Phenomenological Distinction Between Unitary and Antiunitary Symmetry Operators

Francisco Marcelo Fernández, Javier Garcia

Jos� F. Cari�ena, Mariano Santander

Eugene Paul Wigner

It is well known that one always can find as many orthogonal states (i.e., states between which the transition probability is zero) as the Hilbert space has dimensions which are invariant under a given unitary transformation. The corresponding vectors are characteristic vectors of the unitary operator. In contrast, most antiunitary operators leave not more than one state invariant. However, if there are two orthogonal invariant states, a consideration of the states for which the transition probability is ½ into both invariant states surely provides a distinction. In the antiunitary case, one of these states is also invariant, another one is transformed into an orthogonal state, the rest are in between. In the unitary case, the transition probability between original state and transformed state is the same for all states for which the transition probability is ½ into two orthogonal states. This provides a ``directly observable'' distinction between unitary and antiunitary transformations.

Akihiro Hokkyo, Hiroyasu Tajima

Symmetry imposes fundamental constraints on quantum measurement and control. The Wigner-Araki-Yanase theorem and its quantitative extensions capture this restriction for continuous symmetries, in terms of fluctuations of conserved generators. Such generator-based bounds, however, do not provide quantitative limitations for discrete unitary symmetries or for antiunitary symmetries. Here we establish quantitative WAY-type theorems for symmetry-breaking projective measurements and unitary gates under arbitrary unitary and antiunitary symmetries. Our approach is based on a two-target no-programming inequality: if a single processor approximately implements two operations that amplify distinguishability, then the corresponding program states must themselves be distinguishable. Applied to symmetric implementations, this converts the error of an asymmetric measurement or gate directly into a lower bound on the asymmetry of the apparatus state, quantified by its fidelity with its symmetry-transformed copy. Our results apply to discrete and antiunitary symmetries, thereby providing a fundamental limit for symmetry-limited quantum measurement and control beyond the continuous-symmetry regime.

L. Deák

Symmetry transformations are defined by operators in quantum mechanics that preserve the modulus of the scalar product between Hilbert space vectors. According to Wigner's theorem, any such transformation is represented by either a unitary linear operator or an antiunitary (isometric conjugate-linear) operator. Although antiunitary symmetries-most notably time reversal and charge conjugation-are encountered less frequently than unitary ones, they are fundamental to the description of non-conservative and reversible systems. The most frequently treated antiunitary operators are the involutive ones, called conjugations. Any antiunitary operator can be written as a product of a conjugation and a unitary operator. Considering general scattering problems defined by a scattering potential and separating conjugation from the symmetry operator, one can find the role of complex symmetric (in other words self-transpose) unitary operators in physical problems. This approach provides a robust framework for analyzing the role of non-Hermitian symmetries in wave scattering and dissipative dynamics. To demonstrate the practical applicability of these theoretical concepts, we analyze the case of polarized neutron reflectometry (PNR). We show that the scattering potential in PNR, comprising nuclear and magnetic terms, can satisfy the condition of being unitarily equivalent to its transpose, thereby guaranteeing reciprocity under specific orientations of the magnetic field.

Atri Mukhopadhyay