# On Quantum Entropy

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Our Contribution

## 2. Quantum Entropy in Quantum Phase Spaces

#### 2.1. Coordinate-Entropy

#### 2.1.1. Uniqueness of the Phase Space and QFT

#### 2.1.2. Mixed Quantum States

#### 2.2. Spin-Entropy

## 3. Entropy Invariant Properties

#### 3.1. Canonical Transformations

**Theorem**

**1.**

**Proof.**

**Theorem**

**2**

**.**The entropy of a state is invariant under a change of a quantum reference frame by translations along x and along p.

**Proof.**

- (i)
- translations along x by any ${x}_{0}$$$\begin{array}{cc}\hfill {\mathrm{S}}_{x+{x}_{0}}& =-\phantom{\rule{-1.69998pt}{0ex}}{\int}_{-\infty}^{\infty}|\psi (x+{x}_{0},t){|}^{2}ln{\left|\psi (x+{x}_{0},t)\right|}^{2}{\phantom{(}\mathrm{d}}^{}\phantom{\rule{-0.166667em}{0ex}}x={\mathrm{S}}_{x}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- (ii)
- translations along p by any ${p}_{0}$$$\begin{array}{cc}\hfill {\psi}_{{p}_{0}}(x,t)& =\langle x|{\widehat{T}}_{X}\left({p}_{0}\right)|{\psi}_{t}\rangle ={\int}_{-\infty}^{\infty}\langle x|{\widehat{T}}_{X}\left({p}_{0}\right)|p\rangle \langle p|{\psi}_{t}\rangle {\phantom{(}\mathrm{d}}^{}\phantom{\rule{-0.166667em}{0ex}}p\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\int}_{-\infty}^{\infty}\langle x|p+{p}_{0}\rangle \tilde{\varphi}(p,t){\phantom{(}\mathrm{d}}^{}\phantom{\rule{-0.166667em}{0ex}}p={\int}_{-\infty}^{\infty}\frac{1}{\sqrt{2\mathsf{\pi}}}{\mathrm{e}}^{\mathrm{i}x\frac{1}{\hslash}(p+{p}_{0})}\tilde{\varphi}(p,t){\phantom{(}\mathrm{d}}^{}\phantom{\rule{-0.166667em}{0ex}}p\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\psi (x,t){\mathrm{e}}^{\mathrm{i}x\frac{1}{\hslash}{p}_{0}},\hfill \end{array}$$

#### 3.2. CPT Transformations

**Theorem**

**3**

**.**Given a quantum field operator $\Psi (x,t)$, its Fourier transform $\Phi (k,t)$, and its entropy ${\mathrm{S}}_{t}$ associated with any initial state, the entropies of ${\Psi}^{*}(x,t)$, ${\Psi}^{\mathrm{P}}(-x,t)$, ${\Psi}^{\mathrm{C}}(x,t)$, ${\Psi}^{\mathrm{T}}(x,-t)$, ${\Psi}^{\mathrm{CPT}}(-x,-t)$, and their corresponding Fourier transforms are all equal to ${\mathrm{S}}_{t}$.

**Proof.**

#### 3.3. Lorentz Transformations

**Theorem**

**4.**

**Proof.**

## 4. The Minimum Entropy Value

**Theorem**

**5.**

**Proof.**

## 5. Time Evolution of the Entropy

#### 5.1. A Formalism for Entropy Evolution

**Definition**

**1**

**.**Let $\mathcal{E}$ to be the set of all the QCurves. We define a partition of $\mathcal{E}$ based on the entropy evolution into four blocks:

- $\mathcal{C}$:
- The set of QCurves for which the entropy is a constant.
- $\mathcal{I}$:
- The set of QCurves for which the entropy is increasing, but it is not a constant.
- $\mathcal{D}$:
- The set of QCurves for which the entropy is decreasing, but it is not a constant.
- $\mathcal{O}$:
- The set of oscillating QCurves, with the entropy strictly increasing in some subinterval of $[0,\delta t]$ and strictly decreasing in another subinterval of $[0,\delta t]$.

