arXiv subjects

Ian Tan

Publications and source records attributed to Ian Tan.

7 recordsLinked to original sources

arXiv2026

Connecting Quantum Tomography and Quantum Retrodiction

Quantum tomography and quantum retrodiction are traditionally viewed as separate inference tasks: tomography reconstructs quantum states from measurement data, whereas retrodiction infers past quantum states from observed outcomes. We show that the two are manifestations of the same underlying principle. We prove that the Petz recovery map associated with a measurement channel is precisely the gradient update of the log-likelihood used in maximum-likelihood tomography. Consequently, repeated applications of the Petz map monotonically increase the likelihood. Extending beyond measurement channels, we derive a noncommutative generalization of the Petz map from the gradient of a generalized likelihood for arbitrary quantum channels. The resulting iterative procedure maximizes the likelihood and provides a general framework for quantum tomography, establishing a direct bridge between retrodiction, recovery maps, and statistical inference.

quant-ph↗

arXiv2026

Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits

The group of transversal gates and the group of local symmetries are important features of quantum error correcting codes and pure quantum states, respectively; the former provides fault-tolerant operations on a code while the latter tells us about a state's reachability via stochastic local operations with classical communication. We prove that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in $(\mathbb{C}^D)^{\otimes n}$ where $n$ is even, also known as perfect tensors, and LU orbits of $((n-1,D,n/2))_D$ quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state $|\Phi\rangle$ and its corresponding $((3,3,2))_3$ qutrit code $\mathcal{C}$. We show that $|\Phi\rangle$ and $\mathcal{C}$ are both unique up to the action of the LU group. We find generators of the local symmetry group of $|\Phi\rangle$ and the group of transversal gates on $\mathcal{C}$. Our proofs rely on prior results by Huber and Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). We use Vinberg's theory of Lie algebras to study the special case of $|\Phi\rangle$ and $\mathcal{C}$.

quant-ph↗

arXiv2025

Classification of five-qubit absolutely maximally entangled states

We classify the local unitary equivalence classes of absolutely maximally entangled (AME) states of five qubits. We show that every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code $\mathcal{C}$, and that two such states are equivalent if and only if they are related by the action of a transversal gate of $\mathcal{C}$. Furthermore, we exhibit a set of three invariant polynomials that separates these equivalence classes. As auxiliary results, we construct a 3-uniform $n$-qubit state for even $n\geq 6$, determine the local symmetries of the 6-qubit AME state, and explain how these symmetries are related to the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes. Additionally, we demonstrate that every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code. Our approach leverages an embedding of the 4-qubit state space into the Lie algebra of $8\times 8$ skew-symmetric matrices, allowing us to apply results from Vinberg's theory of graded Lie algebras.

quant-ph↗

arXiv2024

Four-qubit critical states

Verstraete, Dehaene, and De Moor (2003) showed that SLOCC invariants provide entanglement monotones. We observe that many highly entangled or useful four-qubit states that appear in prior literature are stationary points of such entanglement measures. This motivates the search for more stationary points. We use the notion of critical points (in the sense of the Kempf-Ness theorem) together with Vinberg theory to reduce the complexity of the problem significantly. We solve the corresponding systems utilizing modern numerical nonlinear algebra methods and reduce the solutions by natural symmetries. This method produces an extended list of four-qubit stationary points, which includes all the critical states in the survey by Enriquez et al (2016). To illustrate the potential for application, we discuss the use of these states to generate pure five-qubit and six-qubit quantum error correcting codes by reversing a construction of Rains (1996).

quant-ph↗

arXiv2024

Classifying topological floppy modes in the continuum

In floppy mechanical lattices, robust edge states and bulk Weyl modes are manifestations of underlying topological invariants. To explore the universality of these phenomena independent of microscopic detail, we formulate topological mechanics in the continuum. By augmenting standard linear elasticity with additional fields of soft modes, we define a continuum version of Maxwell counting, which balances degrees of freedom and mechanical constraints. With one additional field, these augmented elasticity theories can break spatial inversion symmetry and harbor topological edge states. We also show that two additional fields are necessary to harbor Weyl points in two dimensions, and define continuum invariants to classify these states. In addition to constructing the general form of topological elasticity based on symmetries, we derive the coefficients based on the systematic homogenization of microscopic lattices. By solving the resulting partial differential equations, we efficiently predict coarse-grained deformations due to topological floppy modes without the need for a detailed lattice-based simulation. Our discovery formulates novel design principles and efficient computational tools for topological states of matter, and points to their experimental implementation in mechanical metamaterials.

cond-mat.soft↗

arXiv2024

Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all $n$-qubit pure states under the action of the local unitary group. Taking advantage of the double cover $\operatorname{SL}_2(\mathbb{C}) \times \operatorname{SL}_2(\mathbb{C}) \to \operatorname{SO}_4({\mathbb{C}})$, we produce similar algorithms (distinguished by the parity of $n$) that compute normal forms for almost all $n$-qubit pure states under the action of the SLOCC group.

quant-ph↗

arXiv2023

Fast winning strategies in a generalized van der Waerden game

Consider the following Maker-Breaker game. Fix a finite subset $S\subset\mathbb{N}$ of the naturals. The players Maker and Breaker take turns choosing previously unclaimed natural numbers. Maker wins by eventually building a homothetic copy $aS+b$ of $S$, where $a\in\mathbb{N}\setminus\{0\}$ and $b\in\mathbb{Z}$. This is a generalization of the van der Waerden game analyzed by Beck. By the Hales-Jewett theorem, there exists a constant $c$ depending only on $|S|$ such that Maker can win in $c$ or less moves. We show that Maker can win in $|S|$ moves if $|S|\leq 3$. When $|S|=4$, we show that Maker can always win in $5$ or less moves and describe all $S$ such that Maker can win in $4$ moves. If $|S|\geq 5$, Maker has no winning strategy in $|S|$ moves.

math.CO↗