Density Operator Expectation Maximization

• Density Operator Expectation Maximization (DO-EM) is a quantum adaptation of the classical EM algorithm that employs density operators to capture both classical uncertainty and quantum coherence.
• It replaces the classical E-step with a quantum information projection using techniques like the Petz Recovery Map to optimize the quantum evidence lower bound.
• DO-EM is applied in quantum state tomography and generative quantum machine learning, ensuring convergence and resource efficiency while maintaining physical constraints.

Density Operator Expectation Maximization (DO-EM) generalizes the Expectation-Maximization (EM) algorithm to quantum probability spaces. Here, the parameter of interest is a density operator—a positive semidefinite, unit-trace operator—encoding not only classical uncertainties but also genuine quantum coherence and entanglement. DO-EM is motivated by the need for scalable and principled statistical learning in quantum latent variable models (DO-LVMs), which are foundational in generative quantum machine learning and applicable across quantum state tomography, quantum-enhanced generative models, and hybrid classical-quantum architectures (Roche, 2011, Vishnu et al., 30 Jul 2025). The core contribution is a rigorous adaptation of EM to density operators, resolving the absence of a quantum conditional probability by leveraging quantum information projection, often instantiated via the Petz Recovery Map.

1. Mathematical Foundations and Generalization of EM

The classical EM algorithm is designed to maximize the likelihood function $\ell(\theta)$ in latent variable models by introducing an auxiliary function (the so-called $\mathcal{Q}$-function), underpinned by Jensen’s inequality. This yields monotonic ascent of the likelihood and is interpreted geometrically as a proximal point or minorant-maximization iteration (Roche, 2011, Chrétien et al., 2012, Hino et al., 2022). DO-EM extends this structure to the space of density operators $\rho$:

• Parameter Space: The set of $N$-dimensional density operators, i.e., $\rho \geq 0,\, \operatorname{Tr}\,\rho = 1$.
• Measurement Model: Probabilities of outcomes $y$ are given by Born’s rule,

$p(y|\rho) = \operatorname{Tr}[E_y \rho],$

with $\{E_y\}$ a POVM associated to measurement $y$.

• Likelihood: The empirical log-likelihood for frequency data $\{f_y\}$ is

$$\ell(\rho) = \sum_y f_y \log\left[\operatorname{Tr}(E_y \rho)\right].$$

In EM, the latent variable $Z$ is introduced so that optimizing the expected complete-data log-likelihood is tractable. In the density operator context, this “completion” can correspond to latent quantum degrees of freedom, hidden quantum noise, or purification variables (Roche, 2011).

2. Quantum Information Projection and the Petz Map

A central challenge in DO-EM is the absence of a well-defined quantum conditional probability—there is no general quantum analogue of classical Bayes’ rule. The solution replaces the E-step with a quantum information projection (QIP), which identifies, among all density operators with a given marginal, the one minimizing the Umegaki relative entropy to a model state $\rho(\theta)$.

• QIP Formulation: For observed data reduced density operator $\rho_{\text{target}}$, find the feasible extension $\eta$ minimizing

$$D_U(\eta, \rho(\theta)) = \operatorname{Tr}\left[\eta(\log\eta - \log\rho(\theta))\right]$$

subject to $\operatorname{Tr}_L \eta = \rho_{\text{target}}$ for the latent subsystem $L$.

• Petz Recovery Map: Under sufficiency conditions (Condition S), the solution is given by the Petz map,

$$R_\rho(\omega) = \rho^{1/2} \left[\rho_A^{-1/2} \,\omega\, \rho_A^{-1/2} \otimes I_B\right] \rho^{1/2},$$

which reconstructs the global density operator from the marginal (Vishnu et al., 30 Jul 2025).

This replacement allows the E-step to proceed in the quantum domain without invoking an ill-defined quantum conditional.

3. The Iterative Minorant-Maximization (MM) Procedure

The DO-EM algorithm alternates between a quantum E-step (via the QIP) and a quantum M-step:

• Quantum E-Step: Compute the feasible extension

$$\eta^{(t)} = \arg\min_{\eta : \operatorname{Tr}_L \eta = \rho_{\text{target}}} D_U(\eta, \rho(\theta^{(t)}))$$

via the Petz map if sufficiency holds.

• Quantum M-Step: Update parameters to maximize the quantum evidence lower bound (QELBO; Editor's term), i.e.,

$$Q(\theta; \theta^{(t)}) = \operatorname{Tr}\left[\eta^{(t)} \log\rho(\theta)\right] + S(\eta^{(t)}) - S(\rho_{\text{target}}),$$

where $S$ is the von Neumann entropy. For Hamiltonian-based models $\rho(\theta) = e^{H(\theta)}/Z(\theta)$,

$$\theta^{(t+1)} = \arg\max_{\theta} \left[\operatorname{Tr}(\eta^{(t)} H(\theta)) - \log Z(\theta)\right].$$

This MM structure ensures monotonic ascent of the log-likelihood curve at every iteration under appropriate operator-theoretic conditions, e.g., Ruskai’s condition (Vishnu et al., 30 Jul 2025). For diagonal density operators, the algorithm reduces to classical EM, recovering the classical case as a limit (Vishnu et al., 30 Jul 2025, Roche, 2011).

