Imaginarity-Free Multi-States in Quantum Resource Theory

• Imaginarity-free multi-states are defined as quantum state sets that can be entirely represented using real-valued coefficients, ensuring the absence of quantum imaginary features.
• They are characterized using geometric (coplanar Bloch vectors), algebraic (Gram matrix rank conditions), and invariant-based (Bargmann invariants) methods to detect non-real overlaps.
• Their resource-theoretic framework enables practical quantum operations and experimental protocols in metrology and information processing, while highlighting trade-offs such as nonconvexity.

The term "imaginarity-free multi-states" refers to collections of quantum states (pure or mixed), typically considered in the context of quantum resource theory, that can be simultaneously represented with real-valued coefficients in some fixed orthonormal basis. This property signals the absence of genuinely quantum-imaginary features across the set and is relevant in foundational investigations, operational protocols involving only real quantum mechanics, the structure of resource theories, and in experimental simplicity. Modern research has delivered rigorous characterizations, resource-theoretic frameworks, operational protocols, and geometric and invariant-based diagnostic tools for imaginarity-free multi-states, especially for qubit systems and Gaussian states.

1. Resource-Theoretic Foundations and Definition

The resource theory of imaginarity formalizes "free" states and operations as those that do not introduce or require imaginary components in a fixed basis (Hickey et al., 2018, Wu et al., 2020, Wu et al., 2021, Wu et al., 2023, Diaz et al., 14 Jul 2025, Deneris et al., 15 Jul 2025). Explicitly, the set of imaginarity-free states consists of those density matrices with real matrix elements: $$\rho = \sum_{jk}\rho_{jk}|j\rangle\langle k|,\quad \rho_{jk}\in\mathbb{R}$$ These states are invariant under transposition and, in the pure-state case, may be generated from a reference real-valued state (such as $|0^{\otimes n}\rangle$) via the action of the real orthogonal group $O(d)$: $$S_{\mathrm{imag}} = \{ U|0^{\otimes n}\rangle : U\in O(2^n) \}$$ The corresponding "free" operations in this resource theory are channels whose Kraus operators are real matrices (i.e., the operation is covariant under the action of complex conjugation in the fixed basis), and the free unitaries are the real orthogonal unitaries up to a global phase (Hickey et al., 2018).

In McMillan's approach (Diaz et al., 14 Jul 2025), imaginarity-free quantum resource theory is described as resulting from the choice of an algebraic structure $\mathcal{E}$ corresponding to the Lie algebra of real Hermitian matrices, with free operations defined as automorphisms of $\mathcal{E}$. In this formalism, both deterministic orthogonal evolutions and more general "complexified" stochastic resource-non-increasing operations preserve the real-valuedness of the set (Diaz et al., 14 Jul 2025).

2. Geometric and Algebraic Structure

For collections ("multi-states") of qubit states, the most natural geometric characterization is provided in terms of their Bloch vectors. A (set of) qubit state(s) $\rho_k$ is imaginarity-free if and only if all the corresponding Bloch vectors $\vec{r}_k$ are coplanar in $\mathbb{R}^3$ (Li et al., 20 Jul 2025): $$\exists \, \vec{p} \in \mathbb{R}^3,\, \|\vec{p}\|=1 : \forall k,\, \langle \vec{r}_k, \vec{p} \rangle = 0$$ Equivalent algebraic tests are given by considering the Gram matrix $G_{r} = (\langle \vec{r}_k,\vec{r}_l\rangle)_{kl}$: $\mathrm{rank}(G_{r})\leq 2$ if and only if the multi-state set is imaginarity-free (up to global unitary rotation).

In higher dimensions, the condition generalizes but is less tractable geometrically. The resource theory remains well-posed via the structure of real density matrices and operations, or, for multipartite systems, via antiunitary symmetries that act as global conjugation (Miyazaki et al., 2020).

3. Invariant-Based Diagnostics: Bargmann Invariants and Relational Tests

Unitary-invariant and basis-independent detection of imaginarity in multi-state collections relies on higher-order Bargmann invariants $\Delta_n$ (Oszmaniec et al., 2021, Fernandes et al., 22 Mar 2024, Li et al., 11 Dec 2024, Li et al., 20 Jul 2025): $$\Delta_n = \langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle$$ For single-qubit states, all such invariants (up to complex conjugation) are determined by pairwise overlaps—the relational information encoded in the Gram matrix—implying that imaginarity can be fully witnessed by these invariants for up to three states, and, in the case of four or greater, nonreal overlaps witness the failure of a common real-basis representation (Fernandes et al., 22 Mar 2024, Li et al., 11 Dec 2024).

The presence of a nonzero imaginary part of $\Delta_n$ is necessary and sufficient for the set to lack a real-valued basis representation, thus serving as a robust, device-independent witness for multi-state imaginarity and enabling practical measurement protocols (e.g., extensions of SWAP test circuits) for both pure and mixed states.

