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Entanglement and separability of graph Laplacian quantum states

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Abstract

In this article, we study the entanglement properties of multi-qubit quantum states using a graph-theoretic approach. For this, we define entanglement and separability for m-qubit quantum states associated with a weighted graph on \(2^m\) vertices. We further explore the properties of a block graph and a star graph to demonstrate criteria for entanglement and separability of these graphs.

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Data Availability statement

All data generated or analyzed during this study are included in this published article.

Abbreviations

1. \(||x ||\) :

Absolute value of x= (\(\sqrt{x {\bar{x}}}\)).

2. \({\mathrm{Tr}}(A)\) :

Trace of a matrix A.

2. \({\mathrm{det}}(A)\) :

Determinant of a matrix A.

4. D(G):

The degree matrix of (G, a).

5. \({\rho _G}^{PT}\) :

Partial transpose of the density operator of a graph G.

6. \({\bar{\rho _G}}^{PT}\) :

Conjugate partial transpose of the density operator of a graph G.

7. \(A^*\) :

Conjugate transpose of A.

8. \(|{E(G)}|\) :

Number of edges in G.

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Acknowledgements

The authors are grateful to Ranveer Singh and Satish Sangwan for their valuable comments and suggestions. The authors are also grateful to IIT Jodhpur for providing the research infrastructure. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Authors and Affiliations

  1. Indian Institute of Technology Jodhpur, Karwar, Rajasthan, India

    Anoopa Joshi & Atul Kumar

  2. Central University of Punjab, Bathinda, India

    Parvinder Singh

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  1. Anoopa Joshi
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  2. Parvinder Singh
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Correspondence to Atul Kumar.

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Appendix

Appendix

1.1 A brief explanation of Examples (6) and (7)

  1. 1.

    Explanation of the example (6)

figure k

The density operator for the graph \(G_2\) can be represented as

$$\begin{aligned} \rho _{G_2}= \frac{1}{4} \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}-1\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}-1\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -1&{} 0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -1&{} 0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1 \end{array}\right] \end{aligned}$$

Using \(\rho _{G_2}\) , one can verify that \(\rho _{G_2} \ne {{\rho _{G_2}}}^{PT}\). Therefore, interchanging columns \(C_9\) with \(C_{11}\), and \(C_{14}\) with \(C_{16}\) and corresponding rows \(R_9\) with \(R_{11}\) and \(R_{14}\) with \(R_{16}\), we have

$$\begin{aligned}\rho _{G_2} \cong {\rho _{G_2}}'= \frac{1}{4} \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -1&{} 0&{}0&{}0&{}0&{}-1&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0 \end{array}\right] . \end{aligned}$$

Here, we can see that blocks are symmetric, satisfying the Theorem 4.4. Hence, the state associated with the graph \(G_2\) is separable.

  1. 2.

    Explanation of the example (7)

figure l

Similar to the previous case, the density operator for the graph \(G_3\) is expressed as

$$\begin{aligned} \rho _{G_3}= \frac{1}{4} \left[ \begin{array}{rrrrrrrrrrrrrrrr} 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}-1\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}-1\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{} 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 1&{} 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}-1\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -1&{} 0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}-1&{}0&{}0&{}0&{}0&{}1 \end{array}\right] , \end{aligned}$$

In this case, we can see that we cannot get the symmetric blocks by interchanging the columns and corresponding rows. Therefore, the state associated with the graph \(G_3\) is entangled.

1.2 Decomposition of the Laplacian matrix of a simple graph G

Every Laplacian matrix can be decomposed as a sum of Laplacian matrices of subgraphs (\(G_1\), \(G_2\), and \(G_3\)) of a graph G [24].