**Theorem**

**6.**

**Proof.**

#### 5.2. Dispersion of a Fermion Hamiltonian

**Lemma**

**1**

**.**The entropy associated with (11) is equal to the entropy associated with the simplified probability densities

**Proof.**

#### 5.3. The Coordinate-Entropy of Coherent States Increases with Time

**Theorem**

**7.**

**Proof.**

#### 5.4. A Conjecture on Entropy Evolution

**Conjecture**

**1.**

#### 5.5. Time Reflection

**Definition**

**2**

**.**Let the ${\mathrm{CPT}}_{\delta}$ quantum field be

**Definition**

**3**

**.**Let ${Q}_{{\mathrm{CPT}}_{\delta}}$ be $\left(\psi (x,0),U\left(t\right),[0,\delta t]\right)\mapsto \left({\psi}^{{\mathrm{CPT}}_{\delta}}(-x,0),U\left(t\right),[0,\delta t]\right)$.

**Theorem**

**8**

**.**Consider a $\mathrm{CPT}$ invariant quantum field theory (QFT) with energy conservation, such as Standard Model or Wightman axiomatic QFT [28]. Let ${e}_{0}=\left(\Psi (x,0),U\left(t\right),[0,\delta t]\right)$ be a QCurve solution to such QFT. Then, ${e}_{1}={Q}_{{\mathrm{CPT}}_{\delta}}\left({e}_{0}\right)$ is (i) a solution to such QFT, (ii) if ${e}_{0}$ is in $\mathcal{C}$, $\mathcal{D}$, $\mathcal{O}$, $\mathcal{I}$ then ${e}_{1}$ is respectively in $\mathcal{C}$, $\mathcal{I}$, $\mathcal{O}$, $\mathcal{D}$, making $\mathcal{C}$, $\mathcal{I}$, $\mathcal{O}$, $\mathcal{D}$ reflections of $\mathcal{C}$, $\mathcal{D}$, $\mathcal{O}$, $\mathcal{I}$, respectively.

**Proof.**

#### 5.6. Entropy Oscillations

**Theorem**

**9**

**.**Consider the QCurve $\left(|{\psi}_{{E}_{i}}\rangle ,U\left(t\right)={\mathrm{e}}^{-\mathrm{i}\frac{(H+{H}^{\mathrm{I}})}{\hslash}t},T\right)$ with $\hslash {\omega}_{1}$ the ground state value of H and $T=\frac{2\mathsf{\pi}}{|{\omega}_{i}-{\omega}_{1}|}$. Assume that $|{\alpha}_{1}{\left(t\right)|}^{2},|{\alpha}_{i}{\left(t\right)|}^{2}\gg {\left|{\alpha}_{k}\left(t\right)\right|}^{2}$ for $k\ne 1,i$ and $t\in [0,T]$. Then the QCurve is in $\mathcal{O}$.

**Proof.**

**Theorem**

**10**

**.**Consider a particle in an eigenstate $|{\psi}_{{E}_{1}}\rangle $ of a Hamiltonian H that has only two eigenstates $|{\psi}_{{E}_{1}}\rangle $ and $|{\psi}_{{E}_{2}}\rangle $ with eigenvalues ${E}_{1}=\hslash {\omega}_{1}$ and ${E}_{2}=\hslash {\omega}_{2}$, respectively. Let this particle interact with an external field (such as the impact of a Gauge Field), requiring an additional Hamiltonian term ${H}^{\mathrm{I}}$ to describe the evolution of this system.