4. Relationship to Proximal Point and Accelerated Variants

Many EM variants for classical models (e.g., Proximal Point EM, Generalized EM, ECM) have natural quantum analogues (Roche, 2011, Chrétien et al., 2012):

• Quantum Proximal Point: EM is interpreted as a proximal-point algorithm regularized by quantum relative entropy,

$$D(\rho_1\|\rho_2) = \operatorname{Tr}[\rho_1(\log \rho_1 - \log \rho_2)].$$

The update reads

$$\rho^{(k+1)} = \arg\max_{\rho} \left\{ \ell(\rho) - \beta_k D(\rho^{(k)}\|\rho) \right\},$$

where $\beta_k$ governs the step size. As in the classical case, decreasing $\beta_k$ can accelerate convergence, sometimes reaching superlinear rates (Chrétien et al., 2012).

• Block-wise (ECM) or Incremental DO-EM: Decompose parameter updates into blocks (e.g., eigenvalues, eigenvectors) or subsets to improve convergence or computational trade-offs (Roche, 2011).

These extensions leverage the geometry of the density operator space to devise more stable or faster-converging algorithms.

5. Practical Implementation and Applications

DO-EM arises in several quantum learning contexts:

• Quantum State Tomography: The classical R$\rho$R algorithm for likelihood maximization is an instance of DO-EM, updating $\rho$ by

$$\rho^{(n+1)} = \frac{\rho^{(n)} R(\rho^{(n)})}{\operatorname{Tr}[\rho^{(n)} R(\rho^{(n)})]}$$

with $$R(\rho^{(n)}) = \sum_y \frac{f_y}{\operatorname{Tr}[E_y\rho^{(n)}]} E_y$$ (Roche, 2011).

• Generative Models: Quantum Interleaved Deep Boltzmann Machines (QiDBMs; Editor's term) employ DO-EM for scalable training on datasets such as MNIST, achieving a $40$–$60\%$ reduction in the Fréchet Inception Distance relative to larger classical Deep Boltzmann Machines, with reduced memory and runtime requirements (Vishnu et al., 30 Jul 2025).
• Density Operator Latent Variable Models: DO-EM enables log-likelihood ascent and efficient parameter learning in quantum machine learning models, while maintaining physical constraints (positivity, trace $1$).

The following table summarizes operational aspects of standard vs. quantum EM:

Property	Classical EM	DO-EM (Quantum)
Parameters	Vectors/tensors	Density operators (matrices)
E-Step	Conditional expectation	Quantum info projection (QIP)
M-Step	Max ∑Q w.r.t. $\theta$	Max quantum evidence lower bound
Regularization	Kullback-Leibler	Quantum relative entropy
Constraints	Probability simplex	Positive semidefiniteness, unit trace

6. Performance, Scaling, and Theoretical Properties

• Monotonicity and Convergence: The MM structure of DO-EM ensures non-decreasing log-likelihood under broad conditions, extending classical EM’s regularity and convergence properties (Vishnu et al., 30 Jul 2025, Roche, 2011).
• Resource Scaling: For models such as QiDBMs, the required computational resources match those of comparably-sized classical DBMs; in practice, superior sample quality is obtained with fewer parameters (Vishnu et al., 30 Jul 2025).
• Physical Consistency: The quantum formulation strictly enforces positivity and normalization, crucial for interpretability and validation in quantum state and generative learning.

7. Future Directions and Open Problems

DO-EM opens several directions in quantum machine learning and quantum-enhanced generative modeling:

• Implementations on Quantum Hardware: While current results rely on classical simulation, the MM/QIP framework is compatible with near-term quantum processors.
• Infinite-dimensional Extensions: Open questions remain regarding continuous-variable systems or infinite-dimensional Hilbert spaces.
• Generalized Operator Projections: Conditions ensuring the optimality of the Petz map (e.g., sufficiency, data-processing equality) may be relaxed or extended, broadening the applicability of DO-EM (Vishnu et al., 30 Jul 2025).

A plausible implication is that certain quantum models may even outperform larger classical models in generative and inference tasks, suggesting new paradigms for learning in both quantum and hybrid quantum-classical computing environments.

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In summary, Density Operator Expectation Maximization (DO-EM) is a rigorous generalization of EM to quantum probability models, resolving the lack of conditional probabilities through quantum information projection, typically via the Petz map. This adaptation preserves the convergence and monotonic ascent properties of classical EM while enabling scalable, physically consistent learning in generative quantum and hybrid models (Roche, 2011, Vishnu et al., 30 Jul 2025). As demonstrated in applications to QiDBMs, DO-EM provides practical advantages over classical approaches and forms a foundation for further research in quantum statistical inference and machine learning.