High-Order Quantum Coherences

• High-order quantum coherences are multi-particle correlations defined by the decomposition of the density matrix into coherence-order blocks, revealing off-diagonal phase relationships.
• They integrate advanced metrics like α-relative purity, Wigner–Yanase–Dyson skew information, and quantum Fisher information to quantify coherence and entanglement.
• Experimental implementations in NMR, trapped-ion systems, and ultracold gases demonstrate practical methods for probing dynamic quantum processes and metrological advantages.

High-order quantum coherences generalize the concept of single-particle phase coherence to multi-particle or multi-mode quantum systems, encapsulating the structure of off-diagonal density matrix elements and their evolution under interactions and measurements. Originally formalized in the context of quantum optics and nuclear magnetic resonance, high-order (or multiple quantum) coherences now play a foundational role in quantum metrology, quantum information, and many-body physics. Their contemporary mathematical framework connects relative Rényi entropies, Wigner-Yanase-Dyson skew information, and quantum Fisher information, providing a set of intensity observables ("α-multiple quantum coherences") that quantify the distribution and dynamics of quantum correlations across coherence orders. These tools afford both rigorous theoretical and experimentally practical quantification of coherence resources and entanglement in a broad range of physical platforms (Pires et al., 2020).

1. Definitions: α-MQCs, α-Relative Purity, and Coherence Order Decomposition

High-order quantum coherences are encoded in the off-diagonal blocks of the density operator with respect to a reference observable $A$ (with nondegenerate, integer-spaced eigenvalues $\lambda_\ell$). The quantum Rényi relative entropy of order $\alpha \in (0,1)$ between two states $\rho$ and $\sigma$ is given by: $$D_\alpha(\rho\|\sigma) = \frac{1}{\alpha - 1}\ln\, f_\alpha(\rho, \sigma)$$ where the α‐relative purity is

$$f_\alpha(\rho, \sigma) \equiv \mathrm{Tr}\Big[ \rho^\alpha\, \sigma^{1-\alpha} \Big]$$

In phase encoding, one considers $\rho_\phi = e^{-i\phi A}\rho e^{i\phi A}$.

To dissect the coherence content, the fractional-power state $\rho^{(\alpha)} = c_\alpha \rho^\alpha$ (with $c_\alpha = 1/ \mathrm{Tr}\, \rho^\alpha$) is decomposed into coherence-order blocks: $$\rho^{(\alpha)} = \sum_{m \in \mathbb{Z}} \rho^{(\alpha)}_m$$ where $$\langle j | \rho^{(\alpha)}_m | \ell \rangle \neq 0$$ only if $\lambda_j - \lambda_\ell = m$. Each block transforms under $A$ as $e^{-im\phi}$, and the blocks are Hilbert–Schmidt orthogonal.

The α-multiple quantum intensity (α-MQI, or α-MQC) for order $m$ is defined by

$$I_m^\alpha(\rho) \equiv \mathrm{Tr}\left[ (\rho^{(\alpha)}_m)^\dagger\, \rho^{(1-\alpha)}_m \right]$$

so that the α-relative purity is

$$f_\alpha(\rho, \rho_\phi) = (c_\alpha c_{1-\alpha})^{-1} \sum_{m \in \mathbb{Z}} e^{-im\phi} I_m^\alpha(\rho)$$

The set $\{I_m^\alpha\}$ forms the α-MQC spectrum, resolving how coherences of different orders contribute to the quantum state purity landscape (Pires et al., 2020).

2. Connection to Wigner–Yanase–Dyson Skew Information

The α-MQC spectrum is tightly linked to the Wigner–Yanase–Dyson skew information (WYDSI)

$$\mathcal{I}_\alpha(\rho, A) = -\frac{1}{2}\mathrm{Tr} \left[ [A, \rho^\alpha][A, \rho^{1-\alpha}] \right] = \mathrm{Tr}(\rho A^2) - \mathrm{Tr}(\rho^\alpha\, A\, \rho^{1-\alpha} A)$$

which characterizes quantum asymmetry and quantifies sensitivity to phase shifts generated by $A$. Perturbatively,

$$D_\alpha(\rho\|\rho_\phi) = -\frac{\phi^2}{\alpha - 1} \mathcal{I}_\alpha(\rho, A) + O(\phi^3)$$

The second moment of the α-MQC spectrum,

$$F_I^\alpha(\rho, A) \equiv 2\sum_{m \in \mathbb{Z}} m^2 I_m^\alpha(\rho)$$

satisfies

$$F_I^\alpha(\rho, A) = 4 c_\alpha c_{1-\alpha} \mathcal{I}_\alpha(\rho, A)$$

WYDSI thus quantifies the extent to which high-order coherences (large $|m|$) are present, providing a direct operational bridge between the coherence spectrum and the observable consequences of asymmetry (Pires et al., 2020).

3. Hierarchy of Bounds and Quantum Fisher Information

A chain of information-theoretic bounds links the α-MQC second moment to classical commutator information, WYDSI, α-variance, and the quantum Fisher information (QFI). Key bounds include:

• Lower bound via commutator metric:

$$F_I^\alpha(\rho, A) \geq 8\,\alpha(1-\alpha)\,c_\alpha c_{1-\alpha}\, \mathcal{I}^L(\rho, A)$$

where $$\mathcal{I}^L(\rho, A) = -\frac{1}{4}\mathrm{Tr}([ \rho, A ]^2)$$.