**Proof.**

## 6. Entropy Evolution in Physical Scenarios

#### 6.1. A Two-Particle Collision

#### 6.2. The Hydrogen Atom and Photon Emission

- (i)
- The position probability amplitudes described in [31] and the associated entropies are$$\begin{array}{cc}\hfill {\psi}_{2,1,0}(\rho ,\theta ,\varphi )& =\frac{1}{\sqrt{32\mathsf{\pi}}}{\left(\frac{1}{{a}_{0}}\right)}^{\frac{3}{2}}\phantom{\rule{0.222222em}{0ex}}\rho {\mathrm{e}}^{-\frac{\rho}{2}}cos\left(\theta \right)\phantom{\rule{0.222222em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{\mathrm{S}}_{\mathrm{x}}\left({\psi}_{2,1,0}\right)\approx 6.120+ln\mathsf{\pi}+3ln{a}_{0}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {\psi}_{1,0,0}(\rho ,\theta ,\varphi )& =\frac{1}{\sqrt{\mathsf{\pi}}}{\left(\frac{1}{{a}_{0}}\right)}^{\frac{3}{2}}\phantom{\rule{0.222222em}{0ex}}{\mathrm{e}}^{-\rho}\phantom{\rule{0.222222em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{\mathrm{S}}_{\mathrm{x}}\left({\psi}_{1,0,0}\right)\approx 3.000+ln\mathsf{\pi}+3ln{a}_{0}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- (ii)
- The momentum probability amplitudes described in [31] and the associated entropies are$$\begin{array}{cc}\hfill {\Phi}_{2,1,0}(p,{\theta}_{p},{\varphi}_{p})& =\sqrt{\frac{{128}^{2}}{2\mathsf{\pi}{p}_{0}^{3}}}\phantom{\rule{0.166667em}{0ex}}\frac{p}{{p}_{0}}\phantom{\rule{0.166667em}{0ex}}{\left(1+{\left(2\frac{p}{{p}_{0}}\right)}^{2}\right)}^{-3}\phantom{\rule{0.166667em}{0ex}}cos\left({\theta}_{p}\right)\phantom{\rule{0.222222em}{0ex}}\to \phantom{\rule{7.5pt}{0ex}}{\mathrm{S}}_{p}\left({\Phi}_{2,1,0}\right)\approx 0.042+3ln{p}_{0}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {\Phi}_{1,0,0}(p,{\theta}_{p},{\varphi}_{p})& =\sqrt{\frac{32}{\mathsf{\pi}\phantom{\rule{0.166667em}{0ex}}{p}_{0}^{3}}}{\left(1+{\left(\frac{p}{{p}_{0}}\right)}^{2}\right)}^{-2}\phantom{\rule{0.222222em}{0ex}}\to \phantom{\rule{7.5pt}{0ex}}{\mathrm{S}}_{p}\left({\Phi}_{1,0,0}\right)\approx 2.422+3ln{p}_{0}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- (iii)
- Therefore, $\mathsf{\Delta}{\mathrm{S}}_{2,1,0\to 1,0,0}={\mathrm{S}}_{\mathrm{x}}\left({\psi}_{1,0,0}\right)+{\mathrm{S}}_{p}\left({\Phi}_{1,0,0}\right)-{\mathrm{S}}_{\mathrm{x}}\left({\psi}_{2,1,0}\right)-{\mathrm{S}}_{p}\left({\Phi}_{2,1,0}\right)\approx -0.740\phantom{\rule{0.166667em}{0ex}}.$

#### Experiments in a Reflective Cavity

## 7. An Entropy Law and a Time Arrow

**Law**

**.**The entropy of an isolated quantum system is an increasing function of time.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Glossary

Section 2 | |

x | 3D position variable or operator. |

k | spatial frequency variable (the Fourier transform variable of the x variable). |

p | momentum variable or operator, conjugate pair to x |

$|\psi \rangle $ and $\rho =|\psi \rangle \langle \psi |$ | QM state and QM density operator associated with a QM state. |

$|{\psi}_{t}\rangle ={\mathrm{e}}^{-\mathrm{i}\frac{H}{\hslash}t}|\psi \rangle $ and ${\rho}_{t}={\mathrm{e}}^{-\mathrm{i}\frac{H}{\hslash}t}\phantom{\rule{-0.166667em}{0ex}}\rho {\mathrm{e}}^{\mathrm{i}\frac{H}{\hslash}t}$ | Time evolution of a QM state and of a density operator |

$\psi (x,t)=\langle x|{\psi}_{t}\rangle $ and $\varphi \left(p\right)=\langle p|{\psi}_{t}\rangle $ | QM wave functions in each coordinate of phase space $(x,p)$ |

$\mathrm{Entropy}$ | coordinate-entropy in phase space |

${S}_{r},{S}_{p},{S}_{k}$ | coordinate-entropy components for position x, momentum p, and spatial frequency k, respectively. |

$\Psi (x,t)$ and $\Phi (k,t)$ | QFT operators in position and spatial frequency domains, respectively. |

