arXiv subjects

Felix Huber

Publications and source records attributed to Felix Huber.

At least 19 recordsLinked to original sources

arXiv2026

Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography

Fractional graph colorings are useful for the Shadow tomography of Pauli observables. In practice, it is desirable that any experimentally interesting set of Pauli operators has a small fractional chromatic number $\chi_{f}$ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if $B_\epsilon(\varrho)$ is the set of Pauli observables having expectation value magnitude at least $\epsilon$ in some given quantum state $\varrho$, then the fractional chromatic number of the anticommutation graph $G$ induced by $B_\epsilon(\varrho)$ is $O(\epsilon^{-2})$. In other words, it asserts that there exists a constant $C$ such that $\chi_{f} \cdot \epsilon^2 \leq C$ on all states and graphs. If the conjecture were true, it would imply that there exists a triply efficient Pauli shadow tomography algorithm for {\it any} subset $S$ of Pauli observables, provided that there is also an efficient fractional coloring algorithm for the set $B_\epsilon$. Here we show that the conjecture is false by constructing a family of states and observables for which no finite $C$ satisfying the bound exists. We also give a more general construction relying on the commutation index or $\beta$ number of a graph. The key ingredient in the proofs can be seen as an instance of the amplification trick, where fractional chromatic numbers, $\beta$ numbers, and expectation values are amplified through lexicographic graph products.

quant-ph↗

arXiv2026

Iterated Distinct Absolute Differences of Integer Compositions

Starting from an integer composition, form its consecutive absolute differences, provided that they are nonzero and pairwise distinct, and then permute these differences arbitrarily before repeating the operation. The depth of the composition is the maximum possible number of successive iterations. We determine the least positive integer admitting a composition of any prescribed depth. If a(n) is the least positive integer having a composition of depth n, then $a(n)=n+1+\lceil n(n+1)/4\rceil+\lfloor n/2\rfloor$. We also prove that every integer k at least a(n) has a composition of depth at least n. Consequently, if d(k) denotes the maximum depth of a composition of k, then $d(k)=\max\{n\geq 0:a(n)\leq k\}$. Finally, we classify and enumerate all compositions attaining the minimum a(n). Their number is given by four factorial formulas according to n modulo 4.

math.CO↗

arXiv2026

Large Sidon Subsets and Pair-Sum Multiplicities of Distinct Multinomial Coefficients

For a positive integer $n$, let $\mathcal{M}_n$ be the set of distinct multinomial coefficient values $n!/(p_1!\cdots p_t!)$, where $(p_1,\ldots,p_t)$ ranges over the integer partitions of $n$. We study two complementary aspects of the additive structure of $\mathcal{M}_n$: the maximum cardinality $s(n)$ of a Sidon subset and the multiplicities of unordered pair sums in the full set. A product embedding, strongly Sidon subsets, and estimates for prime partitions give \[ \liminf_{n\to\infty}\frac{\log(s(n)/n)\log\log n}{\sqrt{\log n}}\geq\frac{\pi}{\sqrt{3}}. \] Long arithmetic progressions and separated copies yield a complementary lower bound $2/(3\sqrt{3})$ for the normalized Sidon defect. Writing $r_n(t)$ for the number of unordered representations $t=x+y$ with $x,y\in\mathcal{M}_n$, we study the collision excess $C(n)$, the number $D(n)$ of multiply represented sums, the maximum multiplicity $\mu(n)$, and the cumulative profile $T(n,k)$. We prove \[ \liminf\frac{C(n)}{n^2\log n}\geq\frac18,\qquad \liminf\frac{D(n)}{n^{3/2}\sqrt{\log n}}\geq\frac{4}{3\sqrt{3}},\qquad \liminf\frac{\mu(n)}{\sqrt{n\log n}}\geq\frac12. \] We also obtain a scaled lower envelope for the full profile, an additive-energy bound, stabilization with respect to the number of variables, and an explicit family of trinomial collisions. Exact Sidon values through $n=16$, the lower bound $s(17)\geq89$, and pair-sum statistics through $n=20$ are reported with reproducible computational material.

math.CO↗

arXiv2026

On the two-copy distillability of Werner states and a new partial trace inequality

Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state $\varrho(4,-\tfrac12)$ is two-copy distillable, where $\varrho(d,\alpha)=(I+\alpha F)/(d^2+\alpha d)$. We answer it in the negative. To this end, we show the following stronger statement: for all $C\in M_{d_1d_2}(\mathbb{C})$ of rank at most $r \le d_1 d_2$, $\mathrm{tr}_1(C)\|_F^2+\|\mathrm{tr}_2(C)\|_F^2 \le r\|C\|_F^2+\frac{1}{r}|\mathrm{tr}(C)|^2$. A result by Costa Rico on the equivalence of this inequality with two-copy undistillability at $r = 2$ then settles Problem 5: $\varrho(4,-\tfrac{1}{2})$ is not two-copy distillable. Furthermore, we show that $\varrho(d,\alpha)$ is two-copy undistillable for every $d\ge2$, if and only if $\alpha\ge-\tfrac{1}{2}$. Thus, the one and two-copy distillability regions of $\varrho(d,\alpha)$ coincide. These results have been found and written up with AI tools, pointing towards a structural change affecting the field of quantum information and computation.

