# SU(2) Decomposition for the Quantum Information Dynamics in 2d-Partite Two-Level Quantum Systems

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Hamiltonian

## 3. $\mathit{SU}(\mathbf{2})$ Decomposition Generalities

## 4. GBS: A Non-Local Basis Fitting in {$\left(\right)$}

#### 4.1. GBS Basis and Hamiltonian Components

#### 4.2. Case $d=1$

#### 4.3. Case $d>1$

#### 4.3.1. Analysis of $\left(\right)$

#### 4.3.2. Analysis of $\left(\right)$

#### 4.3.3. Analysis of $\left(\right)$

**Correspondent terms**${H}_{{\mathrm{nl}}_{i}}^{(k,k+d)}$. This term in the Hamiltonian ${H}_{{\mathrm{nl}}_{i}}$ contains ${\sigma}_{0}\otimes \dots \otimes {\sigma}_{i}\otimes \dots \otimes {\sigma}_{i}\otimes \dots \otimes {\sigma}_{0}$ with ${\sigma}_{i}$ in positions k and $k+d$, and ${\sigma}_{0}$ in any other. When this term is allocated in $\left(\right)$ in agreement with (18), it does not cancel if each factor in the product become different from zero, implying ${i}_{s}={k}_{s}\phantom{\rule{1.em}{0ex}}\forall s=1,2,\dots ,d$. Thus, this term gives non-zero entries only in the diagonal elements. Thus, each non-zero entry of ${H}_{{\mathrm{nl}}_{i}}$ will have d different terms in each diagonal element (one for each pair of interacting correspondent particles). Those terms will appear with different signs in each diagonal element in spite of ${c}_{{j}_{s},{j}_{d+s}}^{{i}_{s},{k}_{s}}$. At this point, note that results for ${H}_{{\mathrm{nl}}_{i}}^{(k,k+d)}$ and ${H}_{{\mathrm{l}}_{i}}^{(k)}$ were expected due to the results in [11,26] and the separability of the GBS basis in their constitutive entangled pairs.

**Non-correspondent terms**${H}_{{\mathrm{nl}}_{i}}^{(k,{k}^{\prime}\ne k+d)}$. These terms have a different behavior. Each term contains ${\sigma}_{0}\otimes \dots \otimes {\sigma}_{i}\otimes \dots \otimes {\sigma}_{i}\otimes \dots \otimes {\sigma}_{0}$, with ${\sigma}_{i}$ in positions k and ${k}^{\prime}$, and ${\sigma}_{0}$ in any other. It defines two pairs of correspondent parts involving ${\sigma}_{i}$: $[k,k+d,{k}^{\prime},{k}^{\prime}+d]$ if $k<{k}^{\prime}\le d$ or $k,k+d,{k}^{\prime}-d,{k}^{\prime}$ if $k\le d<{k}^{\prime}\le 2d$. Then, each factor in (18) related with those two pairs ($s=k,{k}^{\prime}$ or $s=k,{k}^{\prime}-d$) will now include $\mathrm{Tr}({\sigma}_{{i}_{s}}{\sigma}_{i}{\sigma}_{{k}_{s}})$ (until unitary factors), which is non-zero only if: (a) ${i}_{s}$ or ${k}_{s}$ are one of the pairs 0 and i or i and 0; (b) $i,{i}_{s},{k}_{s}$ are a permutation $i,{i}^{\prime},{i}^{\u2033}$ of $1,2,3$ (having two cases depending on the parity). The last situation is similar to the local terms in the previous subsection, but in two parts simultaneously. The remaining factors for $s\ne k,{k}^{\prime}$ or $s\ne k,{k}^{\prime}-d$ will require ${i}_{s}={k}_{s}$ in order to become non-zero. The latter scenario gives 16 possibilities for each term ${h}_{{(i({4}^{k-1}+{4}^{{k}^{\prime}-1}))}_{4}^{2d}}$, which will appear in diagonal-off positions obtained departing from the diagonal position $({i}_{1},\dots ,{i}_{d};{i}_{1},\dots ,{i}_{d})$ in $\left(\right)$, by changing each index in the pair $({i}_{k},{i}_{k}^{\prime})$ in the row, following the rules depicted in cases A and B. Thus, for each column and with $i,k,{k}^{\prime}$ fixed, only one row becomes non-zero, in agreement with the previous rule. Each entry of this kind involves four terms, including the four combinations of each pair of non-correspondent parts selected from the set $[k,{k}^{\prime},k+d,{k}^{\prime}+d]$. Instead, when all values i and $k,{k}^{\prime}$ are considered, a total of $3\xb7\frac{1}{2}d(d-1)$ non-zero rows appear in each column (clearly, by considering all these terms, $SU(2)$ decomposition is not achieved).

