arXiv subjects

Scott N. Walck

Publications and source records attributed to Scott N. Walck.

At least 19 recordsLinked to original sources

arXiv2024

Functional Programming in Learning Electromagnetic Theory

Electromagnetic theory is central to physics. An undergraduate major in physics typically takes a semester or a year of electromagnetic theory as a junior or senior, and a graduate student in physics typically takes an additional semester or year at a more advanced level. In fall 2023, the author taught his undergraduate electricity and magnetism class using numerical methods in Haskell in parallel with traditional analytical methods. This article describes what functional programming has to offer to physics in general, and electromagnetic theory in particular. We give examples from vector calculus, the mathematical language in which electromagnetic theory is expressed, and electromagnetic theory itself.

cs.PL↗

arXiv2017

Local Pauli stabilizers of symmetric hypergraph states

Hypergraph states of many quantum bits share the rich interplay between simple combinatorial description and nontrivial entanglement properties enjoyed by the graph states that they generalize. In this paper, we consider hypergraph states that are also permutationally invariant. We characterize the states in this class that have nontrivial local Pauli stabilizers and give applications to nonlocality and error correction.

quant-ph↗

arXiv2016

Learn Quantum Mechanics with Haskell

To learn quantum mechanics, one must become adept in the use of various mathematical structures that make up the theory; one must also become familiar with some basic laboratory experiments that the theory is designed to explain. The laboratory ideas are naturally expressed in one language, and the theoretical ideas in another. We present a method for learning quantum mechanics that begins with a laboratory language for the description and simulation of simple but essential laboratory experiments, so that students can gain some intuition about the phenomena that a theory of quantum mechanics needs to explain. Then, in parallel with the introduction of the mathematical framework on which quantum mechanics is based, we introduce a calculational language for describing important mathematical objects and operations, allowing students to do calculations in quantum mechanics, including calculations that cannot be done by hand. Finally, we ask students to use the calculational language to implement a simplified version of the laboratory language, bringing together the theoretical and laboratory ideas.

cs.PL↗

arXiv2015

Local unitary symmetries of hypergraph states

Hypergraph states are multiqubit states whose combinatorial description and entanglement properties generalize the well-studied class of graph states. Graph states are important in applications such as measurement-based quantum computation and quantum error correction. The study of hypergraph states, with their richer multipartite entanglement and other nonlocal properties, has a promising outlook for new insight into multipartite entanglement. We present results analyzing local unitary symmetries of hypergraph states, including both continuous and discrete families of symmetries. In particular, we show how entanglement types can be detected and distinguished by certain configurations in the hypergraphs from which hypergraph states are constructed.

quant-ph↗

arXiv2014

Learn Physics by Programming in Haskell

We describe a method for deepening a student's understanding of basic physics by asking the student to express physical ideas in a functional programming language. The method is implemented in a second-year course in computational physics at Lebanon Valley College. We argue that the structure of Newtonian mechanics is clarified by its expression in a language (Haskell) that supports higher-order functions, types, and type classes. In electromagnetic theory, the type signatures of functions that calculate electric and magnetic fields clearly express the functional dependency on the charge and current distributions that produce the fields. Many of the ideas in basic physics are well-captured by a type or a function.

cs.CY↗

arXiv2013

Entanglement verification using local unitary stabilizers

Local unitary stabilizer subgroups constitute powerful invariants for distinguishing various types of multipartite entanglement. In this paper, we show how stabilizers can be used as a basis for entanglement verification protocols on distributed quantum networks using minimal resources. As an example, we develop and analyze the performance of a protocol to verify membership in the space of Werner states, that is, multi-qubit states that are invariant under the action of any 1-qubit unitary applied to all the qubits.

quant-ph↗

arXiv2012

Werner state structure and entanglement classification

We present applications of the representation theory of Lie groups to the analysis of structure and local unitary classification of Werner states, sometimes called the {\em decoherence-free} states, which are states of $n$ quantum bits left unchanged by local transformations that are the same on each particle. We introduce a multiqubit generalization of the singlet state, and a construction that assembles these into Werner states.

