Quantum Thermodynamic Length

• Quantum thermodynamic length is a geometric measure that quantifies irreversibility by distinguishing nonequilibrium states from equilibrium ones via metrics such as the Bures angle.
• It appears in several formulations—including Bures-angle, Riemannian, and retrodictive approaches—that yield lower bounds on entropy production and work dissipation.
• Optimized geodesic protocols derived from these frameworks minimize dissipation in both closed and open quantum systems, as demonstrated in experiments with quantum dots and Gaussian models.

Searching arXiv for the cited thermodynamic-length papers to ground the article in the primary literature. Quantum thermodynamic length is a geometric notion used to quantify irreversibility, dissipation, or statistical distinguishability in quantum thermodynamic transformations. In the literature it appears in several related forms: as the Bures-angle distance between a nonequilibrium state and its corresponding equilibrium state for arbitrary far-from-equilibrium unitary driving, as a Riemannian length on a control-parameter manifold whose metric is obtained from linear-response dissipation, and, for general open quantum processes, as a Bures distance between an initial Gibbs state and a retrodicted state defined by the Heisenberg-picture map (Deffner et al., 2012, Abiuso et al., 2020, Buscemi et al., 2020). In each formulation, the squared length yields lower bounds on irreversible entropy production or dissipated work, while geodesics of the relevant metric identify minimally dissipative protocols.

1. State-space geometry and the Bures formulation

For mixed states $\rho$ and $\sigma$, the Bures angle is defined by

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$

with fidelity

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$

Equivalently,

$L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$

By construction, $L_B$ is a true distance on the space of density operators, reduces to Wootters’ angle for pure states, and is invariant under unitary transformations (Deffner et al., 2012).

For a closed quantum system driven from $H_0$ to $H_\tau$, with unitary evolution $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$, and with initial thermal state $\rho_0=e^{-\beta H_0}/Z_0$, the equilibrium state associated with the final Hamiltonian is $\sigma$0. In this setting, the irreversible entropy production is exactly

$\sigma$1

where $\sigma$2. Using the Audenaert–Eisert inequality for relative entropy and choosing the Bures angle as the unitarily invariant distance gives the generalized Clausius inequality

$\sigma$3

valid arbitrarily far from equilibrium (Deffner et al., 2012).

This construction assigns thermodynamic meaning directly to a state-space distance. Geometrically, $\sigma$4 measures the minimal geodesic angular distance between the actual final state and the equilibrium state corresponding to the final Hamiltonian. Thermodynamically, it gives a rigorous lower bound on the entropy cost of finite-time driving. In the classical, near-equilibrium limit, the lowest-order term reduces to the familiar Salamon–Berry thermodynamic length theory (Deffner et al., 2012).

The driven harmonic oscillator provides an explicit example. For

$\sigma$5

non-adiabaticity is encoded in the Husimi parameter $\sigma$6. In the zero-temperature limit, the fidelity simplifies to

$\sigma$7

so that

$\sigma$8

while in the high-temperature limit

$\sigma$9

Once $$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$0 is known, the thermodynamic length is $$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$1, and the entropy-production bound follows immediately (Deffner et al., 2012).

2. Control-parameter manifolds and linear-response metrics

A second major formulation treats thermodynamic length as the Riemannian length of a trajectory in the space of externally controlled parameters. For a driven Hamiltonian

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$2

with instantaneous Gibbs state

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$3

the slow-driving, linear-response regime yields a quadratic form for dissipated work,

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$4

where $$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$5 is a positive-definite matrix and therefore a Riemannian metric on the control-parameter manifold (Abiuso et al., 2020).

The associated line element is

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$6

and the thermodynamic length of a trajectory is

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$7

By Cauchy–Schwarz,

$$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$8

A geodesic extremises $$L_B(\rho,\sigma)=\arccos\!\bigl[F(\rho,\sigma)^{1/2}\bigr],$$9 at fixed endpoints and satisfies

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$0

On a geodesic, the quantity $F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$1 is constant in time; minimal dissipation at fixed duration occurs when the system is driven along a geodesic at constant thermodynamic speed (Abiuso et al., 2020).

In open quantum systems, this metric can be derived from a time-dependent Lindblad generator. Let $F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$2 define a manifold of Gibbs states

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$3

On the tangent space one introduces the Kubo–Mori–Bogoliubov inner product

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$4

with

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$5

and equivalently

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$6

If the dynamics is generated by $F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$7 with unique instantaneous steady state $F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$8, then in the quasistatic limit

$F(\rho,\sigma)=\Bigl[\Tr\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\Bigr]^2.$9

where $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$0 is the Drazin inverse. The leading dissipated work is

$L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$1

with

$L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$2

and

$L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$3

For instantaneous full thermalisation, one recovers the KMB metric: $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$4 (Scandi et al., 2018).

This formulation makes explicit that thermodynamic length in open systems is not determined solely by equilibrium fluctuations. The background geometry is the KMB/Fisher metric, while the Drazin-inverse correction biases the metric toward slowly equilibrating directions, so that optimal paths exploit fast modes (Scandi et al., 2018). In the review formulation, the same structure appears through the Lindblad metric

$L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$5

alongside the unitary metric $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$6 and the discrete BKM metric $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$7 (Abiuso et al., 2020).