4. Resource Measures and Conversion

The degree of imaginariy in single or multi-states is quantified via several resource monotones:

• Robustness of imaginarity: $I_R(\rho) = (1/2)\|\rho-\rho^T\|_1$;
• Geometric imaginarity: $I_g(\psi) = (1 - |\langle \psi^*|\psi\rangle|)/2$ for pure states;
• Fidelity of imaginarity distillation: $F_I(\rho) = (1 + I_R(\rho))/2$;
• Measures for Gaussian states: $\mathcal{I}^{G_n}$, based on block-determinants and displacement vectors, efficiently characterizes imaginarity in multimode continuous-variable systems (Zhang et al., 10 Apr 2025).

For collections, the average or maximal robustness is minimized over all possible real-basis representations; in qubit multi-states, lower and upper bounds in terms of the smallest eigenvalue of the Gram matrix are provided (Li et al., 20 Jul 2025).

The resource theory's structure allows for conversion and distillation protocols. For example, in distributed scenarios, imaginarity can be distilled or concentrated on one subsystem using only real operations and classical communication, with tight analytic expressions for achievable fidelity given the initial shared state (Wu et al., 2023).

5. Operational Contexts and Experimental Implications

Imaginarity-free multi-states possess operational and conceptual importance:

• Quantum Metrology: Real density matrices admit target-independent optimal measurements and allow all symmetric logarithmic derivatives to be chosen real, yielding compatible multi-parameter estimation strategies (Miyazaki et al., 2020).
• Optical Implementation: Preparation and manipulation of imaginarity-free (real) quantum states require fewer adjustable parameters or devices in experimental settings, notably reducing the number of unfixed wave plates in linear-optics circuits (Wu et al., 2021). Additionally, in interferometric tasks, the distinction between imaginarity-free and generic quantum states directly affects phenomena such as multi-photon indistinguishability (Oszmaniec et al., 2021).
• Quantum Information Processing: In distributed scenarios (e.g., LOCC-limited discrimination protocols), the presence or absence of imaginarity resources serves as a strict criterion for the operational distinguishability of states or channels (Wu et al., 2023, Wu et al., 2020).
• Quantum Control and Dynamics: Markovian channels typically drive non-initially optimal states towards the boundary of imaginarity-mixedness trade-off, offering possible strategies for engineering or erasing imaginarity in high-dimensional or noisy contexts (Chen et al., 25 Apr 2024, Shi, 22 Jan 2025).

6. Mathematical Properties and Convexity

Unlike many other quantum resource-free sets (e.g., incoherent or separable states), the set of imaginarity-free multi-states is not convex. That is, convex combinations ("classical mixtures") of imaginarity-free ensembles can yield collective state-sets not jointly representable in a real basis, as their constituent Bloch vectors may span a three-dimensional region after mixing, breaking coplanarity (Li et al., 20 Jul 2025). This has implications for state engineering and for the design of resource theories in multi-state scenarios, limiting the effectiveness of simple convexity-based resource witnesses.

In the multipartite and multi-qudit case, similar issues are resolved via polynomial or Boolean function-based constructions (e.g., permutation-symmetric Dicke states or balanced states with functional dependencies) characterized by their real amplitude expansions in the $Z$-basis, which achieve imaginarity freedom while maintaining the operational strength (e.g., logical contextuality or nonlocality) of the resource (Abramsky et al., 2014).

7. Connections to Coherence and Other Quantum Resources

Imaginarity is strictly a subset of coherence: a nonzero imaginary part implies nonzero coherence in any fixed basis. Coherence measures (e.g., $l_1$-norm or relative entropy) are always at least as large as their restriction to the real part; i.e., $C(\rho) - C(\mathrm{Re}\,\rho) \geq 0$ for measures $C$ invariant under complex conjugation (Xu, 9 Apr 2024). Imaginarity-free states are thus characterized by the condition $C(\rho) = C(\mathrm{Re}\,\rho)$ for all such coherence measures.

In cross resource-theory analysis, imaginarity-free states, while resource-free for imaginarity, may be highly resourceful for other quantum resources (such as entanglement or spin coherence), a fact quantitatively explored through group-theoretical averages and numerical simulation (Deneris et al., 15 Jul 2025).

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In summary, imaginarity-free multi-states are rigorously characterized by real-valuedness in a global basis, with necessary and sufficient conditions provided in geometric, algebraic, and invariant-theoretic terms. Their structure underpins broad operational implications in metrology, computation, and experimental design, while also presenting distinctive mathematical properties (e.g., nonconvexity in the multi-state set). Recent work continues to elucidate the interplay of imaginarity with coherence, mixedness, and other quantum resources, establishing imaginarity as a primary and quantifiable resource in the modern landscape of quantum theory.