Entanglement and separability of graph Laplacian quantum states

Anoopa Joshi¹,
Parvinder Singh² &
Atul Kumar¹

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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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Quantum Computing

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.

References

Adhikari, B., Banerjee, S., Adhikari, S., Kumar, A.: Laplacian matrices of weighted digraphs represented as quantum states. Quantum Inf. Process. 16(3), 79 (2017)
Article ADS MathSciNet Google Scholar
Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54(5), 3824 (1996)
Article ADS MathSciNet Google Scholar
Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Number 186. American Mathematical Society (2013)
Braunstein, S.L., Ghosh, S., Mansour, T., Severini, S., Wilson, R.C.: Some families of density matrices for which separability is easily tested. Phys. Rev. A 73(1), 012320 (2006)
Article ADS Google Scholar
Braunstein, S.L., Ghosh, S., Severini, S.: The Laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states. Ann. Comb. 10(3), 291–317 (2006)
Article MathSciNet Google Scholar
Bruss, D., Leuchs, G.: Quantum Information: From Foundations to Quantum Technology Applications. Wiley, New York (2019)
MATH Google Scholar
Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61(5), 052306 (2000)
Article ADS Google Scholar
Dür, W., Aschauer, H., Briegel, H.-J.: Multiparticle entanglement purification for graph states. Phys. Rev. Lett. 91(10), 107903 (2003)
Article ADS MathSciNet Google Scholar
Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62(6), 062314 (2000)
Article ADS MathSciNet Google Scholar
Dutta, S., Adhikari, B., Banerjee, S.: A graph theoretical approach to states and unitary operations. Quantum Inf. Process. 15(5), 2193–2212 (2016)
Article ADS MathSciNet Google Scholar
Dutta, S.: Bipartite separability and nonlocal quantum operations on graphs. Phys. Rev. A 94(1), 012306 (2016)
Article ADS Google Scholar
Eisert, J., Plenio, M.B.: Introduction to the basics of entanglement theory in continuous-variable systems. Int. J. Quantum Inf. 1(04), 479–506 (2003)
Article Google Scholar
Hall, B.C.: Quantum Theory for Mathematicians, vol. 267. Springer, Berlin (2013)
Book Google Scholar
Hall, M.J.W.: Random quantum correlations and density operator distributions. Phys. Lett. A 242(3), 123–129 (1998)
Article ADS MathSciNet Google Scholar
Hassan, A.S.M., Joag, P.S.: A combinatorial approach to multipartite quantum systems: basic formulation. J. Phys. A Math. Theor. 40(33), 10251 (2007)
Article ADS MathSciNet Google Scholar
Hildebrand, R., Mancini, S., Severini, S.: Combinatorial Laplacians and positivity under partial transpose. Math. Struct. Comput. Sci. 18(1), 205–219 (2008)
Article MathSciNet Google Scholar
Horodecki, M., Horodecki, P., Horodecki, R.: Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps. Phys. Lett. A 283(1–2), 1–7 (2001)
Article ADS MathSciNet Google Scholar
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Reviews of modern physics. Quantum Entangl 81(2), 865 (2009)
MATH Google Scholar
Hou, Q.-H., Mansour, T., Severini, S.: Partial transposes of permutation matrices. Integers Electron J Combin Number Theory 8(A49), A49 (2008)
MathSciNet MATH Google Scholar
Joshi, A., Singh, R., Kumar, A.: Concurrence and three-tangle of the graph. Quantum Inf. Process. 17(12), 327 (2018)
Article ADS MathSciNet Google Scholar
Lohmayer, R., Osterloh, A., Siewert, J., Uhlmann, A.: Entangled three-qubit states without concurrence and three-tangle. Phys. Rev. Lett. 97(26), 260502 (2006)
Article ADS MathSciNet Google Scholar
Meenakshi, A.R., Rajian, C.: On a product of positive semidefinite matrices. Linear Algebra Appl. 295(1–3), 3–6 (1999)
Article MathSciNet Google Scholar
Milivojević, M.: Maximal thermal entanglement using three-spin interactions. Quantum Inf. Process. 18(2), 1–14 (2019)
Article MathSciNet Google Scholar
Newman, M.W.: The Laplacian spectrum of graphs (2001)
Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002)
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information (2000)
Rigolin, G.: Superdense coding using multipartite states. arXiv:hep/0407193 (2004)
Rigolin, G.: Quantum teleportation of an arbitrary two-qubit state and its relation to multipartite entanglement. Phys. Rev. A 71(3), 032303 (2005)
Article ADS MathSciNet Google Scholar
Wang, B.-Y., Gong, M.-P.: Some eigenvalue inequalities for positive semidefinite matrix power products. Linear Algebra Appl. 184, 249–260 (1993)
Article MathSciNet Google Scholar
Wang, X., Wilde, M.M.: Cost of quantum entanglement simplified. Phys. Rev. Lett. 125(4), 040502 (2020)
Article ADS MathSciNet Google Scholar
Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40(8), 4277 (1989)
Article ADS Google Scholar
West, D.B., et al.: Introduction to Graph Theory, vol. 2. Prentice Hall, Upper Saddle River (2001)
Google Scholar
Wilson, R.J.: Introduction to Graph Theory. Pearson Education, New York (1979)
MATH Google Scholar
Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)
Article ADS Google Scholar
Wootters, W.K.: Entanglement of formation and concurrence. Quantum Inf. Comput. 1(1), 27–44 (2001)
MathSciNet MATH Google Scholar
Wu, P.Y.: Products of positive semidefinite matrices. Linear Algebra Appl. 111, 53–61 (1988)
Article MathSciNet Google Scholar
Zhang, F., Zhang, Q.: Eigenvalue inequalities for matrix product. IEEE Trans. Autom. Control 51(9), 1506–1509 (2006)
Article ADS MathSciNet Google Scholar
Zhang, G.-F.: Thermal entanglement and teleportation in a two-qubit Heisenberg chain with Dzyaloshinski–Moriya anisotropic antisymmetric interaction. Phys. Rev. A 75(3), 034304 (2007)
Article ADS Google Scholar

Download references

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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Indian Institute of Technology Jodhpur, Karwar, Rajasthan, India
Anoopa Joshi & Atul Kumar
Central University of Punjab, Bathinda, India
Parvinder Singh

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Anoopa Joshi
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Parvinder Singh
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Atul Kumar
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Correspondence to Atul Kumar.

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Appendix

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

1.
Explanation of the example (6)

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.

2.
Explanation of the example (7)

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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Received: 18 October 2021
Accepted: 07 March 2022
Published: 07 April 2022
DOI: https://doi.org/10.1007/s11128-022-03483-z