quant-ph↗

arXiv2026

Holographic quantum codes with trapped ions

Holography is a central concept at the intersection of gravity, condensed matter theory, and quantum information, linking the interior bulk of a system to its boundary. A model realizing key features of holographic systems is the holographic pentagon code by Pastawski et al. Here we experimentally implement instances of the holographic pentagon and heptagon codes with trapped ions and test their properties: For the pentagon code, we recover logical bulk qubits from their nearby boundary and test the Ryu-Takayanagi entanglement area law. For the heptagon code, we show that the transversal Hadamard gate native to the constituent Steane codes induces a single-qubit, correctable error in the holographic code. Our implementation paves the way towards the use of holographic quantum codes for quantum information processing.

quant-ph↗

arXiv2026

A 0.651-approximation to quantum Max Cut via Rydberg atoms

Quantum Max Cut, also known as the anti-ferromagnetic Heisenberg Hamiltonian, is a QMA-complete problem which serves as a benchmark for approximation algorithms in quantum physics. Here we develop a hybrid approximation algorithm to quantum Max Cut, which uses the natural quantum dynamics of Rydberg atom systems in combination with semidefinite programming and randomized rounding. It achieves a conditional approximation ratio of $0.651$, compared to the best-known ratio of $0.614$ that relies on semidefinite programming alone. The algorithm is robust in the sense that the advantage persists even if the annealing procedure of the Rydberg atom system obtains a state whose energy is only $89\%$ of its true ground state energy. Our approach opens a new route for hybrid quantum-classical algorithms that combine quantum with classical optimization methods.

quant-ph↗

arXiv2026

Complete entanglement detection using polynomial invariants

Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.

quant-ph↗

arXiv2026

Combining moment matrices, symmetric extension, and Lov\'asz theta: $\Phi_{\text{E8}}$ is entangled

We solve an open problem in entanglement theory posed by Yu et al., {\it Nature Communications 12, 1012 (2021)}. The problem is to show, via an entanglement witness, that the $14$-qubit state $\Phi_{\text{E8}}$ is entangled. Inspired by a method from quantum codes, we combine symmetric extension with moment matrices to prove that $\Phi_{\text{E8}}$ is entangled. The proof has the form of a rational infeasibility certificate for a semidefinite program, yielding an explicit entanglement witness. Our approach unifies and extends several earlier methods that involve the Lov\'asz theta number of the Pauli anti-commutativity graph, promising scalability and flexibility in further applications.

quant-ph↗

arXiv2026

SDP bounds on quantum codes: rational certificates

A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon $18$ upper bounds on the maximum size of $n$-qubit codes with $6 \leq n \leq 19$. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.

quant-ph↗

arXiv2025

Lov\'asz theta and Shearer lower bounds on Quantum Max Cut

Quantum Max Cut is a problem relevant to computer science and many-body quantum physics due to its links to classical Max Cut and the anti-ferromagnetic Heisenberg Hamiltonian. We prove a lower bound to quantum Max Cut of a graph in terms of the Lov\'asz theta function of its complement. For a graph with $m$ edges, $\text{qmc}(G) \geq \tfrac{m}{4}\big( 1 + \tfrac{8}{3\pi}\tfrac{1}{\vartheta(\bar{G}) -1} \big)$, with the bound achieved by a product state. The proof can be strenghtened by the vector chromatic number and extends a result by Balla, Janzer, and Sudakov on classical Max Cut. A relaxed bound follows from $\vartheta(\bar{G}) - 1 \leq \Delta$ for graphs with maximum degree $\Delta$, making it interesting for practically relevant quantum many-body systems. We also extend results by Carlson et al. and Shearer and show that $\text{qmc}(G) \geq \frac{m}{4} + \frac{2m^{3/4}}{3 \pi}$ for all triangle-free graphs with $m$ edges.

quant-ph↗

arXiv2025

Quasi-Clifford to qubit mappings

Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by $\alpha_1, \dots, \alpha_n$ is is given by the relations $\alpha_i^2 = k_i$ and $\alpha_j \alpha_i = (-1)^{\chi_{ij}} \alpha_i \alpha_j$, where $k_i \in \mathbb{C}$ and $\chi_{ij} \in \{0, 1\}$. We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.

quant-ph↗

arXiv2025

Bayesian preference elicitation for decision support in multiobjective optimization