#### 4.3.4. Analysis of $\left(\right)$

**Correspondent terms**${H}_{{\mathrm{cnl}}_{i}}^{(k,k+d)}$. For each term ${H}_{{\mathrm{cnl}}_{i}}^{(k,k+d)}$, the behavior is similar as for ${H}_{{\mathrm{l}}_{i}}^{({k}^{\prime})}$. Because only one correspondent pair has ${j}_{p}={j}_{s}\ne 0\ne {j}_{d+s}={k}_{p}$ in (18), then ${i}_{s},{k}_{s}$ for $s={k}^{\prime}$ should be $0,i$ or ${j}_{p},{k}_{p}$. For $s\ne {k}^{\prime}$, ${i}_{s}={k}_{s}$. As before, it means that each term is diagonal-off by combining the values of index ${k}^{\prime}$ in $\mathcal{I}$ and $\mathcal{K}$ as before: $0,i$; $i,0$; ${j}_{p},{k}_{p}$; and ${k}_{p},{j}_{p}$. For a fixed column and $i,k$, it will give four possibilities and two $SU(2)$ blocks. Each entry will have two terms corresponding to the different parities p. Note that only one i and ${k}^{\prime}$ can be considered to achieve the $SU(2)$ decomposition. Otherwise, for each column, $3d$ rows different from zero could appear, breaking the $SU(2)$ decomposition as for the local interaction case.

**Non-correspondent terms**${H}_{{\mathrm{cnl}}_{i}}^{(k,{k}^{\prime}\ne k+d)}$. As for ${H}_{{\mathrm{nl}}_{i}}^{(k,{k}^{\prime}\ne k+d)}$, in this case the only non-zero terms have ${i}_{s}={k}_{s}$ for $s\ne k,{k}^{\prime},k-d,{k}^{\prime}-d$. Meanwhile, for the two remaining cases $s\in \{k,{k}^{\prime},k-d,{k}^{\prime}-d\}\cap \{1,2,\dots ,d\}$, each ${i}_{s},{k}_{s}$ should be selected from the set $0,{j}_{p}$; ${j}_{p},0$; $i,{k}_{p}$; ${k}_{p},i$ or $0,{k}_{p}$; ${k}_{p},0$; $i,{j}_{p}$; ${j}_{p},i$. In a specific column and fixing i, it will give 16 possibilities and 8 blocks in $SU(2)$, as for the ${H}_{{\mathrm{nl}}_{i}}^{(k,{k}^{\prime}\ne k+d)}$ case. Note that parity p should be fixed in this case because each one gives a different decomposition. Each entry will contain four terms for each parity p combining the four possible interaction terms. Again, if all options for i and $k,{k}^{\prime},p$ are considered, then $3\xb7d(d-1)$ non-zero rows will appear for each column, breaking the $SU(2)$ decomposition. These terms are not commonly introduced in models such as Heisenberg–Ising and those related. Instead, for magnetic systems they are the first-order approximation in the spin–orbit coupling, introducing antisymmetric exchange as in the Dzyaloshinskii–Moriya model: ${H}_{DM}=\overrightarrow{D}\xb7(\overrightarrow{{\sigma}_{1}}\times \overrightarrow{{\sigma}_{2}})$. There, $\overrightarrow{D}$ is the Dzyaloshinskii–Moriya vector defining the orientation of coupling. Here, as only one term can be included in order to preserve the $SU(2)$ reduction property, this coupling should be strictly oriented.

