Inequivalent Purifications of Thermal Spectra

• The paper demonstrates that a thermal density matrix can be purified into infinitely many pure states, each exhibiting unique entanglement structures and operational behaviors.
• It details canonical and alternative purifications, such as the thermofield double and thermal vacuum, which differ in geometric phases and computational complexities.
• The discussion highlights tensor network methods and MPS optimizations that improve simulations of many-body systems by targeting minimally entangled purifications.

Inequivalent purifications of thermal spectra arise from the fundamental observation that a given thermal (i.e., Gibbs) density matrix on a quantum system admits infinitely many pure-state representations on an enlarged Hilbert space. These purifications are physically and operationally distinct, with differences manifesting in entanglement structure, geometric phases, computational complexity, and even operational behavior in relativistic quantum field theory. The freedom to choose among these purifications underpins both the theoretical richness and the practical challenges of simulating and understanding thermal states, particularly in many-body and open quantum system contexts.

1. Definition of Thermal Density Matrices and Purification Non-uniqueness

For a Hamiltonian $H$ with eigenstates $\{|n\rangle\}$ and eigenvalues $\{E_n\}$, the thermal density matrix at inverse temperature $\beta$ is

$$\rho(\beta) = \frac{e^{-\beta H}}{Z(\beta)}, \qquad Z(\beta)=\mathrm{Tr}(e^{-\beta H}).$$

A purification of $\rho(\beta)$ is a pure state $$|\Psi\rangle \in \mathcal{H}^{P} \otimes \mathcal{H}^{A}$$ such that

$$\rho(\beta) = \mathrm{Tr}_{A} \left(|\Psi\rangle\langle\Psi|\right),$$

where $\mathcal{H}^P$ is the physical Hilbert space and $\mathcal{H}^A$ is an auxiliary ancilla space. The set of purifications is highly non-unique: acting by an arbitrary unitary $U_A$ on $\mathcal{H}^A$ generates a continuum of distinct purifications, all yielding the same reduced physical state after tracing out the ancilla. This gauge freedom encapsulates the core of inequivalent purifications (Hauschild et al., 2017).

2. Canonical and Alternative Purifications

Thermofield Double State and Ancilla-Gauge Freedom

The thermofield double (TFD) is the canonical purification: $$|\mathrm{TFD}(\beta)\rangle = Z(\beta)^{-1/2} \sum_n e^{-\beta E_n/2} | n\rangle_P \otimes | n\rangle_A.$$ Tracing out the ancilla recovers $\rho(\beta)$. However, applying $1 \otimes U_A$ produces an entire family of purifications with potentially very different entanglement structure. For a single harmonic oscillator, all purifications of the same thermal density matrix may be parametrized as two-mode squeezed states $$|\psi(r,\theta)\rangle = \hat S(r,\theta) |0\rangle \otimes |0\rangle_\mathrm{anc}$$, with the squeezing parameter $r$ set by $(\beta, \omega)$ but the angle $\theta$ entirely unconstrained; each $\theta$ generates an inequivalent purification with distinct computational properties (Haque et al., 2021).

Geometric and Operational Inequivalence

Two explicit inequivalent (but unitarily related) purifications are the purified state (TFD) and the thermal vacuum (obtained by a partial transpose on the ancilla):

• $$\Psi_p(\beta) = (\rho^{1/2} \otimes I_A)|\Omega\rangle = \sum_i \sqrt{\lambda_i} |i\rangle \otimes |i\rangle_A$$
• $$\Psi_t(\beta) = (\rho^{1/2} \otimes I_A) T_A |\Omega\rangle = \sum_i \sqrt{\lambda_i} |i\rangle \otimes |i\rangle_A^T$$ where $T_A$ is the partial transpose and $$|\Omega\rangle = \sum_i |i\rangle \otimes |i\rangle_A$$ (Hou et al., 2022).

Importantly, expectation values of system observables $\langle O \rangle$ coincide for both, but Berry phases generalized to mixed states can distinguish these purifications, reflecting operational inequivalence.

3. Entanglement, Matrix Product State Purifications, and Minimality

In one-dimensional quantum systems, the amount of entanglement in the purification fundamentally constrains its efficient representation as a matrix-product state (MPS). The thermofield double is typically not the minimally entangled purification, especially at low temperature. Within the MPS ansatz, each physical site and its ancilla are treated as a single enlarged site, and the purification is written as

$$|\Psi\rangle = \sum_{\{\sigma_i^P, \tau_i^A\}} \Gamma^{\sigma_1^P \tau_1^A}_{\alpha_0\alpha_1} \, \Gamma^{\sigma_2^P \tau_2^A}_{\alpha_1\alpha_2} \, \cdots \, \Gamma^{\sigma_L^P \tau_L^A}_{\alpha_{L-1}\alpha_L} \bigotimes_{i=1}^L |\sigma_i^P\rangle \otimes |\tau_i^A\rangle,$$

where the bond dimension $\chi$ controls the maximal bipartite entanglement. By variationally minimizing the second Rényi entropy $S_2 = -\log \mathrm{Tr}(\rho_{LL'}^2)$ using iterative local optimization over two-site unitaries on the ancilla bonds, $S_2$ can be substantially reduced compared to the TFD, especially at low temperatures, recovering the entanglement of purification $E_p$ in the limit $\beta \to \infty$. This leads to a hierarchy of inequivalent purifications, with practical consequences for simulating dynamics and thermal equilibrium in many-body systems (Hauschild et al., 2017).

