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arXiv:1506.03635

Mixed graph states

Constanza Riera, Ramij Rahaman, Matthew Geoffrey Parker

Published Jun 11, 2015• 0 citations

Abstract

We have generalised the concept of graph states to what we have called mixed graph states, which we define in terms of mixed graphs, that is graphs with both directed and undirected edges, as the density matrix stabilized by the associated stabilizer matrix defined by the mixed graph. We can interpret this matrix as a quantum object by making it part of a larger fully commuting matrix, i.e. where we choose the environment appropriately, and this will imply that our quantum object is a mixed state. We prove that, in the same way as (pure) graph states, the density matrix of a parent of mixed graph state can be written as sum of a few Pauli matrices, well defined from the mixed graph. We have proven that the set of matrices that appear in this sum is fully pair-wise commuting, and form a multiplicative group up to global constants, which is always of maximum size. Furthermore, the cardinality of the set depends solely of the miminum possible number of extension columns/rows, and the number of nodes of the mixed graph. We prove a formula for this cardinality. Finally, in the case of purely undirected graphs, this corresponds to the usual pure graph state. Also, we have developed a way of finding maximal commutative group of such Pauli matrices as a linear subspace problem, for any given mixed graph. We also have proven how the structure of maximal commutative groups is independent of the direction of the arrows of the mixed graph, and also of the undirected edges; this allows the simplification of the problem of finding these groups in general to finding them for a much smaller set of graphs.

DOI: 10.48550/arxiv.1506.03635

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We have generalised the concept of graph states to what we have called mixed graph states, which we define in terms of mixed graphs, that is graphs with both directed and undirected edges, as the density matrix stabilized by the associated stabilizer matrix defined by the mixed graph. We can interpret this matrix as a quantum object by making it part of a larger fully commuting matrix, i.e. where we choose the environment appropriately, and this will imply that our quantum object is a mixed state. We prove that, in the same way as (pure) graph states, the density matrix of a parent of mixed graph state can be written as sum of a few Pauli matrices, well defined from the mixed graph. We have proven that the set of matrices that appear in this sum is fully pair-wise commuting, and form a multiplicative group up to global constants, which is always of maximum size. Furthermore, the cardinality of the set depends solely of the miminum possible number of extension columns/rows, and the number of nodes of the mixed graph. We prove a formula for this cardinality. Finally, in the case of purely undirected graphs, this corresponds to the usual pure graph state. Also, we have developed a way of finding maximal commutative group of such Pauli matrices as a linear subspace problem, for any given mixed graph. We also have proven how the structure of maximal commutative groups is independent of the direction of the arrows of the mixed graph, and also of the undirected edges; this allows the simplification of the problem of finding these groups in general to finding them for a much smaller set of graphs.

Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.

Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the Pauli group, one approach taken is to study non-abelian stabilizers. In this work, we propose to generalize graph states based on the encoding circuit, which is completely determined by the graph and a Hadamard matrix. We study the entanglement structures of these generalized graph states and show that they are all maximally mixed locally. We also explore the relationship between the equivalence of Hadamard matrices and local equivalence of the corresponding generalized graph states. This leads to a natural generalization of the Pauli (X, Z) pairs, which characterizes the local symmetries of these generalized graph states. Our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes.

In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties of clustered graphs which provide a family of commuting normal matrices formed by the blocks of its Laplacian matrices. We apply these results on some important mixed quantum states, such as the Werner, Isotropic, and

X

-states.

Mixed graphs, which generalize both undirected and directed graphs, provide a natural framework for investigating a broader class of quantum transport phenomena, including perfect state transfer (

\mathrm{PST}

) and multiple state transfer (

\mathrm{MST}

). The Hermitian adjacency matrix, incorporating both symmetric and asymmetric edge structures, establishes a unified spectral framework that reveals quantum phenomena absent in undirected graphs. In this paper, we study

\mathrm{PST}

and

\mathrm{MST}

on mixed graphs using the Hermitian adjacency matrix, extending results from undirected graphs and focusing on phenomena such as periodicity, one-way

\mathrm{PST}

, two-way

\mathrm{PST}

, and

\mathrm{MST}

. Additionally, we present novel results that contribute to a deeper understanding of quantum state transfer in mixed graphs.

Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices

(A, B, C)

. The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components:

A

and

C

are described by a classical weighted graph called the strange graph, while

B

provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.

Abstract The objective of Quantum State Verification (QSV) is to evaluate whether an unknown quantum state is sufficiently similar to a predetermined target state at a certain confidence level. The required number of identical and independent copies, N, of an unknown state for such a task is inversely proportional to the spectral gap of a verification operator. Stabilizer states form a foundational class of quantum states, extensively utilized in quantum computing. Defined through a group of stabilizer operators constructed from Pauli matrices in qubit systems, their structural properties not only simplify the mathematical representation and simulation of quantum states but also make them particularly suitable for verification protocols that rely on local measurements. Restricting the verification operator to be those that can be achieved by local measurements, previous researches had determined an upper bound for the spectral gap to be 2/3 in the cases of qubit stabilizer states. In this research, we show a similar upper bound of d/(d+1) in the cases of graph states of d-level systems by a technique from [1] but adapted to d-level graph states. In particular, we show that d-level graph states are maximally entangled across any bipartition of the graph, which allows us to reduce the problem to the case of a bipartite maximally entangled d-level systems for which the maximal spectral gap is known.