arXiv subjects

Nathaniel Johnston

Publications and source records attributed to Nathaniel Johnston.

At least 19 recordsLinked to original sources

arXiv2026

PPT states of almost maximal Schmidt number

We construct PPT states on $\mathbb{C}^m \otimes\mathbb{C}^n$ that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt number at least $$ \left\lceil \frac{m + n - ((m - n)^2 + 4(m + n - 1))^{1/2}}{2} \right\rceil. $$ In the case of equal local dimensions ($m = n$), this becomes $n - \lfloor(2n - 1)^{1/2}\rfloor$, far exceeding the previous constructions which achieved $n/2 + O(1)$. In the case of unequal local dimensions, our result shows that there exists a PPT state on $\mathbb{C}^n \otimes \mathbb{C}^{3n-4}$ with Schmidt number at least $n-1$.

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arXiv2026

Enumeration of Laplacian integral and {-1,0,1}-diagonalizable graphs

A graph with Laplacian matrix $L$ is called Laplacian integral if the eigenvalues of $L$ are all integers, and it is called $\{-1,0,1\}$-diagonalizable if $L$ has a full set of eigenvectors with entries from $\{-1,0,1\}$. We herein develop a structure theorem for both Laplacian integral graphs and $\{-1,0,1\}$-diagonalizable graphs of prime order, and combine it with some novel computational techniques to characterize all such graphs for orders larger than was previously possible. For example, we enumerate all Laplacian integral and $\{-1,0,1\}$-diagonalizable graphs of order $13$ or less, all $\{-1,0,1\}$-diagonalizable graphs of prime order $23$ or less, all regular integral graphs of order $15$ or less, and all regular $\{-1,0,1\}$-diagonalizable graphs of prime order $53$ or less. As an immediate byproduct of our work, we show that the $S_{n,n}$ conjecture for Laplacian integral graphs is true when $n = 12$, thus making $n = 16$ the smallest open case; additionally, we disprove two related conjectures regarding Laplacian spectra. We also establish an exponential lower bound on the number of connected $\{-1,0,1\}$-diagonalizable graphs of order $n$, thus beating the previously best-known (subexponential) lower bound. Finally, we show that every bipartite $\{-1,0,1\}$-diagonalizable graph is regular (a fact that fails to generalize to Laplacian integral graphs).

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arXiv2026

Distinguishability of locally diagonal orthogonally invariant quantum states

We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on locally diagonal orthogonally invariant (LDOI) states -- those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from $n^4$ to $O(n^2)$ in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most $(n-2)/(2n^2)$ for local dimension $n$.

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arXiv2026

Approximating quantum states by states of low rank

Given a positive integer k, it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k. This problem has already been solved when "distance" is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally-mixed state, but this is not true in many other unitary similarity invariant norms.

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arXiv2025

The complexity of perfect quantum state classification

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of $k$-learnability, which captures the ability to identify the correct state using at most $k$ guesses, with zero error. We show that deciding whether a given family of states is $k$-learnable can be solved via semidefinite programming. When there are $n$ states, we present polynomial-time (in $n$) algorithms for determining $k$-learnability for two cases: when $k$ is a fixed constant or the dimension of the states is a fixed constant. When both $k$ and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical $k$-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.

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arXiv2025

Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 - \langle\mathbf{v},\mathbf{w}\rangle^2$ for all $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$. We present three new proofs of this inequality that better illustrate "why" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.

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arXiv2025

Tight bounds for antidistinguishability and circulant sets of pure quantum states

A set of pure quantum states is said to be antidistinguishable if upon sampling one at random, there exists a measurement to perfectly determine some state that was not sampled. We show that antidistinguishability of a set of $n$ pure states is equivalent to a property of its Gram matrix called $(n-1)$-incoherence, thus establishing a connection with quantum resource theories that lets us apply a wide variety of new tools to antidistinguishability. As a particular application of our result, we present an explicit formula (not involving any semidefinite programming) that determines whether or not a set with a circulant Gram matrix is antidistinguishable. We also show that if all inner products are smaller than $\sqrt{(n-2)/(2n-2)}$ then the set must be antidistinguishable, and we show that this bound is tight when $n \leq 4$. We also give a simpler proof that if all the inner products are strictly larger than $(n-2)/(n-1)$, then the set cannot be antidistinguishable, and we show that this bound is tight for all $n$.

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arXiv2025

A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks

We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over $\mathbb{F}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile $\mathbb{R}^n$.

math.CO↗

arXiv2024

X-arability of mixed quantum states

The problem of determining when entanglement is present in a quantum system is one of the most active areas of research in quantum physics. Depending on the setting at hand, different notions of entanglement (or lack thereof) become relevant. Examples include separability (of bosons, fermions, and distinguishable particles), Schmidt number, biseparability, entanglement depth, and bond dimension. In this work, we propose and study a unified notion of separability, which we call X-arability, that captures a wide range of applications including these. For a subset (more specifically, an algebraic variety) of pure states X, we say that a mixed quantum state is X-arable if it lies in the convex hull of X. We develop unified tools and provable guarantees for X-arability, which already give new results for the standard separability problem. Our results include: -- An X-tensions hierarchy of semidefinite programs for X-arability (generalizing the symmetric extensions hierarchy for separability), and a new de Finetti theorem for fermionic separability. -- A hierarchy of eigencomputations for optimizing a Hermitian operator over X, with applications to X-tanglement witnesses and polynomial optimization. -- A hierarchy of linear systems for the X-tangled subspace problem, with improved polynomial time guarantees even for the standard entangled subspace problem, in both the generic and worst case settings.

