Digest for "On Multiquantum Bits, Segre Embeddings and Coxeter Chambers"

Abstract

This work explores the interplay between quantum information theory, algebraic geometry, and number theory, with a particular focus on multiqubit systems, their entanglement structure, and their classification via geometric embeddings. The Segre embedding, a fundamental construction in algebraic geometry, provides an algebraic framework to distinguish separable and entangled states, encoding quantum correlations in projective geometry. We develop a systematic study of qubit moduli spaces, illustrating the geometric structure of entanglement through hypercube constructions and Coxeter chamber decompositions.

We establish a bijection between the Segre embeddings of tensor products of projective spaces and binary words of length \( n-1\) , structured as an \( (n-1)\) -dimensional hypercube, where adjacency corresponds to a single Segre operation. This reveals a combinatorial structure underlying the hierarchy of embeddings, with direct implications for quantum error correction schemes. The symmetry of the Segre variety under the Coxeter group of type \( A\) allows us to analyze quantum states and errors through the lens of reflection groups, viewing separable states as lying in distinct Coxeter chambers on a Segre variety. The transitive action of the permutation group on these chambers provides a natural method for tracking errors in quantum states and potentially reversing them. Beyond foundational aspects, we highlight relations between Segre varieties and Dixon elliptic curves, drawing connections between entanglement and number theory.

Mathematics Subject Classification (MSC 2020): 14M99 (None of the above, but in Algebraic Geometry), 20F55 (Reflection and Coxeter groups), 81P40 (Quantum coherence, entanglement, quantum correlations), 11G05 (Elliptic curves over global fields), 05C12 (Hypercubes in graph theory).

Definition 1

A state is entangled if it cannot be factorized as a simple tensor product.
A separable (i.e. non-entangled) state takes the form:

\[ \begin{equation} | \psi \rangle = (c_{00} |0\rangle + c_{01} |1\rangle) \otimes (c_{10} |0\rangle + c_{11} |1\rangle). \end{equation} \]

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Lemma 1

Let \( |\psi\rangle_{AB} \) be a two-qubit pure state. Let \( \mathcal{H}_A\) and \( \mathcal{H}_B\) be the respective Hilbert spaces of the subsystems \( A\) and \( B\) .

The pure state \( |\psi\rangle_{AB} \) is separable if and only if the corresponding coordinates

\[ [Z^{0}: Z^{1}: Z^{2}: Z^{3}] \in \mathbb{P}^3~ \text{of}~ |\psi\rangle_{AB} \]

lie on a Segre variety, which is the image of the canonical embedding:

\[ \mathbb{P}(\mathcal{H}_A) \times \mathbb{P}(\mathcal{H}_B) \hookrightarrow \mathbb{P}^3. \]

Explicitly, the Segre variety is described as the locus of points that satisfy the quadratic equation:

\[ \det \begin{pmatrix} c_{00} & c_{01} \\ c_{10} & c_{11} \end{pmatrix} = 0, \]

where \( |\psi\rangle_{AB} = c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle \).

Corollary 1

A state corresponding to a 2-qubit is separable if and only if

\[ \begin{equation} Z^0Z^3 - Z^1Z^2 = 0. \end{equation} \]

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This equation defines the Segre variety in \( \mathbb{P}^{3}\) .

Lemma 2

The quantum state \( |\psi\rangle \) of a 3-qubit is said to be separable with respect to the tensor decomposition \( A \otimes B \otimes C \) if and only if its associated point \([ \psi ] \in \mathbb{P}^7\) lies on the generalized Segre variety. This variety arises naturally as the image of the Segre embedding:

\[ \mathbb{P}(\mathcal{H}_A) \times \mathbb{P}(\mathcal{H}_B) \times \mathbb{P}(\mathcal{H}_C) \hookrightarrow \mathbb{P}(\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C), \]

where \( \mathcal{H}_A \), \( \mathcal{H}_B \), and \( \mathcal{H}_C \) are the respective Hilbert spaces of the subsystems \( A \), \( B \), and \( C \).

Theorem 1

Let \( \mathcal{H}_2 \) denote the 2-dimensional Hilbert space of a single qubit.

The pure state \( |\psi\rangle \) of a \( n\) -qubit is a separable state if and only if the associated coordinate points in \( \mathbb{P}^{2^n - 1}\) lie on the generalized Segre variety. Furthermore, it is a product state if it is parametrized by a \(n-1\)-hypercube of Segre embeddings given by:

\[ \mathbb{P}^1 \times \mathbb{P}^1 \times \dots \times \mathbb{P}^1 \hookrightarrow \mathbb{P}^{2^n - 1}. \]

Theorem 2

Assume \( |\psi\rangle\) is a pure separable product state. Suppose that the coefficients of \( |\psi\rangle\) are perturbed by an error occurring, defining thus a new vector \( |\psi'\rangle\) . Then, if \( |\psi'\rangle\) is a pure separable product state then either it belongs to the same Coxeter chamber \( C\) as \( |\psi\rangle\) on the Segre variety or it belongs to a different Coxeter chamber \( C\neq C'\) such that \( C'=gC\) , where \( g\) is an element of the group \( S_{n+1}\) .

Table of contents

Digest for "On Multiquantum Bits, Segre Embeddings and Coxeter Chambers"

Abstract