initial release

8 files changed, 274 insertions, 0 deletions

diff --git a/Readme.md b/Readme.md
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+This is matlab implementation of some function
+from python package [Strawberry Fields](https://strawberryfields.ai/)
+
+The initial conversion was done with Perplexity AI, but
+it required a human intervention for xpxp_to_xxpp code.
+
diff --git a/python_src/williamson.py b/python_src/williamson.py
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+def williamson(V, tol=1e-11):
+    r"""Williamson decomposition of positive-definite (real) symmetric matrix.
+
+    See :ref:`williamson`.
+
+    Note that it is assumed that the symplectic form is
+
+    .. math:: \Omega = \begin{bmatrix}0&I\\-I&0\end{bmatrix}
+
+    where :math:`I` is the identity matrix and :math:`0` is the zero matrix.
+
+    See https://math.stackexchange.com/questions/1171842/finding-the-symplectic-matrix-in-williamsons-theorem/2682630#2682630
+
+    Args:
+        V (array[float]): positive definite symmetric (real) matrix
+        tol (float): the tolerance used when checking if the matrix is symmetric: :math:`|V-V^T| \leq` tol
+
+    Returns:
+        tuple[array,array]: ``(Db, S)`` where ``Db`` is a diagonal matrix
+            and ``S`` is a symplectic matrix such that :math:`V = S^T Db S`
+    """
+    (n, m) = V.shape
+
+    if n != m:
+        raise ValueError("The input matrix is not square")
+
+    diffn = np.linalg.norm(V - np.transpose(V))
+
+    if diffn >= tol:
+        raise ValueError("The input matrix is not symmetric")
+
+    if n % 2 != 0:
+        raise ValueError("The input matrix must have an even number of rows/columns")
+
+    n = n // 2
+    omega = sympmat(n)
+    vals = np.linalg.eigvalsh(V)
+
+    for val in vals:
+        if val <= 0:
+            raise ValueError("Input matrix is not positive definite")
+
+    Mm12 = sqrtm(np.linalg.inv(V)).real
+    r1 = Mm12 @ omega @ Mm12
+    s1, K = schur(r1)
+    X = np.array([[0, 1], [1, 0]])
+    I = np.identity(2)
+    seq = []
+
+    # In what follows I construct a permutation matrix p  so that the Schur matrix has
+    # only positive elements above the diagonal
+    # Also the Schur matrix uses the x_1,p_1, ..., x_n,p_n  ordering thus I use rotmat to
+    # go to the ordering x_1, ..., x_n, p_1, ... , p_n
+
+    for i in range(n):
+        if s1[2 * i, 2 * i + 1] > 0:
+            seq.append(I)
+        else:
+            seq.append(X)
+
+    p = block_diag(*seq)
+    Kt = K @ p
+    s1t = p @ s1 @ p
+    dd = xpxp_to_xxpp(s1t)
+    perm_indices = xpxp_to_xxpp(np.arange(2 * n))
+    Ktt = Kt[:, perm_indices]
+    Db = np.diag([1 / dd[i, i + n] for i in range(n)] + [1 / dd[i, i + n] for i in range(n)])
+    S = Mm12 @ Ktt @ sqrtm(Db)
+    return Db, np.linalg.inv(S).T
+
diff --git a/python_src/xpxp_to_xxpp.py b/python_src/xpxp_to_xxpp.py
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+
+def xpxp_to_xxpp(S):
+    """Permutes the entries of the input from xpxp ordering to xxpp ordering.
+
+    Args:
+        S (array): input even dimensional square matrix or vector
+
+    Returns:
+        (array): permuted matrix or vector
+    """
+    shape = S.shape
+    n = shape[0]
+
+    if n % 2 != 0:
+        raise ValueError("The input array is not even-dimensional")
+
+    n = n // 2
+    ind = np.arange(2 * n).reshape(-1, 2).T.flatten()
+
+    if len(shape) == 2:
+        if shape[0] != shape[1]:
+            raise ValueError("The input matrix is not square")
+        return S[:, ind][ind]
+
+    return S[ind]
+
diff --git a/python_src/xxpp_to_xpxp.py b/python_src/xxpp_to_xpxp.py
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+++ b/python_src/xxpp_to_xpxp.py
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+def xxpp_to_xpxp(S):
+    """Permutes the entries of the input from xxpp ordering to xpxp ordering.
