# The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

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## Abstract

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## 1. Introduction

## 2. Necessary and Sufficient Conditions for Reaching Orthogonality

#### 2.1. Families of Allowed Triads $\left\{{r}_{i}\right\}$

#### 2.2. The Solution Diagram

- $\left\{\frac{1}{2},\frac{1}{2},0\right\}$ corresponds to$${\omega}_{21}\tau =(2l+1)\pi ,\phantom{\rule{1.em}{0ex}}l=0,1,\cdots ,$$
- $\left\{\frac{1}{2},0,\frac{1}{2}\right\}$ corresponds to ${\omega}_{31}\tau =\pi ,3\pi ,5\pi ,\dots $. By writing ${\omega}_{31}={\omega}_{21}(1+\Omega )$, this is equivalently expressed as$${\omega}_{21}\tau =\frac{(2l+1)\pi}{1+\Omega},\phantom{\rule{1.em}{0ex}}l=0,1,\cdots ,$$
- $\left\{0,\frac{1}{2},\frac{1}{2}\right\}$ corresponds to ${\omega}_{32}\tau =\pi ,3\pi ,5\pi ,\dots $. With ${\omega}_{32}=\Omega {\omega}_{21}$, this amounts to$${\omega}_{21}\tau =\frac{(2l+1)\pi}{\Omega},\phantom{\rule{1.em}{0ex}}l=0,1,\cdots ,$$

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- $\left\{\frac{1}{2},r,\frac{1}{2}-r\right\}$ corresponds to$${\omega}_{21}\tau =(2l+1)\pi \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega =\frac{2({l}^{\prime}+1)}{2l+1},$$
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- $\left\{r,\frac{1}{2},\frac{1}{2}-r\right\}$ corresponds to$${\omega}_{21}\tau =(2l+1)\pi \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega =\frac{2{l}^{\prime}+1}{2l+1},$$
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- $\left\{r,\frac{1}{2}-r,\frac{1}{2}\right\}$ corresponds to$${\omega}_{21}\tau =2(l+1)\pi \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\Omega =\frac{2{l}^{\prime}+1}{2(l+1)},$$

#### 2.3. Orthogonality Time

## 3. The Quantum Speed Limit

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- $\left\{\frac{1}{2},r,\frac{1}{2}-r\right\}$, $0<r<\frac{1}{2}$. These points form the edge that goes from the vertex $\left\{\frac{1}{2},0,\frac{1}{2}\right\}$ to $\left\{\frac{1}{2},\frac{1}{2},0\right\}$ of the 2-simplex ${\delta}^{2}$ in Figure 2 and Figure 3, and they correspond to$$\alpha =\sqrt{1+\frac{r(1-2r){\Omega}^{2}}{{\left[r+(\frac{1}{2}-r)(1+\Omega )\right]}^{2}}}>1\phantom{\rule{1.em}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\Omega ,$$
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- $\left\{r,\frac{1}{2}-r,\frac{1}{2}\right\}$, $0<r<\frac{1}{2}$. These points form the edge that goes from the vertex $\left\{\frac{1}{2},0,\frac{1}{2}\right\}$ to $\left\{0,\frac{1}{2},\frac{1}{2}\right\}$ in Figure 2 and Figure 3, and give$$\alpha =\sqrt{1-\frac{(1-2r)(1-r+\Omega )}{{\left[1-r+\frac{1}{2}\Omega \right]}^{2}}}<1\phantom{\rule{1.em}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\Omega .$$
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- $\left\{r,\frac{1}{2},\frac{1}{2}-r\right\}$, $0<r<\frac{1}{2}$. This set of points lies along the edge that goes from the vertex $\left\{0,\frac{1}{2},\frac{1}{2}\right\}$ to $\left\{\frac{1}{2},\frac{1}{2},0\right\}$ in Figure 2 and Figure 3, and their corresponding $\alpha $ reads$$\alpha =\sqrt{1+\frac{(1-2r)(1+\Omega )\left[r(1+\Omega )-1\right]}{{\left[1-r+\Omega \left(\frac{1}{2}-r\right)\right]}^{2}}}.$$Unlike the previous cases, here, the range of $\alpha $ depends on the range of $\Omega $. For $\Omega \le 1$ we have $\alpha <1$, whereas for $\Omega >1$ we find three cases:$$\begin{array}{cc}\hfill \alpha >1& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\frac{1}{1+\Omega}<r<\frac{1}{2},\hfill \end{array}$$$$\begin{array}{cc}\hfill \alpha =1& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r=\frac{1}{1+\Omega},\hfill \end{array}$$$$\begin{array}{cc}\hfill \alpha <1& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0<r<\frac{1}{1+\Omega}.\hfill \end{array}$$In this way, for a given $\Omega $, the edge under consideration is divided into two segments, one cyan and one magenta, separated by the point at $r={(1+\Omega )}^{-1}$. Notice that such segments are not appreciated in Figure 3, since (as explained above) this figure considers a sample of different values of $\Omega $ and not a single one.

