arXiv subjects
Publications and source records attributed to Stefan Trandafir.
We present bijections connecting tree records, the girth of a connected endofunction, and the genesis sequence (the first sequence in OEIS). Using these, we derive generating functions for tree and forest record numbers in terms of Cayley's tree function and give a new proof of Cayley's forest formula.
We present a search for small Kochen-Specker (KS) sets in dimension 3, specifically targeting extensions of the 13-ray Yu-Oh set, which has been proven to be the minimal witness to state-independent contextuality. To enable this search, we introduce a novel SAT-based orderly generation framework integrating recursive canonical labeling (RCL) with the graph isomorphism tool NAUTY. We demonstrate that previous SAT approaches relying on lexicographical canonicity suffer from exponential scaling on canonical graphs. This limitation renders them intractable on the large instances (25 to 33 vertices) encountered in our search, whereas our RCL check maintains consistent millisecond-level performance, effectively eliminating the bottleneck. Overcoming this bottleneck allows us to perform the first exhaustive enumeration of all KS sets with up to 33 rays containing the complete 25-ray state-independent contextuality (SI-C) set obtained by rigid extensions of the Yu-Oh set in 1,641 CPU hours. We found and verified that the 33-ray set discovered by Schütte is the smallest three-dimensional KS set containing the complete 25-ray SI-C set. All non-existence results are backed by independently verifiable proof certificates via an extension of the DRAT proof format.
This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join equidistribution result between a 5-tuple of statistics on rooted trees and a corresponding 5-tuple of statistics on parking functions. Some enumerative questions are also considered.
In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to derive the generating functions for the tree and forest record numbers, expressing them in terms of the Cayley's tree function. Finally, we provide a new proof for Cayley's forest formula.
A record of a rooted Cayley tree is a node whose label is the largest along the unique path to the root. In this work, we find elegant functional equations relating the generating functions for records of rooted Cayley trees and for records of forests of rooted trees with the Cayley tree function, and explore the consequences of our results.
Recent results show that Kochen-Specker (KS) sets of observables are fundamental to quantum information, computation, and foundations beyond previous expectations. Among KS sets, those that are unique up to unitary transformations (i.e., "rigid") are especially important. The problem is that we do not know any rigid KS set in $\mathbb{C}^3$, the smallest quantum system that allows for KS sets. Moreover, none of the existing methods for constructing KS sets leads to rigid KS sets in $\mathbb{C}^3$. Here, we show that two fundamental structures of quantum theory define two rigid KS sets. One of these structures is the super-symmetric informationally complete positive-operator-valued measure. The other is the minimal state-independent contextuality set. The second construction provides a clue to solve the minimal KS problem, the most important open problem in this field. We prove that there is no KS set of 30 elements that can be obtained from the minimal state-independent contextuality set by completing bases and adding elements that are orthogonal to two previous elements. We conjecture that 31 is the solution to the minimal KS set problem.
Classically simulating the quantum contextual correlations produced by sequences of ideal measurements of compatible observables requires the measured system to have an internal memory. Computing the minimum amount of memory needed is, in general, challenging. Here, building upon the work of Kleinmann et al. [New J. Phys. 13, 113011 (2011)], we prove that the memory cost for simulating the contextuality produced by the $10$ three-qubit observables of Mermin's pentagram is only $\log_2(5) \approx 2.32$ bits, but the memory cost for simulating the contextuality produced by all $15$ two-qubit Pauli observables is, at least, $\log_2(6) \approx 2.58$ bits, thus exceeding the classical capacity of the system on which the measurements are performed. We also add results on the memory for simulating some subsets of quantum predictions.
According to Paviči{ć}, Kochen and Specker's 117-observable set is not a ``Kochen-Specker set''. By the same reason, in arXiv:2502.13787, Paviči{ć} claims that 10 statements in our paper ``Optimal conversion of Kochen-Specker sets into bipartite perfect quantum strategies'' [Phys. Rev. A 111, 022408 (2025)] are ``wrong''. In all cases, Paviči{ć}'s claims are based on the fact that he is assuming a different definition of Kochen-Specker (KS) set. Adopting a terminology used by, e.g., Larsson, the sets that Paviči{ć} call KS sets can be called ``extended'' KS sets, since they are constructed by adding observables to the ``original'' KS sets. For example, Paviči{ć} adds 75 observables to the original 117-observable KS set. Beyond terminology, there are fundamental physical reasons for focusing on the original KS sets. One reason is that, for experimentally observing quantum state-independent contextuality, there is no need to measure the observables added in the extended sets. Another reason is that, to produce bipartite perfect quantum strategies, or correlations in a face of the nonsignaling polytope with no local points, or correlations with nonlocal content 1, the two parties do not need to measure any of the observables added in the extended sets. We also respond to other claims made by Paviči{ć} about our work.