3. Retrodicted thermodynamic length for general open processes

A distinct construction applies to arbitrary completely positive trace-preserving maps, without restricting to slow driving or to Lindblad form. Consider a system $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$8 whose reduced dynamics over $L_B(\rho,\sigma)=\arccos\!\Bigl[\Tr\bigl(\sqrt{\sqrt{\rho}\,\sigma\,\sqrt{\rho}}\bigr)\Bigr].$9 is described by a CPTP map $L_B$0, with endpoint Hamiltonians $L_B$1 and $L_B$2 and corresponding Gibbs states

$L_B$3

No assumption is made on the microscopic origin of $L_B$4 except that it arises from unitary coupling to an ancilla (Buscemi et al., 2020).

The key object is not the pair $L_B$5, but the pair formed by the initial equilibrium state and a retrodicted state,

$L_B$6

where $L_B$7 is the trace-dual map. The starting point is the thermodynamic reverse bound, derived from the operator concavity of the logarithm together with the Schwarz inequality for positive maps. For an initial equilibrium state $L_B$8, one obtains

$L_B$9

Since $H_0$0 need not be normalized,

$H_0$1

Substituting this identity and then applying the Audenaert–Eisert bound gives

$H_0$2

This is the open-process thermodynamic-length inequality (Buscemi et al., 2020).

The geometric meaning is explicit: $H_0$3 is the shortest Riemannian Bures distance between the initial equilibrium state and the retrodicted state obtained by pulling back the final Gibbs state through the Heisenberg-picture map. Thermodynamically, this distance controls a lower bound on the irreversible work even for fully open, non-unitary processes. The additional term

$H_0$4

accounts for the non-unit-trace of the pulled-back operator and quantifies the departure from a heat-conserving or unital evolution (Buscemi et al., 2020).

Several special cases clarify the construction. When $H_0$5 is unitary, $H_0$6 and $H_0$7, so one recovers the Deffner–Lutz result

$H_0$8

If $H_0$9 is unital, then $H_\tau$0 is trace-preserving, so again the extra trace term vanishes. In an erasure setting, applying the same steps to an erasure channel $H_\tau$1 on an ancilla initialized in $H_\tau$2 yields

$H_\tau$3

which refines the Landauer bound by an additional divergence term (Buscemi et al., 2020).

4. Work optimisation, variance optimisation, and Gaussian states

In weakly coupled Gaussian open systems, thermodynamic length appears in two inequivalent forms. Let $H_\tau$4 denote the control parameters, and assume slow driving so that the system remains close to its instantaneous thermal state $H_\tau$5 with covariance matrix $H_\tau$6. Then one may define an excess-work length

$H_\tau$7

and a work-variance length

$H_\tau$8

where $H_\tau$9 is the excess-work metric and $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$0 is the variance metric (Mehboudi et al., 2021).

In the slow-driving regime,

$\rho_\tau=U_\tau\rho_0U_\tau^\dagger$1

Extremising either quadratic form with fixed endpoints amounts to finding a geodesic with respect to $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$2 or $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$3. Physically, the $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$4-geodesic minimises average excess work, while the $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$5-geodesic minimises work variance. In a classical limit, or whenever all conjugate forces commute, one recovers

$\rho_\tau=U_\tau\rho_0U_\tau^\dagger$6

so that the two geodesics coincide (Mehboudi et al., 2021).

For an $\rho_\tau=U_\tau\rho_0U_\tau^\dagger$7-mode bosonic system with quadratures

$\rho_\tau=U_\tau\rho_0U_\tau^\dagger$8

and quadratic Hamiltonian

$\rho_\tau=U_\tau\rho_0U_\tau^\dagger$9

Mehboudi and Miller derive explicit expressions for both metric tensors. Defining

$\rho_0=e^{-\beta H_0}/Z_0$0

and introducing

$\rho_0=e^{-\beta H_0}/Z_0$1

with

$\rho_0=e^{-\beta H_0}/Z_0$2

they obtain closed-form expressions for $\rho_0=e^{-\beta H_0}/Z_0$3 and $\rho_0=e^{-\beta H_0}/Z_0$4 in terms of $\rho_0=e^{-\beta H_0}/Z_0$5, $\rho_0=e^{-\beta H_0}/Z_0$6, $\rho_0=e^{-\beta H_0}/Z_0$7, $\rho_0=e^{-\beta H_0}/Z_0$8, $\rho_0=e^{-\beta H_0}/Z_0$9, and $\sigma$00 (Mehboudi et al., 2021).

The damped quantum harmonic oscillator illustrates the resulting optimisation problem. For one mode with controls $\sigma$01 and $\sigma$02, a weak-coupling Lindblad model gives diagonal metrics

$\sigma$03

Because $\sigma$04 is diagonal, the $\sigma$05-geodesic decouples:

$\sigma$06

which integrates to

$\sigma$07

The $\sigma$08-geodesic obeys coupled ordinary differential equations and must be solved numerically. After obtaining the optimal path,

$\sigma$09

Numerical studies reported in the same work show up to $\sigma$10 reduction in dissipation and $\sigma$11 reduction in fluctuations compared to naive linear protocols (Mehboudi et al., 2021).