## 4. Summary and Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Diagram ${\omega}_{21}\tau $ vs. $\Omega $, with $\Omega ={\omega}_{32}/{\omega}_{21}$. The coefficients $\{{r}_{1},{r}_{2},{r}_{3}\}$ giving rise to states (1) that reach an orthogonal state at time $\tau $ are represented by points in the diagram for the corresponding values of ${\omega}_{21}\tau $ and $\Omega $. Blue-shaded regions represent $\left\{{r}_{i}\right\}$ of Family II, satisfying Equation (8). These regions are bordered by the solutions $\left\{{r}_{i}\right\}$ pertaining to Family I as follows. For Subfamily I-qubit: $\left\{\frac{1}{2},\frac{1}{2},0\right\}$ is identified with solid-blue lines, $\left\{0,\frac{1}{2},\frac{1}{2}\right\}$ with solid-green lines and $\left\{\frac{1}{2},0,\frac{1}{2}\right\}$ with red lines. For Subfamily I-b: $\left\{\frac{1}{2},r,\frac{1}{2}-r\right\}$ is identified with a star, $\left\{r,\frac{1}{2},\frac{1}{2}-r\right\}$ with a left-triangle and $\left\{r,\frac{1}{2}-r,\frac{1}{2}\right\}$ with a square. The red-dashed curve indicates the global lower bound for the orthogonality time.

**Figure 2.**The 2-simplex ${\Delta}^{2}$ of ${\mathbb{R}}^{3}$ (light-blue shaded plane), defined by the set of points $({r}_{1},{r}_{2},{r}_{3})$ satisfying ${r}_{i}\ge 0$ and ${\sum}_{i}{r}_{i}=1$. The 2-simplex ${\delta}^{2}$ (colored central triangle) contains the subset of points of ${\Delta}^{2}$ that define the coefficients in the energy-expansion of initial states $\left|\psi \left(0\right)\right.\u232a$ that evolve towards a distinguishable state at time $\tau $. The vertices of ${\delta}^{2}$ correspond to elements of Family I-qubit, its edges to elements of Family I-b and its interior to elements of Family II. The colors in ${\delta}^{2}$ identify the triads according to a RGB map-code defined by its vertices.

**Figure 3.**Map of the quantum speed limit (of the states associated to the triads $\left\{{r}_{i}\right\}$) in the 2-simplex ${\delta}^{2}$. For different values of $\Omega $, the simplex is colored according to the range of $\alpha $ (Equation (26)) as follows: the Mandelstam–Tamm subset ${\delta}_{\mathrm{MT}}^{2}$ ($\alpha <1$, cyan) and the Margolus–Levitin one ${\delta}_{\mathrm{ML}}^{2}$ ($\alpha >1$, magenta).

**Table 1.**Complete set of expansion coefficients $\left\{{r}_{i}\right\}$ that satisfy the orthogonality condition, the corresponding orthogonality time $\tau $ (in terms of the energy-separation levels of the Hamiltonian) and the ratio $\alpha $ that determines whether the quantum speed limit is given by the Mandelstam–Tamm bound ($\alpha <1$) or the Margolus–Levitin bound ($\alpha >1$). We denote $D=sin{\omega}_{23}\tau +sin{\omega}_{31}\tau +sin{\omega}_{12}\tau $.

Family | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | ${\mathit{r}}_{3}$ | $\mathit{\tau}$ | $\mathit{\alpha}$ | |
---|---|---|---|---|---|---|

I-qubit | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{n\pi}{{\omega}_{32}}$, n odd | 1 | |

I-qubit | $\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{n\pi}{{\omega}_{31}}$, n odd | 1 | |

I-qubit | $\frac{1}{2}$ | $\frac{1}{2}$ | 0 | $\frac{n\pi}{{\omega}_{21}}$, n odd | 1 | |

I-b | $\frac{1}{2}$ | r | $\frac{1}{2}-r$ | $(0<r<\frac{1}{2})$ | $\frac{n\pi}{{\omega}_{21}}$ with $\frac{{\omega}_{32}}{{\omega}_{21}}=\frac{m}{n}$, n odd, m even | $>1$ |

I-b | r | $\frac{1}{2}-r$ | $\frac{1}{2}$ | $(0<r<\frac{1}{2})$ | $\frac{n\pi}{{\omega}_{21}}$ with $\frac{{\omega}_{32}}{{\omega}_{21}}=\frac{m}{n}$, n even, m odd | $<1$ |

I-b | r | $\frac{1}{2}$ | $\frac{1}{2}-r$ | $(0<r<\frac{1}{2})$ | $\frac{n\pi}{{\omega}_{21}}$ with $\frac{{\omega}_{32}}{{\omega}_{21}}=\frac{m}{n}$, n odd, m odd | $>1$, for $\frac{1}{1+\frac{{\omega}_{32}}{{\omega}_{21}}}<r<\frac{1}{2}$ $=1$, for $r=\frac{1}{1+\frac{{\omega}_{32}}{{\omega}_{21}}}$ $<1$, for $0<r<\frac{1}{1+\frac{{\omega}_{32}}{{\omega}_{21}}}$ |

II | $\frac{sin{\omega}_{23}\tau}{D}$ | $\frac{sin{\omega}_{31}\tau}{D}$ | $\frac{sin{\omega}_{12}\tau}{D}$ | $(0<{r}_{i}<1)$ | Implicitly defined via Equation (8) | $\u2a8c1$ |

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**MDPI and ACS Style**

Sevilla, F.J.; Valdés-Hernández, A.; Barrios, A.J. The Underlying Order Induced by Orthogonality and the Quantum Speed Limit. *Quantum Rep.* **2021**, *3*, 376-388.
https://doi.org/10.3390/quantum3030024

**AMA Style**

Sevilla FJ, Valdés-Hernández A, Barrios AJ. The Underlying Order Induced by Orthogonality and the Quantum Speed Limit. *Quantum Reports*. 2021; 3(3):376-388.
https://doi.org/10.3390/quantum3030024

**Chicago/Turabian Style**

Sevilla, Francisco J., Andrea Valdés-Hernández, and Alan J. Barrios. 2021. "The Underlying Order Induced by Orthogonality and the Quantum Speed Limit" *Quantum Reports* 3, no. 3: 376-388.
https://doi.org/10.3390/quantum3030024