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On the quantum separability of qubit registers

Szymon Łukaszyk

TL;DR

The paper addresses how to determine bipartite separability of pure $n$-qubit registers by analyzing the computational-basis support. It introduces a Boolean-cube geometry framework with concepts such as common-bit supports (CMB$_c$), APS, PDS$_c$, and ADS to classify states by support structure, not amplitudes, and derives closed-form counts and feasibility conditions (e.g., $| ext{CMB}_c|(n,k)$ and $k = k_B k_C \le 2^{\lceil \log_2(k_B)\rceil + \lceil \log_2(k_C)\rceil}$) for separability across bipartitions. The work shows that the distribution of support-based state types can be computed exactly for small $n$ and that some ADS sizes are forbidden, implying unconditional entanglement, while APS and PDS$_c$ permit fast, amplitude-independent separability checks ($O(nk)$) and enable a fast pre-filter for simulations. Practically, the framework provides fast entanglement diagnostics, demonstrates tracking of entanglement spreading in circuits (e.g., GHZ state preparation), and enables substantial classical speedups for simulating localized-entanglement states, with broad implications for circuit design and quantum error-correcting code analysis.

Abstract

We show that the bipartite separability of a pure qubit state hinges critically on the combinatorial structure of its computational-basis support. Using Boolean cube geometry, we introduce a taxonomy that distinguishes support-guaranteed separability from cases in which entanglement depends on probability amplitudes. We provide closed-form support counts, identify forbidden configurations that enforce multipartite entanglement, and show how these results can enable fast entanglement diagnostics in quantum circuits. The framework offers immediate utility in classical simulation, entanglement-aware circuit design, and quantum error-correcting code analysis. This establishes support geometry as a practical and scalable tool for understanding entanglement in quantum information processing.

On the quantum separability of qubit registers

TL;DR

The paper addresses how to determine bipartite separability of pure -qubit registers by analyzing the computational-basis support. It introduces a Boolean-cube geometry framework with concepts such as common-bit supports (CMB), APS, PDS, and ADS to classify states by support structure, not amplitudes, and derives closed-form counts and feasibility conditions (e.g., and ) for separability across bipartitions. The work shows that the distribution of support-based state types can be computed exactly for small and that some ADS sizes are forbidden, implying unconditional entanglement, while APS and PDS permit fast, amplitude-independent separability checks () and enable a fast pre-filter for simulations. Practically, the framework provides fast entanglement diagnostics, demonstrates tracking of entanglement spreading in circuits (e.g., GHZ state preparation), and enables substantial classical speedups for simulating localized-entanglement states, with broad implications for circuit design and quantum error-correcting code analysis.

Abstract

We show that the bipartite separability of a pure qubit state hinges critically on the combinatorial structure of its computational-basis support. Using Boolean cube geometry, we introduce a taxonomy that distinguishes support-guaranteed separability from cases in which entanglement depends on probability amplitudes. We provide closed-form support counts, identify forbidden configurations that enforce multipartite entanglement, and show how these results can enable fast entanglement diagnostics in quantum circuits. The framework offers immediate utility in classical simulation, entanglement-aware circuit design, and quantum error-correcting code analysis. This establishes support geometry as a practical and scalable tool for understanding entanglement in quantum information processing.
Paper Structure (8 sections, 8 theorems, 44 equations, 3 figures, 3 tables)

This paper contains 8 sections, 8 theorems, 44 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Results
  4. Applications
  5. Tracking entanglement spreading in quantum circuits
  6. Classical simulation speedup for localized-entanglement states
  7. Conclusions
  8. Data Accessibility

Key Result

Lemma 1

The number of bipartitions a quantum register containing $n$ qubits can be separable across is $2^c-1$, where $0 \le c \le n-1$.

Figures (3)

  • Figure 1: Maximum supports of one (a) to four (d) qubit registers in the computational basis. Vertices and blue edges represent any-partition-separable states, green edges --- partition-dependent separable states, and red edges --- states inseparable.
  • Figure 2: Taxonomy of the quantum states and supports introduced in this study.
  • Figure 3: Possible support sizes $k$ (products of the support sizes $k_B$ in the left column and $k_C$ in the bottom row) of ADS$_{c,j}$ states for $2 \le n \le 7$ qubits, indicated with different colors. Forbidden sizes in boxes (OEIS sequence https://oeis.org/A390536) are those where no bipartition can factor the support into $k_B × k_C$ satisfying the equation \ref{['eq:ADSineq']} (see text for details).

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 9 more