${\rho}_{\mathrm{x}}^{\mathrm{QFT}}(x,t)={\Psi}^{\u2020}(x,t)\Psi (x,t)$ and ${\rho}_{\mathrm{k}}^{\mathrm{QFT}}(k,t)={\Phi}^{\u2020}(k,t)\Phi (k,t)$ | QFT probability density operators in position and spatial frequency domains, respectively. |

$|{n}_{{q}_{1}},{n}_{{q}_{2}},,\cdots ,{n}_{{q}_{i}},\cdots {n}_{{q}_{K}}\rangle $ | Fock states with occupation number, where ${n}_{{q}_{i}}$ is the number of particles in a QM state $|{q}_{i}\rangle $. |

$|\mathrm{state}\rangle ={\sum}_{m}{\alpha}_{m}|{n}_{{q}_{1}},{n}_{{q}_{2}},\cdots ,{n}_{{q}_{i}},\cdots \rangle $ | a QFT state in a Fock space where m is an index over configurations of a Fock state, ${\alpha}_{m}\in \mathbb{C}$, and $1={\sum}_{m}{\left|{\alpha}_{m}\right|}^{2}$. |

Below are the same symbols for probability density values for the $x,k$ variables used for the QM and QFT frameworks. Context will disambiguate | ${\rho}_{\mathrm{x}}(x,t)=\left\{\begin{array}{cc}\langle state|{\Psi}^{\u2020}(x,t)\Psi (x,t)|state\rangle :& \mathrm{QFT}\mathrm{for}\mathrm{the}\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{0.166667em}{0ex}}\mathrm{variable}\\ \langle x|{\psi}_{t}\rangle \langle {\psi}_{t}||x\rangle :& \mathrm{QM}\mathrm{for}\mathrm{the}\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{0.166667em}{0ex}}\mathrm{variable}\end{array}\right.$ |

${\rho}_{\mathrm{k}}(k,t)=\left\{\begin{array}{cc}\langle state|{\Phi}^{\u2020}(k,t)\Phi (k,t)|state\rangle :& \mathrm{QFT}\mathrm{for}\mathrm{the}\phantom{\rule{0.166667em}{0ex}}k\phantom{\rule{0.166667em}{0ex}}\mathrm{variable}\\ \langle k||{\psi}_{t}\rangle \langle {\psi}_{t}||k\rangle :& \mathrm{QM}\mathrm{for}\mathrm{the}\phantom{\rule{0.166667em}{0ex}}k\phantom{\rule{0.166667em}{0ex}}\mathrm{variable}\end{array}\right.$ | |

${\lambda}_{i}>0,i=1,\cdots ,m$ | probability coefficients of a mixed state made of m pure states. |

Section 3 | |

$F:(x,k,t)\mapsto ({x}^{\prime},{k}^{\prime},t)$ | a canonical transformation of coordinates |

${J}_{F}(x,k,t)$ to be the Jacobian | |

Given a QFT solution $\Psi (x,t)$ | |

${\Psi}^{\mathrm{C}}(x,t)=C{\overline{\Psi}}^{\mathsf{T}}(x,t)$ | Charge Conjugation satisfying |

$C{\gamma}^{\mu}{C}^{-1}=-{\gamma}^{\mu \mathsf{T}}$, and in the standard representation $C=\mathrm{i}{\gamma}^{2}{\gamma}^{0}$ up to a phase. | |

${\Psi}^{\mathrm{P}}(-x,t)=P\Psi (-x,t)$ | Parity Change, so $P={\gamma}^{0}$ |

${\Psi}^{\mathrm{T}}(x,-t)=T{\Psi}^{*}(x,-t)$ | Time Reversal, 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. |

${\psi}^{\mathrm{CPT}}(-x,-t)=CPT{\overline{\psi}}^{\mathsf{T}}(-x,-t)$ | Charge Conjugation, Parity Change, and Time Reversal. |

Section 5 | |

$U\left(t\right)={\mathrm{e}}^{-\mathrm{i}\frac{H}{\hslash}t}$ | evolution operator for Hamiltonian H |

$\left(\langle x|{\psi}_{0}\rangle ,\phantom{\rule{0.166667em}{0ex}}\langle k|{\psi}_{0}\rangle ,\right)$ | initial QM condition for the unitary evolution |