• For any $\alpha$,

$$2\alpha(1-\alpha) \mathcal{I}^L(\rho, A) \leq F_I^\alpha/(4 c_\alpha c_{1-\alpha}) \leq \mathcal{I}_{1/2}(\rho, A) \leq \mathcal{V}_{1/2}(\rho, A)$$

where $\mathcal{I}_{1/2}(\rho, A)$ is the standard WYDSI and $\mathcal{V}$ denotes variances.

• The quantum Fisher information (for estimation of $\phi$ generated by $A$)

$$\mathcal{F}_Q(\rho, A) = \frac{1}{2} \sum_{j,\ell} \frac{(p_j - p_\ell)^2}{p_j + p_\ell} | \langle \psi_j|A|\psi_\ell \rangle |^2$$

is bounded below by

$$F_I^\alpha(\rho, A)/(4 c_\alpha c_{1-\alpha}) \leq \mathcal{F}_Q(\rho, A)$$

Thus, the experimentally accessible α-MQC spectrum provides a robust lower bound to QFI, and hence to metrological usefulness and certifiable multiparticle entanglement (Pires et al., 2020).

4. Analytic Examples and Scaling Behavior

Closed-form evaluations of α-MQC spectra and their second moments are available for diverse classes of quantum states:

• Single-qubit mixed states: For $\rho = (1/2)(I + \vec{r}\cdot \vec{\sigma})$, and $A=(1/2)\vec{n}\cdot \vec{\sigma}$, the only nonzero $\alpha$-MQIs are for $m=0, \pm 1$. The second moment scales as $$F_I^\alpha(\rho, A) = (2c_\alpha c_{1-\alpha} - 1)[1-(\vec{n}\cdot \hat{r})^2]$$.
• Bell-diagonal two-qubit states: For $\rho_{BD}$, $A = \vec{n} \cdot (\vec{S}_1 + \vec{S}_2)$, coherence orders $m \in \{0, \pm1, \pm2\}$ occur, and $F_I^\alpha$ is a function of state parameters $\{a_j\}$, mixing weights, and direction $\vec{n}$.
• Prototypical N-qubit mixed states:

$$\rho = (1 - p)/d \cdot I + p |\psi\rangle\langle\psi|, \quad d = 2^N$$

For $|\psi\rangle = |+\rangle^{\otimes N}$, $|\mathrm{GHZ}_N\rangle$, or $|\mathrm{W}_N\rangle$, the second moment $F_I^\alpha$ scales as $N$, $N^2$, or $\sim N$ respectively, with explicit formulas contingent on $\alpha$ and $p$. This hierarchy directly reflects how multiparticle entanglement and coherence order grow with system size and mixing (Pires et al., 2020).

5. Dynamics under Many-Body Ising Evolution and Loschmidt Echo Protocols

The build-up and decay of high-order coherences are naturally probed through time-reversal dynamics in fully connected Ising models. The protocol consists of:

1. Preparing a (possibly mixed) state $\rho_0$,
2. Forward evolution under $H_{zz} = (J/N)\sum_{j<k} \sigma_j^z \sigma_k^z$,
3. A global phase rotation $R_\phi = e^{-i\phi S_x}$,
4. Backward evolution under $H_{zz}$.

The output state $\rho_f$ and the time-evolved $\rho_t$ feature in the α-relative purity revival signal

$$f_\alpha(\rho_0, \rho_f) = (c_\alpha c_{1-\alpha})^{-1} \sum_m e^{-im\phi} I_m^\alpha(\rho_t)$$

Time-resolved measurements of $f_\alpha(\phi)$ Fourier-resolve dynamic changes in the α-MQC spectrum $I_m^\alpha(\rho_t)$. For example, for $N=4,5$, density plots show that high-order coherences ($|m| > 0$) develop and decay in periodic oscillations, mirroring the predicted entanglement and phase scrambling/refocusing characteristic of many-body quantum dynamics (Pires et al., 2020).

6. Experimental Realization and Relevance

Measurement of high-order quantum coherences is supported across several platforms:

• NMR systems: Standard multiple quantum coherence (MQC) spectroscopy maps directly onto the α-MQC formalism. Modifications of conventional pulse sequences and readouts (e.g., SWAP test for $\alpha=2$) permit access to the full spectrum.
• Trapped-ion simulators: All-to-all Ising interactions and time reversals are operationally accessible; quantum-fidelity revival protocols ($\alpha=1/2$) have already been leveraged for studying many-body localization and scrambling.
• Ultracold atomic gases: Techniques for measuring higher Rényi entropies (randomized measurements, projective basis statistics) generalize to α-MQC assessment, supporting quantum simulation and thermalization studies.
• Quantum metrology: Because $F_I^\alpha/(4 c_\alpha c_{1-\alpha})$ lower bounds QFI, high-order coherence measurements provide robust certification of metrological advantage and multipartite entanglement without requiring full state tomography.

These experimental capacities confirm the role of high-order quantum coherences as universal witnesses of quantum resources and dynamical complexity (Pires et al., 2020).

7. Unified Framework and Implications

The α-MQC formalism generalizes traditional coherence and out-of-time-ordered correlator (OTO-C) schemes to arbitrary Rényi orders. This unification anchors the operational interchange between quantum asymmetry, coherence spectrums, and quantum metrological advantage. By directly relating experimentally accessible high-order coherence intensities to both WYD skew information and QFI bounds, the formalism provides new tools and witnesses for entanglement, phase sensitivity, and thermalization across platforms. It also enables systematic exploration of Rényi-family resource theories and entropic phase estimation strategies in many-body quantum simulators (Pires et al., 2020).