Notably, for matrix-product density operators (MPDOs) of bond dimension $D$, there does not exist a bounded function relating the minimal bond dimension $D'$ required for a corresponding purified MPS representation, i.e., $D'$ may be unbounded even when $D$ is fixed (Cuevas et al., 2013), demonstrating a sharp form of inequivalence at the level of tensor network complexity.

4. Quantum Information, Geometric Phases, and Complexity

While all inequivalent purifications lead to the same physical expectation values, they can possess distinct geometric and computational attributes.

• Geometric Phases: The Berry phase and its mixed-state generalizations (thermal Berry phase, parallel-transport geometric phase) can distinguish inequivalent purifications, e.g. TFD versus thermal vacuum, even in basic two-level systems. These phases differ by $\pi$ under certain protocols and can be measured in quantum circuits (Hou et al., 2022).
• Circuit Complexity: For Gaussian states (e.g., thermal states of a harmonic oscillator), the circuit complexity of purification—i.e., the minimal cost to prepare the purified state from a reference—depends non-trivially on the purification angle. While the TFD has unbounded complexity in certain limits, minimizing over the purification freedom (e.g., squeezing angle) leads to a unique, least-complex purification with a complexity that saturates at high temperature to an $\mathcal{O}(1)$ value (Haque et al., 2021). Similar minimization applies to Hawking/Unruh and cosmological settings.

5. Non-Gaussian and Many-Body Settings: Tensor Network Approaches

For general many-body mixed states, purifications can be constructed via two methods:

• Sum-of-squares (SOS) polynomial purifications: Approximates the desired mixed state by a polynomial expansion, controlling the bond dimension $D'$ via the expansion order; exact purifications are often exponentially costly, but efficient approximate schemes exist.
• Eigenbasis-compression: Truncates to leading spectral components, yielding approximate purifications with bond dimension $D' \sim D[\ln(1/\epsilon)]^2$ for exponentially decaying spectra (Cuevas et al., 2013).

Both methods illustrate that operational and computational inequivalence—manifest in resource requirements—persists across practical purification schemes.

6. Operational Consequences in Open and Relativistic Quantum Settings

Open Quantum Systems

A thermal reservoir can be regarded as the reduced state of a pure two-mode squeezed vacuum, with a tunable parameter $\eta$ interpolating between no purification ($\eta=0$) and total purification ($\eta=1$). Filtering, squeezing, and entanglement in the system can be substantially improved by accessing degrees of the purification, with optimal squeezing achieved only for $\eta=1$ (Genoni et al., 2014). Thus, inequivalent purifications manifest distinct steady-state properties in measurement-based feedback and reservoir engineering.

Relativistic Quantum Field Theory

In quantum field theory in curved or accelerated spacetimes, the operational distinction between inequivalent purifications is even sharper. For instance, in the Unruh effect for Rindler wedges, four inequivalent "paths to thermality" yield the same Bose–Einstein occupation numbers but realize them via genuinely distinct quantum states. Standard wedge restriction with a causal horizon produces a Gibbs mixed (thermofield double) state, while null-shifted wedge constructions instead induce pure tensor-product states with thermal spectra arising from Bogoliubov mixing, not entanglement across a horizon (Jha, 28 Jan 2026). This shows thermality (Planck spectrum) can emerge independently of entanglement entropy, with fundamental implications for black hole, cosmological, and horizon thermodynamics.

Scenario	Spectrum (per mode)	State Type	Entanglement Pattern
Unruh effect (horizon tracing)	$n(\nu)=1/(e^{2\pi\nu/a}-1)$	Mixed Gibbs (KMS)	Maximal (TFD)
Null-shifted wedge (no horizon)	$n(\nu)=1/(e^{2\pi\nu/a}-1)$	Pure tensor product	None

7. Implications and Significance

The existence of inequivalent purifications for a given thermal spectrum demonstrates a rich "gauge freedom" in the pure-state representation of mixed states. This freedom has far-reaching consequences:

• For quantum simulation, it enables more efficient tensor-network algorithms by targeting minimal-entanglement purifications, lowering computational cost and enabling simulations at lower truncation error and longer times (Hauschild et al., 2017).
• In quantum information, it reveals subtleties in physical distinguishability, with some observables (e.g., geometric phases) able to discriminate purifications that are otherwise indistinguishable by traditional observables (Hou et al., 2022).
• In open and relativistic systems, it clarifies the roles of environment monitoring and observer horizons, distinguishing between operationally accessible quantum states with identical spectra but fundamentally different internal structure (Genoni et al., 2014, Jha, 28 Jan 2026).
• For quantum complexity theory and holographic duality, the minimization over purifications explains the bounded complexity of thermal states at high temperature, preventing unphysical divergence of operator cost (Haque et al., 2021).

In sum, inequivalent purifications are not merely a mathematical curiosity; they are central to the structure, simulation, and operational interpretation of thermal quantum systems across all scales of physics.