$\left({\Psi}_{0}=\Psi (x,0),\phantom{\rule{0.166667em}{0ex}}{\Phi}_{0}=\Phi (k,0)\right)$ | initial QFT condition for the unitary evolution |

QCurve | |

$\mathcal{C}$ | The set of QCurves for which the entropy is a constant |

$\mathcal{I}$ | The set of QCurves for which the entropy is increasing, but it is not a constant |

$\mathcal{D}$ | The set of QCurves for which the entropy is decreasing, but it is not a constant |

$\mathcal{O}$ | The set of oscillating QCurves, with the entropy strictly increasing in some subinterval of $[0,\delta t]$ and strictly decreasing in another subinterval of $[0,\delta t]$ |

$\mathcal{N}\left(x\mid c,\Sigma \right)$ | Normal distribution for variable x, centered in c, and with covariance $\Sigma $ |

${\alpha}_{k}\left(t\right)$ | Coefficients of expansion of a QM state $|{\psi}_{t}\rangle $ into the energy eigenstates $|{\psi}_{{E}_{k}}\rangle $ |

Section 6 | |

${A}^{i}\left(k\right)$ | Electromagnetic potential 3D components, $i=1,2,3$, in spatial frequency space |

${a}_{\lambda}\left(k\right)$ | annihilation operator per polarization $\lambda =1,2$, in spatial frequency space |

${a}_{\lambda}^{\u2020}\left(k\right)$ | creation operator per polarization $\lambda =1,2$, in spatial frequency space |

${\u03f5}_{\lambda}^{i}\left(k\right)$ | polarization 3D orientation components, $i-1,2,3$, per polarization $\lambda =1,2$ |

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**Figure 1.**A visualization of the Time Reflection Theorem. (i) Axis t: A QCurve ${e}_{1}=\left({\Psi}_{0}=\Psi (x,0),{\mathrm{e}}^{-\mathrm{i}Ht},\delta t\right)$. (ii) Axis ${t}^{\prime}=\delta t-t$: The antiparticle QCurve is created as ${e}_{2}={Q}_{{\mathrm{CPT}}_{\delta}}\left({e}_{1}\right)=\left({\Psi}^{{\mathrm{CPT}}_{\delta}}(-x,{t}^{\prime}=0),{\mathrm{e}}^{-\mathrm{i}H{t}^{\prime}},\delta t\right)$. Axis ${t}^{\prime}$ shows the evolution as going forward in time ${t}^{\prime}$. The evolution of ${\Psi}^{{\mathrm{CPT}}_{\delta}}(-x,{t}^{\prime})=\eta {\gamma}^{5}{\left({\Psi}^{\u2020}\right)}^{\mathsf{T}}(x,\delta t-{t}^{\prime})$ is mirroring the evolution of $\Psi (x,t)$, with $t={t}^{\prime}$ evolving from 0 to $\delta t$. If ${e}_{1}\in \mathcal{D}$, then ${e}_{2}\in \mathcal{I}$.

**Figure 2.**Collision of two fermions with individual amplitudes (18), parameters ${k}_{0}=1$, ${c}_{2}=-{c}_{1}=300$, speed of light $c=1$, a grid of $1\phantom{\rule{0.166667em}{0ex}}000$ points for ${x}_{1},{x}_{2},{k}_{1},{k}_{2}$. The left column shows entropy vs. time. The right column shows snapshots of the density at initial time, final time, and intervals of 100 time units, overlaid on single plots. The z-axis represents the density, and the x and y axes represent the ${x}_{1}$ and ${x}_{2}$ values, respectively. As the particles approach each other, their individual densities disperse, the maximum values are reduced, and the entropy increases. Only when the particles are close to each other, the interference reduces the total entropy.

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Geiger, D.; Kedem, Z.M. On Quantum Entropy. *Entropy* **2022**, *24*, 1341.
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Geiger D, Kedem ZM. On Quantum Entropy. *Entropy*. 2022; 24(10):1341.
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Geiger, Davi, and Zvi M. Kedem. 2022. "On Quantum Entropy" *Entropy* 24, no. 10: 1341.
https://doi.org/10.3390/e24101341