A common source of ambiguity is that the phrase “quantum thermodynamic length” does not denote a single metric object even within slow-driving theory. The Gaussian formulation makes this explicit by exhibiting two distinct Riemannian lengths tied to two different optimisation criteria (Mehboudi et al., 2021).

5. Measurement, finite-time extrapolation, and thermometric geometry

Thermodynamic length need not remain a purely formal geometric quantity. In quantum Markovian dynamics, for a single control parameter $\sigma$12, one can write

$\sigma$13

where $\sigma$14, $\sigma$15 is the Liouvillian, and $\sigma$16 is its Drazin inverse on the subspace orthogonal to $\sigma$17 (Chen et al., 2021).

For a finite-time isothermal process of duration $\sigma$18, one defines the instantaneous excess power

$\sigma$19

with

$\sigma$20

and introduces the finite-time thermodynamic length

$\sigma$21

Using the adiabatic expansion

$\sigma$22

one finds that $\sigma$23 as $\sigma$24, and for a single control parameter this limit is protocol-independent. Measuring $\sigma$25 at several finite durations and fitting

$\sigma$26

therefore allows extrapolation of the true thermodynamic length from finite-time data (Chen et al., 2021).

The same geometric idea appears in quantum thermometry, but with the metric interpreted as a distinguishability measure rather than a dissipation metric. For a thermal family

$\sigma$27

the symmetric logarithmic derivative satisfies

$\sigma$28

and for thermal states

$\sigma$29

The quantum Fisher information metric is then

$\sigma$30

Equivalently,

$\sigma$31

where $\sigma$32 is the heat capacity (Jørgensen et al., 2021).

The thermodynamic length between $\sigma$33 and $\sigma$34 is

$\sigma$35

or in the temperature parametrization,

$\sigma$36

Introducing the length coordinate

$\sigma$37

straightens the metric to $\sigma$38, and Bayesian estimators may then be built by minimizing mean-square distance in $\sigma$39 (Jørgensen et al., 2021).

For a spin-$\sigma$40 with

$\sigma$41

one has

$\sigma$42

and the length variable becomes

$\sigma$43

which runs from $\sigma$44 as $\sigma$45 to $\sigma$46 as $\sigma$47 (Jørgensen et al., 2021). This use of thermodynamic length is formally different from dissipation minimisation, but it relies on the same basic idea: a physically meaningful metric endows the manifold of thermal states with a reparametrization-invariant distance.

6. Experimental validation and conceptual status

A direct open-system implementation was demonstrated in Landauer erasure with a driven electron level in a semiconductor quantum dot. In this setting the Hamiltonian is

$\sigma$48

with occupation $\sigma$49, and the irreversible entropy production in the slow-driving expansion takes the form

$\sigma$50

where

$\sigma$51

is the Hessian of the relative entropy at equilibrium. For Markovian dynamics this may also be written in Kubo–Mori form,

$\sigma$52

In the simplified analytical model for the quantum dot,

$\sigma$53

The corresponding thermodynamic length is

$\sigma$54

and the dissipation satisfies

$\sigma$55

in the slow-drive limit (Scandi et al., 2022).

For the single control parameter $\sigma$56, the geodesic equation becomes

$\sigma$57

with

$\sigma$58

and the analytical solution satisfying $\sigma$59 and $\sigma$60 is

$\sigma$61

By construction, this protocol keeps the instantaneous entropy-production rate constant along the process (Scandi et al., 2022).

Experimentally, two drives were compared: the linear ramp

$\sigma$62

and the geodesic drive $\sigma$63. In the slow-drive regime, with $\sigma$64 and $\sigma$65, the geodesic protocol reduced $\sigma$66 by up to $\sigma$67 compared to the linear ramp, with the largest improvement at high erasure fidelities. The instantaneous entropy-production rate was nearly constant for the geodesic and highly nonuniform for the linear drive. Even for $\sigma$68 and $\sigma$69, the geodesic remained advantageous, although with a small reduction in final empty-state probability that could be compensated by a brief thermalization step (Scandi et al., 2022).

Taken together, these formulations show that the literature uses “quantum thermodynamic length” for several related constructions rather than for a single universal metric. The Bures-angle approach addresses arbitrary far-from-equilibrium state-space distinguishability (Deffner et al., 2012); the KMB, BKM, Lindblad, and Drazin-inverse formulations address control-space optimisation in linear response (Scandi et al., 2018, Abiuso et al., 2020); the retrodictive construction extends Bures-type bounds to arbitrary open CPTP maps (Buscemi et al., 2020); the Gaussian framework distinguishes dissipation-minimising and fluctuation-minimising metrics (Mehboudi et al., 2021); and the QFI formulation underlies Bayesian thermometry (Jørgensen et al., 2021). This suggests a family resemblance rather than a unique definition: in each case, a physically motivated metric induces a length whose square controls irreversibility, finite-time cost, or estimation error, and whose geodesics identify optimal transformations within the regime of validity of the corresponding theory.