In this paper, we investigate the number of independent vector variables in an [Formula: see text]-qubit Orthogonal Product Basis (OPB). Here, the term “independent vector variable” refers to a structural element in the construction of a Unextendible Orthogonal Matrix (UOM). Intuitively, since fixing one state [Formula: see text] in a qubit basis [Formula: see text] uniquely determines its orthogonal partner [Formula: see text], we treat the pair as a single degree of freedom — one variable — rather than two. Based on this counting measure, we propose the conjecture that the number is upper bounded by [Formula: see text]. We show that if the number of orthogonal pairs of variables is minimum, then the matrix corresponding to the OPB contains a column consisting of exactly one variable. In this case the conjecture holds. We also demonstrate more OPBs true for the conjecture, when they have a column containing a small number of independent variables.
We use edge-colored complete multigraphs to study complete orthogonal product bases (OPBs) in
n-qubit systems. We prove that two OPBs are equivalent if and only if their associated multigraphs are isomorphic, thereby reducing OPB classification to graph isomorphism. Within this framework, we establish the upper bound
v≤2n−1 on the number of variables of an
n-qubit OPB. We also derive
(2an−1+1)≤an≤B2n−1n for the number
an of equivalence classes of
n-qubit OPBs, where
Bm denotes the number of partitions of an
m-element set. These bounds imply the asymptotic behavior
an=22n+o(n). For every OPB, the connectivity pattern of its color layers characterizes local irreducibility, which in turn implies indistinguishability by finite-round local operations and classical communication (LOCC); the existence of a complete color-splitting tree characterizes perfect distinguishability by finite-round LOCC. Finally, we give an algorithm for testing OPB equivalence and a recursive graph algorithm that constructs a finite-round LOCC protocol whenever such perfect discrimination is possible.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.
In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as
∣φ1⟩⊗⋯⊗∣φp⟩ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász.
Let
fm(d1,…,dp) be the minimum cardinality of a UPB in the space, and let
fLB(d1,…,dp)=1+∑j=1p(dj−1) be the natural lower bound. Alon and Lovász determined exactly when
fm attains the lower bound
fLB, but the obstructed multipartite cases remained open in general.
We prove a stabilization theorem: for every non-all-qubit system with
p≥3, whenever the natural lower bound
fLB is unattainable, the true minimum is exactly
fLB+1. Precisely, if the number of even local dimensions
di is positive and even, and at least one local dimension is greater than two, then
fm(d1,…,dp)=fLB(d1,…,dp)+1. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.
Four basis vectors of the Hilbert space of two qubits have the property that if three of them are product states, then the fourth one has to be a product state as well. We address the following situation: Consider a set of orthogonal vectors, each exhibiting a certain degree of entanglement. What is the bound on entanglement of the rest of the basis vectors to form a complete orthonormal basis? Specifically, we present an orthonormal basis, the $\ensuremath{\Xi}$ basis in the Hilbert space of two qubits, with one product state and three equally entangled states. The maximum of the so available entanglement is quantified. A close-to-optimal protocol is presented for entanglement purification via entanglement swapping of two-qubit states. It is based on a suitably chosen nonmaximally entangled basis and carried out in a single step without any ancillas. A similar application of the $\ensuremath{\Xi}$ basis is examined. In this latter case, all the involved entangled states have different and nonorthogonal Schmidt decompositions and, except for some possibly resulting states, none of them are maximally entangled. Entanglement of single pair purification is not conserved on average in this case.