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On the distillability problem of two-copy \(4\times 4\) Werner states: matrices A, B of at most four nonzero entries

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Abstract

In entanglement theory, the long-standing distillability problem asks whether the two-copy \(4\times 4\) Werner states are distillable under local operations and classical communications. We report the partial solution to the problem when the involved \(4\times 4\) matrices A and B, respectively, have one and three nonzero entries. As a corollary, the problem is solved when A and B have at most four nonzero entries. Our result presents the latest progress on the problem.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

LC and HXH were supported by the National Key R &D Program of China under Grant No. 2018YFC2001400. LC and CCF were supported by the NNSF of China (Grant No. 11871089) and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12080401 and ZG216S1902). HXH was supported by the NNSF of China (Grant No. 12271038).

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Lin Chen, Changchun Feng & Huixia He

  2. International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China

    Lin Chen

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  1. Lin Chen
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  2. Changchun Feng
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  3. Huixia He
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Correspondence to Huixia He.

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Appendix A: The proof of Lemma 4

Appendix A: The proof of Lemma 4

Proof

  1. (i)

    The proof is done in [1].

  2. (ii)

    Similarly to the proof of (i), one can show that the \(4\times 4\) matrix of three nonzero entries and trace zero is permutationally equivalent to one of the following matrices:

    $$\begin{aligned}{} & {} \begin{bmatrix} a_1 &{} 0 &{} 0 &{} 0\\ 0 &{} a_2 &{} 0 &{} 0\\ 0 &{} 0 &{} -a_1-a_2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} b_1 &{} b_2 &{} 0 &{} 0\\ 0 &{} -b_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} c_1 &{} 0 &{} c_2 &{} 0\\ 0 &{} -c_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} d_1 &{} 0 &{} 0 &{} 0\\ 0 &{} -d_1 &{} 0 &{} 0\\ d_2 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix},\nonumber \\ \end{aligned}$$
    (A1)
    $$\begin{aligned}{} & {} \begin{bmatrix} f_1 &{} 0 &{} 0 &{} 0\\ 0 &{} -f_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} f_2\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} g_1 &{} g_2 &{} 0\\ g_3 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} h_1 &{} h_2 &{} 0\\ 0 &{} 0 &{} h_3 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} i_1 &{} i_2 &{} 0\\ 0 &{} 0 &{} 0 &{} i_3\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \end{aligned}$$
    (A2)
    $$\begin{aligned}{} & {} \begin{bmatrix} 0 &{} i_1' &{} i_2' &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ i_3' &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} i_1'' &{} i_2'' &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} i_3'' &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} i_1''' &{} i_2''' &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} i_3'''\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} j_1 &{} j_2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ j_3 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \end{aligned}$$
    (A3)
    $$\begin{aligned}{} & {} \begin{bmatrix} 0 &{} k_1 &{} k_2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} k_3 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} l_1 &{} 0 &{} 0\\ l_2 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} l_3\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} m_1 &{} 0 &{} 0\\ 0 &{} 0 &{} m_2 &{} 0\\ m_3 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} n_1 &{} 0 &{} 0\\ 0 &{} 0 &{} n_2 &{} 0\\ 0 &{} 0 &{} 0 &{} n_3\\ 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \end{aligned}$$
    (A4)
    $$\begin{aligned}{} & {} \begin{bmatrix} 0 &{} p_1 &{} 0 &{} 0\\ 0 &{} 0 &{} p_2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ p_3 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} q_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ q_2 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} q_3 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} s_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} s_2\\ s_3 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}, \quad \begin{bmatrix} 0 &{} t_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} t_2\\ 0 &{} 0 &{} t_3 &{} 0\\ \end{bmatrix}. \end{aligned}$$
    (A5)

    One can show that the third and fourth matrices in (A1) are unitarily equivalent, the first three matrices in (A3) are unitarily equivalent to the matrices in (A1), and the fourth matrix in (A3) is unitarily equivalent to a monomial matrix of at least two nonzero entries. Further, the fourth matrix in (A4) and the first matrix in (A5) are unitarily equivalent, the second matrix in (A4) and the second matrix in (A5) are unitarily equivalent, the second and third matrices in (A5) are permutationally equivalent, and the fourth matrix in (A5) is permutationally equivalent to the second matrix in (A4). We have proven the assertion.

\(\square \)

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Chen, L., Feng, C. & He, H. On the distillability problem of two-copy \(4\times 4\) Werner states: matrices A, B of at most four nonzero entries. Quantum Inf Process 22, 108 (2023). https://doi.org/10.1007/s11128-023-03850-4

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