Can one quantum wave function stand in for thermal equilibrium?

A whole state, viewed through small windows

A quantum system at thermal equilibrium is conventionally described by a Gibbs state: a statistical mixture whose weights depend on energy and temperature. Representing this mixture directly becomes demanding as the number of particles grows.

There is another possibility. A single pure quantum state can yield approximately the same measurement averages on small subsystems. Although the whole state is pure, a small part can have a mixed description because it is entangled with the rest.

Existing thermal pure quantum methods exploit this by starting with a random state and mathematically cooling it. Different starting states produce different estimates. Those sampling fluctuations can become troublesome in small systems and at low temperatures.

Iwaki and Hotta propose a different construction, which they call a Markov-shielded typical, or MST, state. Rather than randomly choosing a representative, they optimize one, beginning at the ground state and gradually raising the target temperature.

Replacing an unusable entropy

The usual equilibrium variational principle minimizes free energy: energy minus temperature times entropy. But applying that expression directly to a pure state creates a problem. Its total von Neumann entropy is zero, so the entropy term cannot guide it toward finite-temperature behavior.

The authors instead use Markov entropy, assembled from the entropies of overlapping small regions. Each site is paired with a nearby region called its Markov shield. The construction estimates how much additional uncertainty the site contributes once information about that shield is available.

The relevant physical idea is conditional independence: for an approximately Markovian state, knowing the intervening region leaves relatively little additional information to be gained from more distant sites. This makes local entropy calculations useful for approximating thermal free energy.

The algorithm minimizes energy minus temperature times this substitute entropy. At each temperature step, the preceding optimized wave function supplies the next starting point. This is a computational continuation procedure, not a simulation of a material physically warming over time.

What the mathematical bounds establish

For one-dimensional systems with finite-range interactions, the paper derives upper and lower bounds on the optimized Markov free energy per site relative to the true thermal value.

Those bounds are conditional. One assumes that a pure state exists whose density matrices on the relevant small regions closely match the Gibbs state's. Another assumes an approximate Markov property for the physical density matrix minimizing the Markov functional over mixed states.

Under these assumptions, increasing the shield size suppresses one contribution to the error exponentially. Another contribution depends on how accurately the candidate pure state matches the local thermal density matrices.

This supports the optimization target, but it is not an unconditional proof that the practical algorithm always finds the desired state. Nor does a close free-energy value alone establish agreement for every observable. The authors separately test local density matrices numerically.

Testing a chain of quantum spins

The demonstrations use the one-dimensional transverse-field Ising model, in which neighboring spins interact while a magnetic field acts in a competing direction.

For a 16-spin chain, the authors compare their method with ten randomly initialized thermal pure quantum calculations and an exact diagonalization reference. With a three-site shield, the optimized energy and magnetization curves closely follow the reference, avoiding the spread between random samples.

Larger calculations use 48 physical spins and a matrix product state, a structured wave-function representation. Auxiliary degrees of freedom at both ends provide additional entanglement capacity; the pure state in this implementation includes those auxiliaries.

Increasing the shield to four sites improves agreement with high-precision thermal reference calculations. Comparisons of local density matrices also show close agreement over small regions, including tests at low and higher temperatures. Supplementary calculations examine the model's quantum critical field as well.

Local success has a boundary

The distinction between local and global agreement matters. Specific heat estimated from the temperature dependence of energy agrees better with the reference than an estimate based on fluctuations of the whole Hamiltonian. The latter involves correlations beyond what local matching necessarily controls.

Accuracy also costs computation. The reported sweep cost grows quadratically with chain length and exponentially with shield size. A four-site-shield sweep took about three hours on the authors' workstation.

Finally, the optimization landscape has many local minima. Ground-state initialization and gradual temperature increases provide a practical strategy, not a general guarantee of a unique solution or global convergence.

The result is a promising demonstration of deliberately constructed local thermal representatives—not a universal replacement for the Gibbs state. Extensions to higher dimensions and applications on quantum hardware remain proposed directions rather than demonstrations in this study.