quant-ph↗

arXiv2011

Entanglement classes of symmetric Werner states

The symmetric Werner states for $n$ qubits, important in the study of quantum nonlocality and useful for applications in quantum information, have a surprisingly simple and elegant structure in terms of tensor products of Pauli matrices. Further, each of these states forms a unique local unitary equivalence class, that is, no two of these states are interconvertible by local unitary operations.

quant-ph↗

arXiv2011

Symmetric mixed states of $n$ qubits: local unitary stabilizers and entanglement classes

We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states into six classes. These include the stabilizer types of the Werner states, the GHZ state and its generalizations, and Dicke states. For all but the zero algebra, we classify entanglement types (local unitary equivalence classes) of symmetric mixed states that have those stabilizers. We make use of the identification of symmetric density matrices with polynomials in three variables with real coefficients and apply the representation theory of SO(3) on this space of polynomials.

quant-ph↗

arXiv2010

Local unitary group stabilizers and entanglement for multiqubit symmetric states

We refine recent local unitary entanglement classification for symmetric pure states of $n$ qubits (that is, states invariant under permutations of qubits) using local unitary stabilizer subgroups and Majorana configurations. Stabilizer subgroups carry more entanglement distinguishing power than do the stabilizer subalgebras used in our previous work. We extend to mixed states recent results about local operations on pure symmetric states by showing that if two symmetric density operators are equivalent by a local unitary operation, then they are equivalent via a local unitary operation that is the {\em same} in each qubit. A geometric consequence, used in our entanglement classification, is that two symmetric pure states are local unitary equivalent if and only if their Majorana configurations can be interchanged by a rotation of the Bloch sphere.

quant-ph↗

arXiv2010

Symmetric states: local unitary equivalence via stabilizers

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group SO(3), and give examples of states with discrete stabilizers.

quant-ph↗

arXiv2008

Multiparty quantum states stabilized by the diagonal subgroup of the local unitary group

We classify, up to local unitary equivalence, the set of $n$-qubit states that is stabilized by the diagonal subgroup of the local unitary group. We exhibit a basis for this set, parameterized by diagrams of nonintersecting chords connecting pairs of points on a circle, and give a criterion for when the stabilizer is precisely the diagonal subgroup and not larger. This investigation is part of a larger program to partially classify entanglement type (local unitary equivalence class) via analysis of stabilizer structure.

quant-ph↗

arXiv2008

The Parts Determine the Whole except for n-Qubit Greenberger-Horne-Zeilinger States

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of n qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of n-1 qubits. Thus, the generalized GHZ states are the only ones containing information at the n-party level.

quant-ph↗

arXiv2008

Only n-Qubit Greenberger-Horne-Zeilinger States are Undetermined by their Reduced Density Matrices

The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of n qubits that are not uniquely determined among pure states by their reduced density matrices of n-1 qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the n-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.

quant-ph↗

arXiv2008

Classification of nonproduct states with maximum stabilizer dimension

Nonproduct n-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized n-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for n greater than or equal to 3 but not equal to 4. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.

quant-ph↗

arXiv2007

Maximum stabilizer dimension for nonproduct states

Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for n-qubit pure states, and find its maximum dimension to be n-1 for nonproduct states of three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: we show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the n-qubit GHZ state.

quant-ph↗

arXiv2006

Classification of n-qubit states with minimum orbit dimension

The group of local unitary transformations acts on the space of n-qubit pure states, decomposing it into orbits. In a previous paper we proved that a product of singlet states (together with an unentangled qubit for a system with an odd number of qubits) achieves the smallest possible orbit dimension, equal to 3n/2 for n even and (3n + 1)/2 for n odd, where n is the number of qubits. In this paper we show that any state with minimum orbit dimension must be of this form, and furthermore, such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets.

quant-ph↗

arXiv2005

Minimum orbit dimension for local unitary action on n-qubit pure states

The group of local unitary transformations partitions the space of n-qubit quantum states into orbits, each of which is a differentiable manifold of some dimension. We prove that all orbits of the n-qubit quantum state space have dimension greater than or equal to 3n/2 for n even and greater than or equal to (3n + 1)/2 for n odd. This lower bound on orbit dimension is sharp, since n-qubit states composed of products of singlets achieve these lowest orbit dimensions.

quant-ph↗