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arXiv2024

Laplacian $\{-1,0,1\}$- and $\{-1,1\}$-diagonalizable graphs

A graph is called "Laplacian integral" if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets $\{-1,0,1\}$ or $\{-1,1\}$. Such graphs include as special cases the recently-investigated families of "Hadamard-diagonalizable" and "weakly Hadamard-diagonalizable" graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call "balanced", which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are $\{-1,0,1\}$-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the $\{-1,0,1\}$- and $\{-1,1\}$-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are $\{-1,0,1\}$- or $\{-1,1\}$-diagonalizable.

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arXiv2024

The Factor Width Rank of a Matrix

A matrix is said to have factor width at most $k$ if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single $k \times k$ principal submatrix. We explore the ``factor-width-$k$ rank'' of a matrix, which is the minimum number of rank-$1$ matrices that can be used in such a factor-width-at-most-$k$ decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-$k$ rank and the $k$-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.

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arXiv2023

A hierarchy of eigencomputations for polynomial optimization on the sphere

We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) hierarchy is that the lower bound at each level of our hierarchy is obtained by a minimum eigenvalue computation, as opposed to the full semidefinite program (SDP) required at each level of RSOS. In practice, this allows us to compute bounds on much larger forms than are computationally feasible for RSOS. Our hierarchy outperforms previous alternatives to RSOS, both asymptotically and in numerical experiments. We obtain our hierarchy by proving a reduction from real optimization on the sphere to Hermitian optimization on the sphere, and invoking the Hermitian sum-of-squares (HSOS) hierarchy. This opens the door to using other Hermitian optimization techniques for real optimization, and gives a path towards developing spectral hierarchies for more general constrained real optimization problems. To this end, we use our techniques to develop a hierarchy of eigencomputations for computing the real tensor spectral norm.

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arXiv2022

Bounding Real Tensor Optimizations via the Numerical Range

We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed significantly faster than bounds provided by semidefinite programming relaxations. We discuss numerous applications to other hard linear algebra problems, such as showing that a real subspace of matrices contains no rank-one matrix, and showing that a linear map acting on matrices is positive.

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arXiv2022

Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond

We study the problem of finding elements in the intersection of an arbitrary conic variety in $\mathbb{F}^n$ with a given linear subspace (where $\mathbb{F}$ can be the real or complex field). This problem captures a rich family of algorithmic problems under different choices of the variety. The special case of the variety consisting of rank-1 matrices already has strong connections to central problems in different areas like quantum information theory and tensor decompositions. This problem is known to be NP-hard in the worst case, even for the variety of rank-1 matrices. Surprisingly, despite these hardness results we develop an algorithm that solves this problem efficiently for "typical" subspaces. Here, the subspace $U \subseteq \mathbb{F}^n$ is chosen generically of a certain dimension, potentially with some generic elements of the variety contained in it. Our main result is a guarantee that our algorithm recovers all the elements of $U$ that lie in the variety, under some mild non-degeneracy assumptions on the variety. As corollaries, we obtain the following new results: $\bullet$ Polynomial time algorithms for several entangled subspaces problems in quantum entanglement, including determining r-entanglement, complete entanglement, and genuine entanglement of a subspace. While all of these problems are NP-hard in the worst case, our algorithm solves them in polynomial time for generic subspaces of dimension up to a constant multiple of the maximum possible. $\bullet$ Uniqueness results and polynomial time algorithmic guarantees for generic instances of a broad class of low-rank decomposition problems that go beyond tensor decompositions. Here, we recover a decomposition of the form $\sum_{i=1}^R v_i \otimes w_i$, where the $v_i$ are elements of the variety $X$. This implies new uniqueness results and genericity guarantees even in the special case of tensor decompositions.

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arXiv2022

A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces

We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It generalizes straightforwardly to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously-known hierarchies.

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arXiv2022

Absolutely $k$-Incoherent Quantum States and Spectral Inequalities for Factor Width of a Matrix

We investigate the set of quantum states that can be shown to be $k$-incoherent based only on their eigenvalues (equivalently, we explore which Hermitian matrices can be shown to have small factor width based only on their eigenvalues). In analogy with the absolute separability problem in quantum resource theory, we call these states "absolutely $k$-incoherent", and we derive several necessary and sufficient conditions for membership in this set. We obtain many of our results by making use of recent results concerning hyperbolicity cones associated with elementary symmetric polynomials.

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arXiv2022

Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)

In this work we examine quantum states which have non-negative amplitudes (in a fixed basis) and the channels which preserve them. These states include the ground states of stoquastic Hamiltonians and they are of interest since they avoid the Sign Problem and can thus be efficiently simulated. In optimization theory, the convex cone generated by such states is called the set of completely positive (CP) matrices (not be confused with completely positive superoperators). We introduce quantum channels which preserve these states and call them completely positive completely positive. To study these states and channels, we use the framework of resource theories and investigate how to measure and quantify this resource.

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arXiv2022

Entangled subspaces and generic local state discrimination with pre-shared entanglement

Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resource state maximizes this number. Local state discrimination is closely related to the topic of entangled subspaces, which we study in its own right. We introduce $r$-entangled subspaces, which naturally generalize previously studied spaces to higher multipartite entanglement. We use algebraic-geometric methods to determine the maximum dimension of an $r$-entangled subspace, and present novel explicit constructions of such spaces. We obtain similar results for symmetric and antisymmetric $r$-entangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.

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