Imaginarity measures induced by real part states and the complementarity relations

Technical Summary: Imaginarity Measures Induced by Real Part States and Complementarity Relations

Problem Statement
Complex numbers are fundamental to quantum mechanics, yet the resource theory of "imaginarity"—quantifying the non-vanishing imaginary components of a quantum state's density matrix—has only recently been formulated. While several imaginarity measures exist (e.g., trace norm, relative entropy, geometric), they are often defined via optimization problems or specific distance metrics. A central question addressed in this work is whether the "real part state," defined as $\text{Re}(\rho) = \frac{1}{2}(\rho + \rho^*)$, can serve as a general foundation for constructing imaginarity measures. Specifically, the authors investigate if the distance between a state $\rho$ and its real part $\text{Re}(\rho)$ constitutes a valid resource measure and explore the physical constraints (complementarity) imaginarity faces across mutually unbiased bases (MUBs).

Methodology
The paper proposes a general framework for constructing imaginarity measures based on the distance between a quantum state $\rho$ and its real part state $\text{Re}(\rho)$.

1. General Construction: The authors define a function $D(\rho, \sigma)$ satisfying non-negativity, contractivity under completely positive trace-preserving (CPTP) maps, and additivity under direct sums. They prove that $M(\rho) = D(\rho, \text{Re}(\rho))$ satisfies the five axioms of an imaginarity measure (Faithfulness, Monotonicity, Probabilistic Monotonicity, Convexity, and Additivity).
2. Specific Measure Proposal: By selecting the function $D(\rho, \sigma) = 1 - F(\rho, \sigma)$, where $F$ is the fidelity, the authors introduce a new measure:
   $$M_{\text{Re}}(\rho) = 1 - F(\rho, \text{Re}(\rho))$$
3. Analytical Derivation: The paper derives the closed-form analytical expression for $M_{\text{Re}}$ in qubit systems using the Bloch vector representation.
4. Comparative Analysis: The new measure is compared against existing measures (geometric, Tsallis relative entropy, trace norm) to establish inequalities and relationships.
5. Complementarity Analysis: The study investigates the sum of imaginarity measures under complete sets of MUBs in low-dimensional systems (qubits and qutrits) to determine physical constraints on the distribution of imaginarity.

Key Contributions and Results

• General Construction Method: Theorem 1 establishes that any distance function $D$ satisfying specific physical conditions (non-negativity, contractivity, additivity) yields a valid imaginarity measure when applied to the pair $(\rho, \text{Re}(\rho))$. This unifies existing measures like trace norm, relative entropy, and Tsallis relative entropy as specific instances of this framework.
• New Fidelity-Based Measure ($M_{\text{Re}}$): The authors propose $M_{\text{Re}}$ and prove it is a well-defined measure. Key properties include:
  • Bounds: $0 \leq M_{\text{Re}}(\rho) \leq 1 - \frac{\sqrt{2}}{2}$. The upper bound is attained by the maximally imaginary state $|+i\rangle = \frac{|0\rangle + i|1\rangle}{\sqrt{2}}$.
  • Pure States: For pure states $|\psi\rangle$, $M_{\text{Re}}(|\psi\rangle) = 1 - \frac{1}{\sqrt{2}}\sqrt{1 + |\langle\psi|\psi^*\rangle|^2}$.
  • Qubit Analytical Formula: An explicit formula is derived for qubit states $\rho = \frac{1}{2}(I + \mathbf{r} \cdot \boldsymbol{\sigma})$, showing that imaginarity increases with the $y$-component of the Bloch vector ($r_y$).
• Relations to Other Measures:
  • Geometric Imaginarity ($M_g$): The paper establishes the inequality $M_{\text{Re}}(\rho) \leq M_g(\rho) \leq 1 - [1 - M_{\text{Re}}(\rho)]^2$. Equality holds if and only if $\rho$ is a real state or the optimal real state for $M_g$ coincides with $\text{Re}(\rho)$.
  • Optimal Real State: The authors determine that for geometric imaginarity in qubit systems, the optimal real state is generally not $\text{Re}(\rho)$, except in specific cases (e.g., when $r_x=r_z=0$ or $r_y=0$).
  • Trace Norm and Purity: A bound relating $M_{\text{Re}}$, trace norm imaginarity ($M_{\text{tr}}$), and purity $P(\rho)$ is derived: $[1 - M_{\text{Re}}(\rho)]^2 + M_{\text{tr}}(\rho)^2 \geq P(\rho)$.
• Complementarity Relations:
  • Qubits: For any qubit state, the imaginarity measures under the three Pauli eigenbases ($B_1, B_2, B_3$) satisfy a complementarity relation bounded by the Bloch vector length $|\mathbf{r}|$. For pure states, this relation becomes an equality.
  • Qutrits: For qutrits, a complementarity relation is established for a complete set of four MUBs, showing that the sum of squared deviations from unity is constrained by the state's purity: $\sum_{k=1}^4 (1 - M_{\text{Re}}^{Z_k})^2 > \frac{31}{14}P(\rho)$.

Significance and Claims
The paper claims to highlight the "prominent role" of the real part state $\text{Re}(\rho)$ in imaginarity resource theory, demonstrating that it essentially characterizes the resource. By providing a method to construct measures without optimization (unlike the geometric measure's initial definition), the work offers a more computationally convenient approach.

Furthermore, the derived complementarity relations reveal a physical constraint: the amount of imaginarity a quantum state possesses cannot be simultaneously large across a complete set of mutually unbiased bases. This dependence on the measurement basis underscores the intrinsic nature of imaginarity as a quantum resource. The work concludes that these findings deepen the interpretation of real part states in characterizing quantum states and elucidate the constraints imaginarity faces in low-dimensional systems.