#### 4.4. Explicit Analytical Formulas for Hamiltonian Components

## 5. Specific Interactions Generating $\mathit{SU}(\mathbf{2})$ Decomposition

#### 5.1. General Depiction of Interactions Having $SU(2)$ Decomposition for the GBS Basis

- A:
- Curved arrows point out those qubits related through entangling operations in any case.
- B:
- All curved arrows in the bottom refer to Heisenberg–Ising-like (non-crossed) interactions involving the three possible spatial directions together. Those interaction relations set the correspondent pairs.
- C:
- For the curved arrows in the top, two kinds of entangling operations can be considered according to the text: Heisenberg–Ising-like (non-crossed) interactions or Dzyaloshinskii–Moriya-like (crossed) interactions. Only one characteristic spatial direction is allowed.
- D:
- Type II interactions can be split into Type IIa and Type IIb if interactions in the top are non-crossed or crossed (between parts of two different correspondent pairs), respectively. Type IIb interactions in the top admits only one possible parity from the two possible.
- E:
- Type III interactions admit only crossed interactions in the top between parts of one specific correspondent pair, but the two possible parities together are allowed.
- F:
- For Figure 3a, the right arrows correspond to external local interactions such as those generated by magnetic fields on spin-based qubits. Due to their locality, they are referred to as driven interactions, although it actually depends on the available control of the interactions.

#### 5.2. General Structure of $SU(2)$ Blocks

#### 5.3. Structure of $SU(2)$ Blocks for Each Interaction

#### 5.3.1. Blocks in Type I Interaction

#### 5.3.2. Blocks in Type II Interaction

**Type IIa:**In this case, the interaction is completely non-local between correspondent pairs to generate the diagonal entries, and in only one direction between non-correspondent parts in two correspondent pairs to generate the diagonal-off entries. The Hamiltonian becomes:

**Type IIb:**For this interaction, the non-diagonal part generated by the non-local interaction between non-correspondent parts is supplied by a non-local and crossed interaction among non-correspondent parts of two correspondent pairs:

#### 5.3.3. Blocks in Type III Interaction

#### 5.4. Available Parameters and Structure of Entries

#### 5.4.1. Structure of Diagonal Entries Belonging to a Specific Block

#### 5.4.2. Structure of Diagonal-Off Entries Belonging to a Specific Block

#### 5.4.3. Block Entries of ${H}_{I}$

#### 5.4.4. Block Entries of ${H}_{IIa}$

#### 5.4.5. Block Entries of ${H}_{IIb}$

#### 5.4.6. Block Entries of ${H}_{III}$

## 6. Connectedness, Superposition, Entanglement and Separability

#### 6.1. Exchange Connectedness under Interactions

#### 6.2. Notable Quantum Processing Operations Achievable under $SU(2)$ Decomposition

#### 6.3. $SU(2)$ Decomposition in the Context of $n-$Qubit Controlled Gates

#### 6.4. Generating Superposition and Entanglement

#### 6.4.1. Generating $2-$Separable Superposition

#### 6.4.2. Entanglement Dynamics under Interactions

#### 6.4.3. Generating Larger Maximal Entangled Systems

#### 6.4.4. Recursive Generation of Larger Maximal Entangled Systems

#### 6.5. Multipartite Entanglement and General States

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Generic Hamiltonian Expressed in Terms of Pauli Operators

#### Appendix A.2. Group Theory Basics in the Context of the SU(2) Decomposition

#### Appendix A.3. Generalized Bell States Basis in Context

#### Appendix A.4. Illustrative Examples of SU(2) Decomposition

#### Appendix A.4.1. Case d = 1

#### Appendix A.4.2. Case d = 2

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**Figure 1.**First case for a pair of entries in which $\left(\right)$ is non-zero. In them, for a fixed position $s=k$ in the row and the column labels appears i or 0, while the other corresponding positions in the row and in the column have the same values.