Bipartite perfect quantum strategies (BPQSs) allow two players isolated from each other to win every trial of a nonlocal game. BPQSs have crucial roles in recent developments in quantum information and quantum computation. However, only few BPQSs with a small number of inputs are known and only one of them has been experimentally tested. It has recently been shown that every BPQS has an associated Kochen-Specker (KS) set. Here, we first prove that any BPQS of minimum input cardinality that can be obtained from a generalized KS set can also be obtained from a KS set of pure states. Then, we address the problem of finding BPQSs of small input cardinality starting from KS sets. We introduce an algorithm that identifies the BPQS with the minimum number of settings for any given KS set. We apply it to many well-known KS sets of small cardinality in dimensions 3, 4, 5, 6, 7, and 8. In each dimension, the algorithm either recovers the best BPQS known or find one with fewer inputs.
A perfect strategy is one that allows the mutually in-communicated players of a nonlocal game to win every trial of the game. Perfect strategies are basic tools for some fundamental results in quantum computation and crucial resources for some applications in quantum information. Here, we address the problem of producing qudit-qudit perfect quantum strategies with a small number of settings. For that, we exploit a recent result showing that any perfect quantum strategy induces a Kochen-Specker set. We identify a family of KS sets in even dimension $d \ge 6$ that, for many dimensions, require the smallest number of orthogonal bases known: $d+1$. This family was only defined for some $d$. We first extend the family to infinitely many more dimensions. Then, we show the optimal way to use each of these sets to produce a bipartite perfect strategy with minimum input cardinality. As a result, we present a family of perfect quantum strategies in any $(2,d-1,d)$ Bell scenario, with $d = 2^kp^m$ for $p$ prime, $m \geq k \geq 0$ (excluding $m=k=0$), $d = 8p$ for $p \geq 19$, $d=kp$ for $p > ((k-2)2^{k-2})^2$ whenever there exists a Hadamard matrix of order $k$, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions $d$ from any dimension $d'$ with a perfect quantum strategy. We identify their associated Bell inequalities and prove that they are not tight, which provides a second counterexample to a conjecture of 2007.
The vector partition function $p_A$ associated to a $d \times n$ matrix $A$ with integer entries is the function $\mathbb{Z}^d \to \mathbb{N}$ defined by $\mathbf{b} \to \#\{\mathbf{x} \in \mathbb{N}^n : A\mathbf{x} = \mathbf{b}\}$. It is known that vector partition functions are piecewise quasi-polynomials whose domains of quasi-polynomiality are maximal cones (chambers) of a fan called the chamber complex of $A$. In this article we introduce \emph{external columns} and \emph{external chambers} of vector partition functions. Our main result is that (up to a saturation condition) the quasi-polynomial associated to a chamber containing external columns arises from a vector partition function with $k$ fewer equations and variables. In the case that the chamber is external -- that is, when the number of external columns in a chamber is as large as possible without being trivial -- the quasi-polynomial arises from a coin exchange problem. By exploiting this we are able to obtain a determinantal formula, characterize when the quasi-polynomial is polynomial, and show that in this case it is actually given by a negative binomial coefficient. We then apply these results to the enumeration of loopless multigraphs satisfying some degree conditions. Finally, we suggest a generalization to a result of Baldoni and Vergne for polynomials arising from chambers that we call \emph{semi-external chambers}.
The $\mathit{SU}(3)$ tensor multiplicities are piecewise polynomial of degree $1$ in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order $144$) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.
Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of $n\ge 2$ qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that $n$-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)]. An open question is whether there are magic sets that cannot be reduced to the square or the pentagram. If they exist, a second key question is whether they require $n >3$ qubits, since, if this is the case, these magic sets would capture minimal state independent quantum advantage that is specific for $n$-qubit systems with specific values of $n$. Here, we answer both questions affirmatively. We identify magic sets which cannot be reduced to the square or the pentagram and require $n=3,4,5$, or $6$ qubits. In addition, we prove a generalized version of Arkhipov's theorem providing an efficient algorithm for, given a hypergraph, deciding whether or not it can accommodate a magic set, and solve another open problem, namely, given a magic set, obtaining the tight bound of its associated noncontextuality inequality.
We study the Kronecker coefficients $g_{λ, μ, ν}$ via a formula that was described by Mishna, Rosas, and Sundaram, in which the coefficients are expressed as a signed sum of vector partition function evaluations. In particular, we use this formula to determine formulas to evaluate, bound, and estimate $g_{λ, μ, ν}$ in terms of the lengths of the partitions $λ, μ$, and $ν$. We describe a computational tool to compute Kronecker coefficients $g_{λ, μ, ν}$ with $\ell(μ) \leq 2,\ \ell(ν) \leq 4,\ \ell(λ) \leq 8$. We present a set of new vanishing conditions for the Kronecker coefficients by relating to the vanishing of the related atomic Kronecker coefficients, themselves given by a single vector partition function evaluation. We give a stable face of the Kronecker polyhedron for any positive integers $m,n$. Finally, we give upper bounds on both the atomic Kronecker coefficients and Kronecker coefficients.