+
+    Args:
+        S (array): input even dimensional square matrix or array
+
+    Returns:
+        (array): permuted matrix or array
+    """
+    shape = S.shape
+    n = shape[0]
+
+    if n % 2 != 0:
+        raise ValueError("The input array is not even-dimensional")
+
+    n = n // 2
+    ind = np.arange(2 * n).reshape(2, -1).T.flatten()
+
+    if len(shape) == 2:
+        if shape[0] != shape[1]:
+            raise ValueError("The input matrix is not square")
+        return S[:, ind][ind]
+
+    return S[ind]
+
diff --git a/sympmat.m b/sympmat.m
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+++ b/sympmat.m
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+function omega = sympmat(n)
+    % Create a symplectic form matrix
+    % Note that it is assumed that the symplectic form is
+    %
+    %.. math:: \Omega = \begin{bmatrix}0&I\\-I&0\end{bmatrix}
+    %
+    % where :math:`I` is the identity matrix and :math:`0` is the zero matrix.
+    I = eye(n);
+    Z = zeros(n);
+    omega = [Z I; -I Z];
+end
+
diff --git a/williamson.m b/williamson.m
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+function [Db, S] = williamson(V, tol)
+    % Williamson decomposition of positive-definite (real) symmetric matrix.
+    %
+    % Args:
+    %   V (matrix): positive definite symmetric (real) matrix
+    %   tol (scalar): tolerance for symmetry check (default: 1e-11)
+    %
+    % Returns:
+    %   Db (matrix): diagonal matrix
+    %   S (matrix): symplectic matrix such that V = S' * Db * S
+
+    if nargin < 2
+        tol = 1e-11;
+    end
+
+    [n, m] = size(V);
+
+    if n ~= m
+        error('The input matrix is not square');
+    end
+
+    diffn = norm(V - V');
+    if diffn >= tol
+        error('The input matrix is not symmetric');
+    end
+
+    if mod(n, 2) ~= 0
+        error('The input matrix must have an even number of rows/columns');
+    end
+
+    n = n / 2;
+
+    omega = sympmat(n);
+
+    vals = eig(V);
+    if any(vals <= 0)
+        error('Input matrix is not positive definite');
+    end
+
+    Mm12 = real(sqrtm(inv(V)));
+    r1 = Mm12 * omega * Mm12;
+    [K, s1] = schur(r1);
+
+    X = [0 1; 1 0];
+    I = eye(2);
+    seq = cell(1, n);
+
+    % Construct a permutation matrix p
+    for i = 1:n
+        if s1(2*i-1, 2*i) > 0
+            seq{i} = I;
+        else
+            seq{i} = X;
+        end
+    end
+    p = blkdiag(seq{:});
+
+    Kt = K * p;
+    s1t = p * s1 * p;
+    s1t
+
+    dd = xpxp_to_xxpp(s1t);
+    perm_indices = xpxp_to_xxpp(1:2*n);
+    Ktt = Kt(:, perm_indices);
+
+    Db_diag = [1 ./ diag(dd(1:n, n+1:end)); 1 ./ diag(dd(1:n, n+1:end))];
+    Db = diag(Db_diag);
+
+    S = Mm12 * Ktt * sqrtm(Db);
+    S = inv(S)';
+end
+
+
diff --git a/xpxp_to_xxpp.m b/xpxp_to_xxpp.m
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+function S_permuted = xpxp_to_xxpp(S)
+    % Permutes the entries of the input from xpxp ordering to xxpp ordering.
+    %
+    % Args:
+    %   S (array): input even dimensional square matrix or a row vector
+    %
+    % Returns:
+    %   S_permuted (array): permuted matrix or vector
+
+    shape = size(S);
+    if length(shape) > 2
+        error('The input matrix is not 2 dimensional');
+    end
+
+    n = shape(2);
+
+    if mod(n, 2) ~= 0
+        error('The input array is not even-dimensional');
+    end
+
+    n = n / 2;
+    ind = reshape(1:2*n, 2, [])';
+    ind = ind(:)';
+
+    
+    if shape(1) == shape(2)
+        S_permuted = S(ind, ind);
+    else
+        S_permuted = S(ind);
+    end
+end
+
diff --git a/xxpp_to_xpxp.m b/xxpp_to_xpxp.m
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+++ b/xxpp_to_xpxp.m
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+function S_permuted = xxpp_to_xpxp(S)
+    % Permutes the entries of the input from xxpp ordering to xpxp ordering.
+    %
+    % Args:
+    %   S (array): input even dimensional square matrix or array
+    %
+    % Returns:
+    %   S_permuted (array): permuted matrix or array
+
+    shape = size(S);
+    n = shape(1);
+
+    if mod(n, 2) ~= 0
+        error('The input array is not even-dimensional');
+    end
+
+    n = n / 2;
+    ind = reshape(1:2*n, 2, [])';
+    ind = ind(:);
+
+    if length(shape) == 2
+        if shape(1) ~= shape(2)
+            error('The input matrix is not square');
+        end
+        S_permuted = S(ind, ind);
+    else
+        S_permuted = S(ind);
+    end
+end
+