**Figure 2.**Second case for a pair of entries in which $\left(\right)$ is non-zero. In them, for a fixed position $s=k$ in the row and the column labels appears ${i}^{\prime}$ or ${i}^{\u2033}$ alternatively ($i,{i}^{\prime},{i}^{\u2033}$ being a permutation of $1,2,3$), while other corresponding positions in the row and in the column have the same values.

**Figure 3.**Three types of physical interactions able to generate the block decomposition. Non-local and non-crossed interactions among any correspondent parts combined with: (

**a**) local interactions on only two correspondent parts (${k}_{l},{k}_{l}+d$); (

**b**) any two non-correspondent parts in only two specific pairs of correspondent parts of only one subtype, non-crossed or crossed; and (

**c**) crossed interactions between a specific pair of correspondent parts.

**Figure 4.**Representation of qubit interactions able to generate $SU(2)$ decomposition: (

**a**) Type I, II, and III interactions among $2d$ qubits (Type III assumes the inclusion of crossed interactions in the pair ${k}^{\prime}$); and (

**b**) Distributed evolution on ${2}^{2d-1}$ Bloch spheres, each one for the states $\left(\right)$.

**Figure 5.**Exchange index relations involved for each interaction and highlighted properties for their correspondent ${\mathbb{S}}_{{U}_{\mathcal{I},{\mathcal{I}}^{\prime}}}^{0}$: (

**a**) ${H}_{I}$; (

**b**) ${H}_{IIa}$; (

**c**) ${H}_{IIb}$; and (

**d**) ${H}_{III}$. Exchange relations in (

**b**,

**d**) are doubled by considering the vertical switching in one of the indexes for each pair shown.

**Figure 6.**Connectedness graphs between states under $SU(2)$ decomposition for one (green) and two (red) exchange scripts for all generalized Bell state (GBS) basis states: (

**a**) $d=1$; (

**b**) $d=2$; and (

**c**) $d=3$.

**Figure 10.**Solutions for ${\u03f5}_{i}$ in ${{\mathbb{H}}_{0}^{0}({\delta}_{i},{\u03f5}_{i})}_{\mathcal{I},{\mathcal{I}}^{\prime}}$ involved in the enlargement of ${\left|W\right.\u232a}^{d}$ into ${\left|W\right.\u232a}^{d+1}$ for values of $d\in \{2,\dots ,60\}$.

**Table 1.**Basis pairs and Hamiltonian required to get the $SU(2)$ block decomposition for case $d=1$.

Basis Arrangement | Hamiltonian |
---|---|

$\{\{\left(\right)open="|"\; close="\rangle ">{\beta}_{00},\left(\right)open="|"\; close="\rangle ">{\beta}_{01},\{\left(\right)open="|"\; close="\rangle ">{\beta}_{11},\left(\right)open="|"\; close="\rangle ">{\beta}_{10}$ | $H={H}_{0}+{h}_{01}{\sigma}_{0}\otimes {\sigma}_{1}+{h}_{10}{\sigma}_{1}\otimes {\sigma}_{0}+{h}_{23}{\sigma}_{2}\otimes {\sigma}_{3}+{h}_{32}{\sigma}_{3}\otimes {\sigma}_{2}$ |

$\{\{\left(\right)open="|"\; close="\rangle ">{\beta}_{00},\left(\right)open="|"\; close="\rangle ">{\beta}_{11},\{\left(\right)open="|"\; close="\rangle ">{\beta}_{01},\left(\right)open="|"\; close="\rangle ">{\beta}_{10}$ | $H={H}_{0}+{h}_{02}{\sigma}_{0}\otimes {\sigma}_{2}+{h}_{20}{\sigma}_{2}\otimes {\sigma}_{0}+{h}_{13}{\sigma}_{1}\otimes {\sigma}_{3}+{h}_{31}{\sigma}_{3}\otimes {\sigma}_{1}$ |

