Detection of $d_{1}\otimes d_{2}$ Dimensional Bipartite Entangled State: A Graph Theoretical Approach

• URL: http://arxiv.org/abs/2202.13963v2
• Date: Wed, 8 Feb 2023 10:48:22 GMT
• Title: Detection of $d_{1}\otimes d_{2}$ Dimensional Bipartite Entangled State: A Graph Theoretical Approach
• Authors: Rohit Kumar, Satyabrata Adhikari
• Abstract summary: We show that the constructed unital map $phi$ characterize the quantum state with respect to its purity. We derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph to detect the entanglement of $d_1 otimes d_2$ dimensional bipartite quantum states.
• Score: 1.5762281194023464
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Braunstein et. al. have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph $G$ and then they have defined the quantum state from the Laplacian of the graph $G$. A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map $\phi$ such that $\phi(\rho)=L_{\rho}+\rho$, where the quantum state is described by the density operator $\rho$. The entries of $L_{\rho}$ depends on the entries of the quantum state $\rho$ and the entries are taken in such a way that $L_{\rho}$ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian $L_{\rho}$. We show that the constructed unital map $\phi$ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix $\phi(\rho)-I$ is positive then the quantum state $\rho$ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of $d_{1} \otimes d_{2}$ dimensional bipartite quantum states. Lastly, We have illustrated our results with few examples.

Related papers

• A polynomial-time dissipation-based quantum algorithm for solving the ground states of a class of classically hard Hamiltonians [4.500918096201963]
  We give a quantum algorithm for solving the ground states of a class of Hamiltonians. The mechanism of the exponential speedup that appeared in our algorithm comes from dissipation in open quantum systems.
  arXiv  Detail & Related papers  (2024-01-25T05:01:02Z)
• Calculating composite-particle spectra in Hamiltonian formalism and demonstration in 2-flavor QED$_{1+1\text{d}}$ [0.0]
  We consider three distinct methods to compute the mass spectrum of gauge theories in the Hamiltonian formalism. We find that the mass of $sigma$ meson is lighter than twice the pion mass, and thus $sigma$ is stable against the decay process. Our numerical results are so close to the WKB-based formula between the pion and sigma-meson masses, $M_sigma/M_pi=sqrt3$.
  arXiv  Detail & Related papers  (2023-07-31T13:38:23Z)
• Detection of Dense Subhypergraphs by Low-Degree Polynomials [72.4451045270967]
  Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem. We consider detecting the presence of a planted $Gr(ngamma, n-alpha)$ subhypergraph in a $Gr(n, n-beta) hypergraph. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known.
  arXiv  Detail & Related papers  (2023-04-17T10:38:08Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Efficient Signed Graph Sampling via Balancing & Gershgorin Disc Perfect Alignment [51.74913666829224]
  We show that for datasets with strong inherent anti-correlations, a suitable graph contains both positive and negative edge weights. We propose a linear-time signed graph sampling method centered on the concept of balanced signed graphs. Experimental results show that our signed graph sampling method outperformed existing fast sampling schemes noticeably on various datasets.
  arXiv  Detail & Related papers  (2022-08-18T09:19:01Z)
• Enlarging the notion of additivity of resource quantifiers [62.997667081978825]
  Given a quantum state $varrho$ and a quantifier $cal E(varrho), it is a hard task to determine $cal E(varrhootimes N)$. We show that the one shot distillable entanglement of certain spherically symmetric states can be quantitatively approximated by such an augmented additivity.
  arXiv  Detail & Related papers  (2022-07-31T00:23:10Z)
• Fermionic tomography and learning [0.5482532589225553]
  Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We show how this approach leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state. We use these tools to show that an $n$-electron, $m$-mode Slater can be learned to within $epsilon$ fidelity given $O(n2 m7 log(m / delta) / epsilon2)$ samples of the determinant.
  arXiv  Detail & Related papers  (2022-07-29T17:12:53Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and Projection [58.5350491065936]
  We consider a structural assumption on the graph Laplacian matrix $L$. The first $K$ eigenvectors of $L$ are pre-selected, e.g., based on domain-specific criteria. We design an efficient hybrid graphical lasso/projection algorithm to compute the most suitable graph Laplacian matrix $L* in H_u+$ given $barC$.
  arXiv  Detail & Related papers  (2020-10-25T18:12:50Z)
• Emergent universality in critical quantum spin chains: entanglement Virasoro algebra [1.9336815376402714]
  Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. We show that the Schmidt vectors $|v_alpharangle$ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT.
  arXiv  Detail & Related papers  (2020-09-23T21:22:51Z)
• Entanglement spectrum of geometric states [0.0]
  We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state $rho_A,m$. We reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs.
  arXiv  Detail & Related papers  (2020-08-28T01:40:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.