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Infeasibility of constructing a special orthogonal matrix for the deterministic remote preparation of arbitrary n-qubit state

Wenjie Liu, Zixian Li, Gonglin Yuan

TL;DR

The paper investigates the deterministic remote state preparation (DRSP) of arbitrary $n$-qubit states via a special $2^n\times2^n$ real-parameter orthogonal matrix whose columns are sign-permutations of a fixed vector. It introduces a polynomial-time route that reduces orthogonality to cooperation among matching operators and encodes the construction as an XOR Gaussian-elimination problem, thereby obtaining a polynomial-time subroutine for semi-orthogonal matrices. A key theoretical advance is the reduction of any semi-orthogonal matrix to a unique ordered type, enabling a rigorous infeasibility proof: such matrices exist only for $N=2,4,8$ (i.e., $n=1,2,3$) and do not exist for $N\ge16$ ($n\ge4$). Consequently, the study proves that constructing a universal special orthogonal matrix for DRSP of an arbitrary $n$-qubit state is infeasible when $n>3$. The work thereby motivates pursuing alternative DRSP schemes for larger systems and clarifies the fundamental limits of this matrix-based approach in quantum information transfer.

Abstract

In this paper, we present a polynomial-complexity algorithm to construct a special orthogonal matrix for the deterministic remote state preparation (DRSP) of an arbitrary n-qubit state, and prove that if n>3, such matrices do not exist. Firstly, the construction problem is split into two sub-problems, i.e., finding a solution of a semi-orthogonal matrix and generating all semi-orthogonal matrices. Through giving the definitions and properties of the matching operators, it is proved that the orthogonality of a special matrix is equivalent to the cooperation of multiple matching operators, and then the construction problem is reduced to the problem of solving an XOR linear equation system, which reduces the construction complexity from exponential to polynomial level. Having proved that each semi-orthogonal matrix can be simplified into a unique form, we use the proposed algorithm to confirm that the unique form does not have any solution when n>3, which means it is infeasible to construct such a special orthogonal matrix for the DRSP of an arbitrary n-qubit state.

Infeasibility of constructing a special orthogonal matrix for the deterministic remote preparation of arbitrary n-qubit state

TL;DR

The paper investigates the deterministic remote state preparation (DRSP) of arbitrary -qubit states via a special real-parameter orthogonal matrix whose columns are sign-permutations of a fixed vector. It introduces a polynomial-time route that reduces orthogonality to cooperation among matching operators and encodes the construction as an XOR Gaussian-elimination problem, thereby obtaining a polynomial-time subroutine for semi-orthogonal matrices. A key theoretical advance is the reduction of any semi-orthogonal matrix to a unique ordered type, enabling a rigorous infeasibility proof: such matrices exist only for (i.e., ) and do not exist for (). Consequently, the study proves that constructing a universal special orthogonal matrix for DRSP of an arbitrary -qubit state is infeasible when . The work thereby motivates pursuing alternative DRSP schemes for larger systems and clarifies the fundamental limits of this matrix-based approach in quantum information transfer.

Abstract

In this paper, we present a polynomial-complexity algorithm to construct a special orthogonal matrix for the deterministic remote state preparation (DRSP) of an arbitrary n-qubit state, and prove that if n>3, such matrices do not exist. Firstly, the construction problem is split into two sub-problems, i.e., finding a solution of a semi-orthogonal matrix and generating all semi-orthogonal matrices. Through giving the definitions and properties of the matching operators, it is proved that the orthogonality of a special matrix is equivalent to the cooperation of multiple matching operators, and then the construction problem is reduced to the problem of solving an XOR linear equation system, which reduces the construction complexity from exponential to polynomial level. Having proved that each semi-orthogonal matrix can be simplified into a unique form, we use the proposed algorithm to confirm that the unique form does not have any solution when n>3, which means it is infeasible to construct such a special orthogonal matrix for the DRSP of an arbitrary n-qubit state.
Paper Structure (19 sections, 21 theorems, 21 equations, 1 table, 6 algorithms)

This paper contains 19 sections, 21 theorems, 21 equations, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Schemes review and analysis
  3. DRSP of a real-parameter state via maximally two-qubit states
  4. Wei et al.'s algorithm to construct a special orthogonal matrix
  5. Analysis of Wei et al.'s algorithm
  6. Problem definition and split of constructing a special orthogonal matrix
  7. Problem definition
  8. Problem split
  9. Proposed algorithm to construct a special orthogonal matrix
  10. Matching operator
  11. Cooperation and matching equation
  12. Algorithm to generate a special orthogonal matrix
  13. Infeasibility proof of constructing a special orthogonal matrix for the DRSP of an arbitrary $n$-qubit state
  14. Semi-matching operators and semi-cooperative group
  15. Bijections and semi-cooperative generator set
  16. ...and 4 more sections

Key Result

Proposition 1

Given two column vectors then the necessary and sufficient condition for $\textbf{s}^{(i)}\perp \textbf{s}^{(j)}$ is that all elements of $\textbf{s}^{(i)}$ can be divided into several $2$-tuples, and in $\textbf{s}^{(j)}$, the elements in each 2-tuple are exchanged and one of them is multiplied by $-1$. i.e.,

Theorems & Definitions (69)

  • Definition 1: Special orthogonal matrix
  • Definition 2: Semi-orthogonal matrix
  • Proposition 1
  • proof
  • Remark 1
  • Definition 3: Matching operation
  • Definition 4: Couple
  • Definition 5: Division
  • Definition 6: Scattered matrix
  • Remark 2
  • ...and 59 more