$\{\{\left(\right)open="|"\; close="\rangle ">{\beta}_{00},\left(\right)open="|"\; close="\rangle ">{\beta}_{10},\{\left(\right)open="|"\; close="\rangle ">{\beta}_{01},\left(\right)open="|"\; close="\rangle ">{\beta}_{11}$ | $H={H}_{0}+{h}_{03}{\sigma}_{0}\otimes {\sigma}_{3}+{h}_{30}{\sigma}_{3}\otimes {\sigma}_{0}+{h}_{12}{\sigma}_{1}\otimes {\sigma}_{2}+{h}_{21}{\sigma}_{2}\otimes {\sigma}_{1}$ |

Hamiltonian | Entries Type | Entries by Column/Row | Parameters by Entry |
---|---|---|---|

${H}_{0}$ | Diagonal | 1 | $D\le 3$ |

${H}_{{\mathrm{l}}_{i}}$ | Non-diagonal | d | 2 |

${H}_{{\mathrm{nl}}_{i}}^{c}$ | Diagonal | 1 | d |

${H}_{{\mathrm{nl}}_{i}}^{nc}$ | Non-diagonal | $\frac{1}{2}d(d-1)$ | 4 |

${H}_{{\mathrm{cnl}}_{i}}^{c}$ | Non-diagonal | d | 2 |

${H}_{{\mathrm{cnl}}_{i}}^{nc}$ | Non-diagonal | $d(d-1)$ | 4 |

${H}_{I}$ | 2 × 2 block | 2 | $2+Dd\le 2+3d$ |

${H}_{IIa,b}$ | 2 × 2 block | 2 | $4+Dd\le 4+3d$ |

${H}_{III}$ | 2 × 2 block | 2 | $2+Dd\le 2+3d$ |

**Table 3.**Values of ${c}_{{j}_{s},{j}_{d+s}}^{{i}_{s},{k}_{s}}$ for all exchange scripts in ${H}_{I},{H}_{IIa,b},{H}_{III}$. $i,j,k$ is an even permutation of $1,2,3$.

$({\mathit{j}}_{\mathit{s}},{\mathit{j}}_{\mathit{s}+\mathit{d}})$ | $({\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}})$ | ${\mathit{c}}_{{\mathit{j}}_{\mathit{s}},{\mathit{j}}_{\mathit{d}+\mathit{s}}}^{{\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}}}$ | $({\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}})$ | ${\mathit{c}}_{{\mathit{j}}_{\mathit{s}},{\mathit{j}}_{\mathit{d}+\mathit{s}}}^{{\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}}}$ | $({\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}})$ | ${\mathit{c}}_{{\mathit{j}}_{\mathit{s}},{\mathit{j}}_{\mathit{d}+\mathit{s}}}^{{\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}}}$ | $({\mathit{i}}_{\mathit{s}},{\mathit{k}}_{\mathit{s}})$ | |
---|---|---|---|---|---|---|---|---|

$(0,2)$ | $(0,2)$ | $-i$ | $(2,0)$ | i | $(1,3)$ | i | $(3,1)$ | $-i$ |

$(2,0)$ | $(0,2)$ | i | $(2,0)$ | $-i$ | $(1,3)$ | i | $(3,1)$ | $-i$ |

$(0,j\ne 2)$ | $(0,j)$ | 1 | $(j,0)$ | 1 | $(i,k)$ | $-{(-1)}^{{\delta}_{2k}}$ | $(k,i)$ | $-{(-1)}^{{\delta}_{2k}}$ |

$(j\ne 2,0)$ | $(0,j)$ | 1 | $(j,0)$ | 1 | $(i,k)$ | ${(-1)}^{{\delta}_{2k}}$ | $(k,i)$ | ${(-1)}^{{\delta}_{2k}}$ |

