Geometry of Kirkwood–Dirac classical states: a case study based on discrete Fourier transform

Abstract The characterization of Kirkwood–Dirac (KD) classicality or non-classicality is very important in quantum information processing. In general, the set of KD classical states with respect to two bases is not a convex polytope (Langrene et al 2024 J. Math. Phys. 65 072201), which makes us interested in finding out in which circumstances they do form a polytope. In this paper, we focus on the characterization of KD classicality of mixed states for the case where the transition matrix between two bases is a discrete Fourier transform (DFT) matrix in Hilbert space with dimensions p 2 and pq , respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math> are prime. For the two particular cases we investigate, the sets of extremal points are finite, implying that the set of KD classical states we characterize forms a convex polytope. We show that for p 2 dimensional system, the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>KD</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi class="MJX-tex-calligraphic">A</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi class="MJX-tex-calligraphic">B</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msubsup></mml:mrow></mml:math> is a convex hull of the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi>pure</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>KD</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi class="MJX-tex-calligraphic">A</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi class="MJX-tex-calligraphic">B</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> based on DFT, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>KD</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi class="MJX-tex-calligraphic">A</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi class="MJX-tex-calligraphic">B</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msubsup></mml:mrow></mml:math> is the set of KD classical states with respect to two bases and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi>pure</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>KD</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi class="MJX-tex-calligraphic">A</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi class="MJX-tex-calligraphic">B</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> is the set of all the rank-one projectors of KD classical pure states with respect to two bases. In pq dimensional system, we believe that this result also holds. Unfortunately, we do not completely prove it, but some meaningful conclusions are obtained about the characterization of KD classicality.