We have

$$\begin{aligned} L= \begin{bmatrix} \sum _{j=1}^{n} a_{1j}&{} -a_{12}&{} \ldots &{} -a_{1n}\\ -a_{21}&{} \sum _{j=1}^{n}a_{2j}&{} \ldots &{} -a_{2n}\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ -a_{n1}&{} -a_{n2}&{} \ldots &{}\sum _{j=1}^{n} a_{nj} \end{bmatrix}. \end{aligned}$$

It is easy to see that L(G) can be rewritten as

$$\begin{aligned} L= & {} \begin{bmatrix} \sum _{j=1}^{\frac{n}{2}} a_{1j} &{}-a_{12} &{} \ldots -a_{1\frac{n}{2}}&{} 0&{} 0&{} \ldots &{} 0\\ -a_{21} &{}\sum _{j=1}^{\frac{n}{2}} a_{2j} &{} \ldots -a_{2\frac{n}{2}}&{} 0&{} 0 &{} \ldots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ -a_{\frac{n}{2}1} &{} -a_{\frac{n}{2}2} &{} \ldots \sum _{j=1}^{\frac{n}{2}} a_{\frac{n}{2}j} &{} 0 &{} 0 &{} \ldots &{} 0\\ 0 &{} 0&{} \ldots &{} 0 &{} 0 &{} \ldots &{} 0\\ 0 &{} 0 &{} \ldots &{} 0 &{} 0 &{} \ldots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} 0 &{} 0 &{} \ldots &{} 0 \end{bmatrix} \\&+\begin{bmatrix} 0 &{} 0 &{} \ldots 0 &{} 0&{} 0&{} \ldots &{} 0\\ 0 &{} 0 &{} \ldots 0 &{} 0&{} 0 &{} \ldots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \ldots 0 &{} 0 &{} \ldots &{} 0\\ 0 &{} 0&{} \ldots &{}0&{} \sum _{ j=\frac{n}{2}+1 }^{n} a_{ \frac{n}{2}+1 j } &{} -a_{ \frac{n}{2}+1 \frac{n}{2}+2 } &{} \ldots &{} -a_{ \frac{n}{2}+1 n}\\ 0 &{} 0 &{} \ldots &{}0 &{} -a_{\frac{n}{2}+2 \frac{n}{2}+1} &{} \sum _{j=\frac{n}{2}+1}^{n} a_{\frac{n}{2}+2 j} &{} \ldots &{} -a_{\frac{n}{2}+2 n}\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \ldots &{} 0 &{} -a_{n \frac{n}{2}+1} &{} -a_{n \frac{n}{2}+2} &{} \ldots &{}0 &{} \sum _{j=\frac{n}{2}+1}^{n} a_{n j} \end{bmatrix} \\&+ \begin{bmatrix} \sum _{j=\frac{n}{2}+1}^{n} a_{1j}&{} \ldots &{} 0&{}-a_{1 \frac{n}{2}+1 } &{} \ldots &{} -a_{1n}\\ \vdots &{} \vdots &{}\vdots &{} \vdots &{} \vdots &{} \vdots \\ 0&{} \ldots &{} \sum _{j=\frac{n}{2}+1}^{n} a_{\frac{n}{2}j}&{} -a_{ \frac{n}{2} \frac{n}{2}+1 } &{} \ldots &{} -a_{ \frac{n}{2}n}\\ -a_{\frac{n}{2}+1 1} &{} \ldots &{} -a_{ \frac{n}{2}+1 \frac{n}{2}} &{} \sum _{j=1}^{\frac{n}{2}} a_{\frac{n}{2}+1 j} &{} \ldots &{} 0\\ \vdots &{} \vdots &{}\vdots &{} \vdots &{} \vdots &{} \vdots \\ -a_{n1} &{} \ldots &{} -a_{n \frac{n}{2}} &{} 0 &{} \ldots &{} \sum _{j=1}^{\frac{n}{2}} a_{n j}\\ \end{bmatrix}. \end{aligned}$$

To summarize, \(L(G)=\begin{bmatrix} L_1&{}0\\ 0&{}0 \end{bmatrix}+ \begin{bmatrix} 0&{}0\\ 0&{}L_2 \end{bmatrix} + \begin{bmatrix} D_1&{}B_1\\ {B_2} &{}D_2 \end{bmatrix}\),

where \(L_1\) and \(L_2\) are also Laplacian of simple graphs and \(D_1\) and \(D_2\) are diagonal matrices with the i-th diagonal entry equals to the sum of absolute value of i-th row sum of \(B_1\) and \(B_2\).

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Joshi, A., Singh, P. & Kumar, A. Entanglement and separability of graph Laplacian quantum states. Quantum Inf Process 21, 152 (2022). https://doi.org/10.1007/s11128-022-03483-z

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