# Quantum Knowledge in Phase Space

## Abstract

**:**

## 1. Introduction

#### 1.1. Bayesian Knowledge in Quantum Physics

#### 1.2. Measurements and Knowledge

#### 1.3. Phase Space

#### 1.4. Entropy, Interference, and Entanglement

#### 1.5. Paper Organization

## 2. A Bayesian View of Quantum Phase Space

**or**and

**and**become additions and products of states analogous to operations in classical probability theory. For example, in the double slit experiment, according to classical logic an electron can pass through slit 1

**or**slit 2, so the quantum state at slit 1 will add with the quantum state at slit 2 to form the final state. To represent a particle A

**and**a particle B, one takes the product of the two quantum states (a state in a product of Hilbert Spaces). After the operations with the state occur, quantum probabilities are then associated with the final state $|{\mathsf{\Psi}}_{t}\rangle $ via the probability density matrix ${\rho}_{t}=|{\mathsf{\Psi}}_{t}\rangle \langle {\mathsf{\Psi}}_{t}|$.

**Lemma**

**1**

**Proof.**

**Theorem**

**1**

**Proof.**

#### Entropy in Phase Space

## 3. Interference

**Definition**

**1**

- the functions’ support in phase space do not overlap, i.e.,$$\begin{array}{cc}\hfill |{\psi}^{\mathrm{A}}\left(x\right)\left|\phantom{\rule{0.166667em}{0ex}}\right|{\psi}^{\mathrm{B}}\left(x\right)|& =0;\phantom{\rule{1.em}{0ex}}\forall x\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}|{\varphi}^{\mathrm{A}}\left(k\right)\left|\phantom{\rule{0.166667em}{0ex}}\right|{\varphi}^{\mathrm{B}}\left(k\right)|=0;\phantom{\rule{1.em}{0ex}}\forall k\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- the complex phases are aligned up to a constant multiple of $\frac{\pi}{2}$, i.e.,$$\begin{array}{c}\hfill {\xi}_{A}\left(x\right)-{\xi}_{B}\left(x\right)-{\phi}_{1}=n\frac{\pi}{2};\phantom{\rule{1.em}{0ex}}n\in {\mathrm{Z}}^{+}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\chi}_{A}\left(k\right)-{\chi}_{B}\left(k\right)-{\phi}_{1}=m\frac{\pi}{2};\phantom{\rule{1.em}{0ex}}m\in {\mathrm{Z}}^{+}\phantom{\rule{0.166667em}{0ex}},\end{array}$$
- either ${\theta}_{1}=0,\frac{\pi}{2}$, since then there is no superposition of states. This will effectively occur when ${\psi}^{\mathrm{A}}\left(x\right)={\psi}^{\mathrm{B}}\left(x\right)$.

## 4. Entanglement

**Proposition**

**1.**

**Proof.**

**Definition**

**2**

- ${\theta}_{2}=0,\frac{\pi}{2}$
- $|{\mathsf{\Psi}}^{\mathrm{A}}\rangle =|{\mathsf{\Psi}}^{\mathrm{B}}\rangle $.

## 5. Entanglement for Spin or Qbit Phase Space

**Definition**

**3**

#### Expansion to Mixed States

## 6. Conclusions

**or**and

**and**logical operations were applied to quantum states and not to quantum probabilities.

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Combining Two States

## Appendix B. Entropy Concepts

## References

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**Figure 1.**Normal distribution in phase space for two coherent states,

**A**and

**B**in 1D with centers and variances as follows. For all (

**a**–

**c**) position space probabilities, with

**${\mu}_{A}=27$**with

**${\sigma}_{A}=5.6$**and for (

**a**)

**${\mu}_{B}=27$**, (

**b**)

**${\mu}_{B}=37$**, (

**c**)

**${\mu}_{B}\approx 48$**, all with

**${\sigma}_{B}=4.2$**. Note that for each coherent state, the spatial frequency value is the phase of the coherent state in position space. For (

