(restricted) Defect of Unitary Matrix

description

Defect of a unitary matrix U gives an upper bound for the dimensionality of a smooth orbit of inequivalent unitary matrices stemming from U. Roughly speaking, two matrices are said to be equivalent if one can be transformed into the other via special unitary operations. In particular it may be used as a one-way criterion

if defect of a matrix is zero then this matrix is an isolated point

which means that in the neighbourhood of a given matrix there are no other inequivalent matrices. This algebraic tool proves to be very useful when searching for potential candidates for families in the set of complex Hadamard matrices.

One can also define restricted defect for a unitary matrix U subjected to additional constraints:

• U is hermitian
• U has constant (and real) diagonal
• U has constant off-diagonal moduli

Motivation behind considering restricted defect lies in the idea of generalised quantum measurement POVM and associated Gram matrices. It turns out that Gram matrix can be expressed as a unitary and hermitian object with constant diagonal, so that the defect is applicable to it. Given a POVM with predefined geometrical structure - either SIC-POVM or MUB, one can ask if it is possible to extend corresponding objects so that a smooth family of quantum measurements exists.

implementation

Defect can be calculated in many ways. Let R be a matrix of a special system of linear equations associated with matrix U [1]. Here we present three possible implementations, appropriately named:

• 'R' - defect of U as the rank of R
• 'S' - defect of U as a function of non-zero singular values of R
• 'T' - defect of U as the dimension of the image of a tangent space to the manifold of unitaries under a certain tangent map... [2]

Methods 'R' and 'T' work for matrices given with the highest possible numerical precision. If matrix U is provided only in approximate form one can try to use the 'S' method. However, a special attention is needed when setting SV_TOLERANCE - a kind of threshold to distinguish between "zero" and "non-zero" singular values of the matrix R!

usage

>> defect_u(U, [, METHOD [, SV_TOLERANCE]])

>> defect_u(U)             % implicit call of 'R' method (default)
>> defect_u(U, 'R')        % explicit call of 'R' method
>> defect_u(U, 'S')        % SVD with default SV_TOLERANCE
>> defect_u(U, 'S', 1e-12) % SVD with custom SV_TOLERANCE
>> defect_u(U, 'T')        % method of tangent spaces...
>> defect_u(U, 'R', 1e-12) % SV_TOLERANCE is ignored with 'R'
>> defect_u(U, 'T', 1e-12) % SV_TOLERANCE is ignored with 'T'

defect_h.m works similarly...

Mathematica users can find two notebooks: defect_u.nb.txt and defect_h.nb.

references

• ^[1] W. Bruzda, D. Goyeneche, K. Życzkowski, "Quantum measurements with prescribed symmetry", Phys. Rev. A 96, 022105 (2017)
• ^[2] W. Tadej, K. Życzkowski, "Defect of a Unitary Matrix", Lin. Alg. Appl., 429, pp. 447-481 (2008)