We present a novel approach to help decision-makers efficiently identify preferred solutions from the Pareto set of a multi-objective optimization problem. Our method uses a Bayesian model to estimate the decision-maker's utility function based on pairwise comparisons. Aided by this model, a principled elicitation strategy selects queries interactively to balance exploration and exploitation, guiding the discovery of high-utility solutions. The approach is flexible: it can be used interactively or a posteriori after estimating the Pareto front through standard multi-objective optimization techniques. Additionally, at the end of the elicitation phase, it generates a reduced menu of high-quality solutions, simplifying the decision-making process. Through experiments on test problems with up to nine objectives, our method demonstrates superior performance in finding high-utility solutions with a small number of queries. We also provide an open-source implementation of our method to support its adoption by the broader community.

stat.ML↗

arXiv2025

On two maximally entangled couples

In a seminal article, Higuchi and Sudbery showed that a pure four-qubit state can not be maximally entangled across every bipartition. Such states are now known as absolutely maximally entangled (AME) states. Here we give a series of old and new proofs of the fact that no four-qubit AME state exists. These are based on invariant theory, methods from coding theory, and basic properties from linear algebra such as the Pauli commutation relations.

quant-ph↗

arXiv2024

Positivity of state, trace, and moment polynomials, and applications in quantum information

State, trace, and moment polynomials are polynomial expressions in several operator or random variables and positive functionals on their products (states, traces or expectations). While these concepts, and in particular their positivity and optimization, arose from problems in quantum information theory, yet they naturally fit under the umbrella of multivariate operator theory. This survey presents state, trace, and moment polynomials in a concise and unified way, and highlights their similarities and differences. The focal point is their positivity and optimization. Sums of squares certificates for unconstrained and constrained positivity (Positivstellens\"atze) are given, and parallels with their commutative and freely noncommutative analogs are discussed. They are used to design a convergent hierarchy of semidefinite programs for optimization of state, trace, and moment polynomials. Finally, circling back to the original motivation behind the derived theory, multiple applications in quantum information theory are outlined.

quant-ph↗

arXiv2024

Second order cone relaxations for quantum Max Cut

Quantum Max Cut (QMC), also known as the quantum anti-ferromagnetic Heisenberg model, is a QMA-complete problem relevant to quantum many-body physics and computer science. Semidefinite programming relaxations have been fruitful in designing theoretical approximation algorithms for QMC, but are computationally expensive for systems beyond tens of qubits. We give a second order cone relaxation for QMC, which optimizes over the set of mutually consistent three-qubit reduced density matrices. In combination with Pauli level-$1$ of the quantum Lasserre hierarchy, the relaxation achieves an approximation ratio of $0.526$ to the ground state energy. Our relaxation is solvable on systems with hundreds of qubits and paves the way to computationally efficient lower and upper bounds on the ground state energy of large-scale quantum spin systems.

quant-ph↗

arXiv2024

SDP bounds on quantum codes

This paper provides a semidefinite programming hierarchy based on state polynomial optimization to determine the existence of quantum codes with given parameters. The hierarchy is complete, in the sense that a $(\!(n, K, {\delta})\!)_2$ code exists if and only if every level of the hierarchy is feasible. It is not limited to stabilizer codes and thus is applicable generally. While the machinery is formally dimension-free, we restrict it to qubit codes through quasi-Clifford algebras. We derive the quantum analog of a range of classical results: first, from an intermediate level a Lov\'asz bound for self-dual quantum codes is recovered. Second, a symmetrization of a minor variation of this Lov\'asz bound recovers the quantum Delsarte bound. Third, a symmetry reduction using the Terwilliger algebra leads to semidefinite programming bounds of size $O(n^4)$. With this we give an alternative proof that there is no $(\!(7, 1, 4)\!)_2$ quantum code, and show that $(\!(8, 9, 3)\!)_2$ and $(\!(10, 5, 4)\!)_2$ codes do not exist.

quant-ph↗

arXiv2023

Uncertainty relations from state polynomial optimization

Uncertainty relations are a fundamental feature of quantum mechanics. How can these relations be found systematically? Here we develop a semidefinite programming hierarchy for additive uncertainty relations in the variances of non-commuting observables. Our hierarchy is built on the state polynomial optimization framework, also known as scalar extension. The hierarchy is complete, in the sense that it converges to tight uncertainty relations. We improve upon upper bounds for all 1292 additive uncertainty relations on up to nine operators for which a tight bound is not known. The bounds are dimension-free and depend entirely on the algebraic relations among the operators. The techniques apply to a range of scenarios, including Pauli, Heisenberg-Weyl, and fermionic operators, and generalize to higher order moments and multiplicative uncertainty relations.

quant-ph↗

arXiv2023

Entanglement detection with trace polynomials

We provide a systematic method for nonlinear entanglement detection based on trace polynomial inequalities. In particular, this allows to employ multi-partite witnesses for the detection of bipartite states, and vice versa. We identify witnesses for which linear detection of an entangled state fails, but for which nonlinear detection succeeds. With the trace polynomial formulation a great variety of witnesses arise from immamant inequalities, which can be implemented in the laboratory through randomized measurements.

quant-ph↗