**d**–

**f**), spatial frequency space probabilities, with

**${k}_{A}\approx 0.35$**and (

**d**)

**${k}_{B}\approx 0.35$**, (

**e**)

**${k}_{B}\approx 0.75$**, (

**f**)

**${k}_{B}\approx 1.15$**. For (

**g**–

**i**), spatial frequency space probabilities, with

**${k}_{A}\approx 1.04$**and (

**g**)

**${k}_{B}\approx 1.04$**, (

**h**)

**${k}_{B}\approx 1.55$**, (

**i**)

**${k}_{B}\approx 2.05$**.

**Figure 2.**Interference Simulations for a superposition of two coherent states as shown in Figure 1. The coherent state $|{\mathsf{\Psi}}^{\mathrm{A}}\rangle $ has ${\mu}_{A}=27$, ${\sigma}_{A}=5.6$, and for (

**a**,

**b**) the phase is ${k}_{A}=0.35$ while for (

**c**,

**d**), the phase is ${k}_{A}=1.04$. The other coherent state $|{\mathsf{\Psi}}^{\mathrm{B}}\rangle $ in position has fixed ${\sigma}_{B}=4.2$ and the position center and phase vary in 48 increments each, as follows: ${\mu}_{B}=[27,\dots ,48]$, and for (

**a**,

**b**), the phase varies as ${k}_{B}=[0.35,\dots ,1.15]$, while for (

**c**,

**d**), the phase varies as ${k}_{B}=[1.04,\dots ,2.05]$. The plots axis are all with $\Delta \mu ={\mu}_{B}-{\mu}_{A}$ vs. $\Delta k={k}_{B}-{k}_{A}$. The KLD and the entropy become small as the two states closely overlap, i.e., where $\delta \mu \approx \delta k\approx 0$. However, the KLD becomes small as the states do not overlap while the entropy gets to be larger. As the phase increases from (

**a**,

**b**) to (

**c**,

**d**) oscillation increases for both (KLD and Entropy) as periods reduce. Entropy seems to be a good estimation for the interference behavior when the two states overlap either in spatial frequency or in position. However, the more the overlap in both spaces is reduced the more the two quantities differ in behavior.

**Figure 3.**Entanglement simulations from two coherent states shown in Figure 1, with normal probability distributions. Note that the phase of the coherent state projected to position space is the center of the coherent state projected in the spatial frequency space, and vice versa. The coherent state $|{\mathsf{\Psi}}^{\mathrm{A}}\rangle $ has a fixed set of parameters, ${\mu}_{A}=27$, ${\sigma}_{A}=5.6$ in position space, and for (

**a**,

**b**) the phase is ${k}_{A}=0.35$ while for (

**c**–

**f**) the phase is ${k}_{A}=1.04$. The coherent state $|{\mathsf{\Psi}}^{\mathrm{B}}\rangle $ in position has fixed ${\sigma}_{B}=4.2$ and the center and phase vary in 48 increments each, as follows: ${\mu}_{B}\in [27,\dots ,48]$, and for (

**a**,

**b**) ${k}_{B}\in [0.35,\dots ,1.15]$, while for (

**c**–

**f**) the phase varies as ${k}_{B}\in [1.04,\dots ,2.05]$. The parameter ${\theta}_{2}=\frac{pi}{4}$ is fixed when entangling the two states. Cases (

**a**–

**d**) show KLD and Entropy, respectively, for a symmetric entanglement where phase ${\phi}_{2}=0$. Cases (

**e**,

**f**) show KLD and Entropy, respectively, for an anti-symmetric entanglement where phase ${\phi}_{2}=\pi $. The effect of the phase ${\phi}_{2}$ is only noticeable when the two states are very similar to each other and then both, KLD and entropy, yield large values for the anti-symmetric case (after all anti-symmetric functions must vanish in these cases, while the product of states does not). While the KLD has a smoother behavior, both increase as the separation of the two coherent state parameters increases. The larger values of the phase parameters in (

**c**–

**f**) clearly cause a periodic behavior. Entropy behavior seems to be a good estimation for entanglement.

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Geiger, D.
Quantum Knowledge in Phase Space. *Entropy* **2023**, *25*, 1227.
https://doi.org/10.3390/e25081227

**AMA Style**

Geiger D.
Quantum Knowledge in Phase Space. *Entropy*. 2023; 25(8):1227.
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**Chicago/Turabian Style**

Geiger, Davi.
2023. "Quantum Knowledge in Phase Space" *Entropy* 25, no. 8: 1227.
https://doi.org/10.3390/e25081227