$2\in (j,k)$ | $(j,k)$ | $-i$ | $(k,j)$ | i | $(0,i)$ | $-i{(-1)}^{{\delta}_{2k}}$ | $(i,0)$ | $i{(-1)}^{{\delta}_{2k}}$ |

$2\in (k,j)$ | $(j,k)$ | i | $(k,j)$ | $-i$ | $(0,i)$ | $-i{(-1)}^{{\delta}_{2k}}$ | $(i,0)$ | $i{(-1)}^{{\delta}_{2k}}$ |

$(1,3)$ | $(1,3)$ | 1 | $(3,1)$ | 1 | $(0,2)$ | $-1$ | $(2,0)$ | $-1$ |

$(3,1)$ | $(1,3)$ | 1 | $(3,1)$ | 1 | $(0,2)$ | 1 | $(2,0)$ | 1 |

**Table 4.**Bipartite concurrence ${\mathcal{C}}^{2}({\mathrm{Tr}}^{S}({\rho}_{\mathcal{I}\mathcal{J}}))$ for several subsystems in the $SU(2)$ mixing of some pairs of GBS basis states.

Case | S | ${\mathcal{C}}^{2}({\mathbf{Tr}}^{\mathit{S}}({\mathit{\rho}}_{\mathcal{I}\mathcal{J}}))$ |
---|---|---|

(a) $\left(\right)$ | $[s\notin \{{k}^{\prime},{k}^{\prime}+d\}]$ | 1 |

(b) $\left(\right)$ | $[s\in \{{k}^{\prime},{k}^{\prime}+d\}]$ | $1-{sin}^{2}\theta {(cos{\varphi}^{\prime}{\delta}_{0,{i}_{{k}^{\prime}}\xb7{j}_{{k}^{\prime}}}+{(-1)}^{{\u03f5}_{{i}_{{k}^{\prime}}{j}_{{k}^{\prime}}j}}(1-{\delta}_{0,{i}_{{k}^{\prime}}\xb7{j}_{{k}^{\prime}}})sin{\varphi}^{\prime})}^{2}$ |

(c) $\left(\right)$ | $[{k}^{\prime},{k}^{\prime}+d]$ | 0 |

(d) $\left(\right)$ | $[{k}^{\prime},{k}^{\prime}+d]$ | ${sin}^{2}\theta $ |

(e) $\left(\right)$ | $[{k}^{\prime},{k}^{\u2033}]$ | $\frac{3}{2}-\frac{1}{2}{sin}^{2}\theta ({cos}^{2}{\varphi}^{\prime}{\delta}_{{i}_{{k}^{\prime}}{j}_{{k}^{\prime}}}{\delta}_{{i}_{{k}^{\u2033}}{j}_{{k}^{\u2033}}}+{sin}^{2}{\varphi}^{\prime}(1-{\delta}_{{i}_{{k}^{\prime}}{j}_{{k}^{\prime}}}{\delta}_{{i}_{{k}^{\u2033}}{j}_{{k}^{\u2033}}}))$ |

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**MDPI and ACS Style**

Delgado, F.
*SU*(2) Decomposition for the Quantum Information Dynamics in 2*d*-Partite Two-Level Quantum Systems. *Entropy* **2018**, *20*, 610.
https://doi.org/10.3390/e20080610

**AMA Style**

Delgado F.
*SU*(2) Decomposition for the Quantum Information Dynamics in 2*d*-Partite Two-Level Quantum Systems. *Entropy*. 2018; 20(8):610.
https://doi.org/10.3390/e20080610

**Chicago/Turabian Style**

Delgado, Francisco.
2018. "*SU*(2) Decomposition for the Quantum Information Dynamics in 2*d*-Partite Two-Level Quantum Systems" *Entropy* 20, no. 8: 610.
https://doi.org/10.3390/e20080610