathrm{i}{p}_0)/\sqrt{2}$. Squeeze states extend coherent states to all eigenstate solutions of the annihilator operator by allowing different variances to the Gaussian solution, and together their representation in 3D position and momentum space are where $\Sigma$ is the spatial covariance matrix.

The theorem suggests that quantum physics has an inherent mechanism to increase entropy for free particles, due to the spatial dispersion property of the Hamiltonian. Note that at $t=0$ a coherent state (13) reaches the minimum possible coordinate-entropy value.

The dispersion properties of the Dirac and Schrödinger Hamiltonian equations have been studied in the past, and are already present in Feynmann path integral formulation for the free particle [27], where an analytical solution is derived showing the dispersion of the initial localized particle.

We present a motivation for this conjecture. The dispersion relation $\omega \left(k\right)$ for the free fermion Hamiltonian has a positive Hessian (7). Note that Schrödinger Hamiltonian also has a positive Hessian. We also observe some mathematical scenarios where the entropy can decrease temporarily. Given a fermion in a coherent state solution $|{\Psi }_0\rangle$ evolving backwards for a time period T. It will yield a solution $|{\Psi }_{-T}\rangle ={\mathrm{e}}^{\mathrm{i}\frac{H}{\hslash }T}|{\Psi }_0\rangle$, with larger entropy than $|{\Psi }_0\rangle$. Then a starting solution $|{\Psi }_{-T}\rangle$ will evolve forward for a period $t\in (0,T]$, until it reaches back to ${|\Psi \rangle }_0$, with entropy decreasing. However, for $t>T$ the entropy of the evolution will increase forever. We discuss this scenario next in Section 5.5. Another physical scenario where the entropy can decrease for a period T is when the initial state is a sum of two coherent states, $\langle x|{\Psi }_0\rangle ={\Psi }_{k_0}(x-x_0)+{\Psi }_{-k_0}(x+x_0)$, that is, these parameters model two components away from each other and moving towards each other. For large distances $2x_0$, where the overlap of the two components is negligible, the entropy evolution of each component increases due to dispersion, and thus, the entropy increases. As the two components come closer to each other, the overlap increases, and the final probability contain a significant term from the interference of the components. Then, due to the interference, the entropy can decrease. Continuing the evolution, as the two components “pass through each other” and start to move away from each other, the entropy will again start to increase, and will increase forever. The parameter T in this case represents the period of a large overlap between the two components up to when they “pass through each other”.

Note that for physical scenarios where the overlap is large enough for causing the entropy to decrease, there is a possible mechanism in nature, outside of the motion equation, to annihilate such solutions (such particles) and create new particles that satisfy the conservation laws that could have the entropy of the evolution to always increase.

Consider a time-independent Hamiltonian. We investigate the discrete symmetries C and P, and propose that Time Reversal be augmented with Time Translation, say by $\delta t$. We refer to the mapping $t\mapsto -t+\delta t$ as Time Reflection, because as t varies from 0 to $\delta t$, $t^{\prime }\left(t\right)=-t+\delta t$ varies as a reflection from $\delta t$ to 0. We define the Time Reflection quantum field

Note that in contrast to the case of Time Reversal, ${\Psi }^{{\mathrm{T}}_{\delta }}(x,t^{\prime })=\mathcal{T}\Psi (x,-t^{\prime }+\delta t)$, and the entropies associated with $\Psi (x,t)$ and ${\Psi }^{{\mathrm{T}}_{\delta }}(x,t)$ are generally not equal. Thus, an instantaneous Time Reflection transformation will cause entropy changes.

We next consider a composition of the three transformation, Charge Conjugation, Parity Change, and Time Reflection.

Using (14) we see that,