arXiv subjects

Zsombor Szilágyi

Publications and source records attributed to Zsombor Szilágyi.

5 recordsLinked to original sources

arXiv2025

Intrinsic volumes of the quantum state space and mutually unbiased bases

Previous studies on the geometrical properties of the state space of a finite-level quantum system have determined its volume and surface area. Building on this foundation, we derive explicit formulas for two additional intrinsic volume quantities. The question of whether a complete set of mutually unbiased bases exists in dimension $d$ can be equivalently framed as whether a specific convex polytope can be inscribed within the state space of a $d$-level quantum system. One motivation for our work was the hypothesis that a smaller intrinsic volume of the state space compared to the corresponding intrinsic volume of the mentioned polytope could rule out such an inscription. While our computations of these two intrinsic volumes do not lead to this conclusion, they nonetheless provide fundamental insights into the geometric structure of quantum state spaces. In particular, we show that these quantities can be used to rule out the existence of some unit-vector ``configurations'' (though not the one formed by the bases vectors of a complete set of mutually unbiased bases).

math-ph↗

arXiv2025

Higher rank antipodality

Motivated by general probability theory, we say that the set $S$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in S$, there is an affine map from $\mathrm{conv}(S)$ to the $k$-dimensional simplex $Δ_k$ that maps $q_1,\ldots q_{k+1}$ bijectively onto the $k+1$ vertices of $Δ_k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $k$ in $\mathbb{R}^d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Grünbaum originally developed for the $k=1$ case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-$k$ antipodality to $k$-neighborly polytopes, we obtain another upper bound when $k>d/2$.

math.MG↗

arXiv2021

On the error exponents of binary state discrimination with composite hypotheses

The trade-off between the two types of errors in binary state discrimination may be quantified in the asymptotics by various error exponents. In the case of simple i.i.d. hypotheses, each of these exponents is equal to a divergence (pseudo-distance) of the two states. In the case of composite hypotheses, represented by sets of states $R,S$, one always has the inequality $\mathrm{e}(R\|S)\le \mathrm{E}(R\|S)$, where $\mathrm{e}$ is the exponent, $\mathrm{E}$ is the corresponding divergence, and the question is whether equality holds. The relation between the composite exponents and the worst pairwise exponents may be influenced by a number of factors: the type of exponents considered; whether the problem is classical or quantum; the cardinality and the geometric properties of the sets representing the hypotheses; and, on top of the above, possibly whether the underlying Hilbert space is finite- or infinite-dimensional. Our main contribution in this paper is clarifying this landscape considerably: We exhibit explicit examples for hitherto unstudied cases where the above inequality fails to hold with equality, while we also prove equality for various general classes of state discrimination problems. In particular, we show that equality may fail for any of the error exponents even in the classical case, if the system is allowed to be infinite-dimensional, and the alternative hypothesis contains countably infinitely many states. Moreover, we show that in the quantum case strict inequality is the generic behavior in the sense that, starting from any pair of non-commuting density operators of any dimension, and for any of the exponents, it is possible to construct an example with a simple null-hypothesis and an alternative hypothesis consisting of only two states, such that strict inequality holds for the given exponent.

quant-ph↗

arXiv2020

On a conjecture regarding quantum hypothesis testing

In this MSc thesis I consider the asymptotic behaviour of the symmetric error in composite hypothesis testing. In the classical case, when the null and alternative hypothesis are finite sets of states, the best achievable symmetric error exponent is simply the the best achievable symmetric error exponent corresponding to the "worst case pair". The conjecture -- raised several years ago -- is that this remains true in the quantum case, too. This is known to be true in some special case. However, all of the known special cases are in some sense "too nice", e.g., have certain symmetries. Hoping to find a counter-example, in my thesis I consider a new special case, which on one hand is as "asymmetrical" as possible, yet still analytically computable. However, as it turns out from some involved computation, the conjecture is also true in this case, and thus gives further evidence to this conjecture.

quant-ph↗

arXiv2019

Rigidity and a common framework for mutually unbiased bases and k-nets

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In the commutative case, we recover (classical) $k$-nets, while choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for $k$-nets that can be completed to affine planes using a completely different, combinatorial argument. For $k$-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but $k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are the elements of the remaining $k$ bases (up to phase factors). In general, this is false when $k$ is just the next integer after $\sqrt{d}$; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

math-ph↗