Matrix Product States as Observations of Entangled Hidden Markov Models

TL;DR

The paper reveals a rigorous link between Matrix Product States (MPS) and Entangled Hidden Markov Models (EHMMs), showing that a broad class of MPS can be interpreted as partial observations of EHMMs and that every MPS admits a canonical EHMM representation. By formulating EHMMs in the Schrödinger picture and deriving a gauge-consistent mapping between MPS tensors and EHMM partial isometries, the authors establish two key results: (i) a direct construction of EHMMs from MPS (Theorem 2), and (ii) a bidirectional relationship between MPS and EHMMs via partial measurements (Theorem 1). They also derive a quantitative bound on the relative entropy $S( ho_N \| \rho_{O;N})$ between the MPS density operator and the EHMM observation density, providing an information-theoretic measure of their divergence. The framework is illustrated with GHZ, cluster, and AKLT states, highlighting when MPS can be viewed as EHMM observations and when EHMMs can be naturally associated to MPS, thereby connecting tensor-network methods to quantum stochastic processes with potential implications for entanglement analysis and state compression.

Abstract

This paper reveals the intrinsic structure of Matrix Product States (MPS) by establishing their deep connection to entangled hidden Markov models (EHMMs). It is demonstrated that a significant class of MPS can be derived as the outcomes of EHMMs, showcasing their underlying quantum correlations. Additionally, a lower bound is derived for the relative entropy between the EHMM-observation process and the corresponding MPS, providing a quantitative measure of their informational divergence. Conversely, it is shown that every MPS is naturally associated with an EHMM, further highlighting the interplay between these frameworks. These results are supported by illustrative examples from quantum information, emphasizing their importance in understanding entanglement, quantum correlations, and tensor network representations.

Figures (3)

• Figure 1: Hidden Markov Model (HMM) for the GHZ state with an identity transition matrix $\Pi = I_2$ and identity observation matrix $Q = I_2$. Each hidden state remains unchanged ($e_0$ stays as $e_0$, and $e_1$ stays as $e_1$), and each deterministically maps to its corresponding observable state ($e_0 \rightarrow |0\rangle$, $e_1 \rightarrow |1\rangle$).
• Figure 2: Hidden Markov Model (HMM) for the 1D cluster state with hidden transition matrix $\Pi = \frac{1}{2} 1111$ between the two states $e_0$ and $e_1$ and an identity emission matrix $Q = I_2$. Each hidden state deterministically emits a corresponding observable state, meaning $e_0$ always maps to $|0\rangle$ and $e_1$ always maps to $|1\rangle$. The transitions between hidden states occur with equal probability $\frac{1}{2}$.
• Figure 3: This diagram represents the HMM for an AKLT state. The hidden states $e_1$ and $e_2$ transition with probabilities as described by the matrix $\Pi$, shown in blue to match the hidden states. The observations $|+\rangle$, $|0\rangle$, and $|-\rangle$ are emitted with probabilities given by the matrix $Q$, highlighted in orange for distinction.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof : Proof Sketch
• Lemma 3.4
• proof
• Theorem 3.5