arXiv:2008.08683

Geometric Quantum Thermodynamics

Fabio Anza∗ and James P. Crutchfield†Complexity Sciences Center and Physics Department,

University of California at Davis, One Shields Avenue, Davis, CA 95616(Dated: February 3, 2021)

Building on parallels between geometric quantum mechanics and classical mechanics, we explorean alternative basis for quantum thermodynamics that exploits the differential geometry of theunderlying state space. We develop both microcanonical and canonical ensembles, introducingdistributions on the manifold of quantum states. We call out the experimental consequences fora gas of qudits. We define quantum heat and work in an intrinsic way, including single-trajectorywork, and reformulate thermodynamic entropy in a way that accords with classical, quantum, andinformation-theoretic entropies. We give both the First and Second Laws of Thermodynamics andJarzynki’s Fluctuation Theorem. The result is a more transparent physics, than conventionallyavailable, in which the mathematical structure and physical intuitions underlying classical andquantum dynamics are seen to be closely aligned.

PACS numbers: 05.45.-a 89.75.Kd 89.70.+c 05.45.TpKeywords: quantum thermodynamics, differential geometry, classical thermodynamics, statistical mechanics

I. INTRODUCTION

The standard formulation of quantum mechanics definesa system’s (pure) states as normalized vectors |ψ〉 in acomplex Hilbert space H. There are alternative startingpoints, though.One is provided by the geometric formalism that describesstates as points on a manifold. For finite-dimensionalquantum systems—that we focus on here—the manifoldsare complex projective spaces CPn of dimension n = D−1,where D := dim H. Our goal is to highlight the geo-metric approach’s advantages when describing quantumdynamics and quantum thermodynamics. In particular,structural and informational properties can be properlyformulated, since close parallels in the mathematical foun-dations of classical and quantum dynamics become clear.This builds on a companion work that argues the geo-metric quantum state, not a density matrix, completelycharacterize the state of a quantum system [1].To the best of our knowledge, such geometric formalismsstarted with early insights from Strocchi [2] and thenwork by Kibble [3], Marsden [4], Heslot [5], Gibbons[6], Ashtekar and Shilling [7, 8], and a host of others[9–19]. Although geometric tools for quantum mechanicsare an interesting topic in their own right, the followingexplores their consequences for statistical mechanics andnonequilibrium thermodynamics.As one example in this direction, Brody and Hughston[20–22] showed that a statistical mechanics treatment ofquantum systems based on the geometric formulationdiffers from standard quantum statistical mechanics: Theformer can describe phase transitions away from the ther-modynamic limit, the latter not [23]. This arises, most

∗ fanza@ucdavis.edu† chaos@ucdavis.edu

directly, since the geometric formulation puts quantummechanics on the same footing as the classical mechanicsof phase space [2, 5], bringing to light the symplectic ge-ometry of quantum state space. It is then straightforwardto build on the principles of classical statistical mechanicsto layout a version of quantum statistical mechanics thattakes advantage of such state-space features.That said, it does not arrive for free. It poses the co-nundrum of a consistent foundation of thermodynamicbehavior. On the one hand, we have quantum statisticalmechanics—a description of macroscopic behavior that,despite limitations, has proven to be remarkably success-ful. On the other, transitioning from microphysics tomacrophysics via quantum mechanics is conceptually dif-ferent than via classical mechanics. Consistency betweenthese approaches begs for a conceptually unique routefrom microphysics to macrophysics.With this broad perspective in mind, unifying the twocoexisting statistical mechanics of quantum systems ap-pears as a challenging topic deserving further attention.To address this, the following advocates a geometric devel-opment of a practical, macroscopic companion of geomet-ric quantum statistical mechanics—a geometric quantumthermodynamics.Geometric quantum thermodynamics is all the moretimely due to recent success in driving thermodynamicsdown to the mesoscopic scale, where statistical fluctua-tions, quantum fluctuations, and collective behavior notonly cannot be neglected, but are essential. Largely, thispush is articulated in two research thrusts: stochasticthermodynamics [24, 25] and quantum thermodynamics[26, 27]. The following draws ideas and tools from both,in effect showing that geometric tools provide a robustand conceptually-clean crossover between them.The development unfolds as follows. First, we recall thebasic elements of geometric quantum mechanics. Sec-ond, we show how this formalism emerges naturally in

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a thermodynamic context. Third, we describe our ver-sion of the statistical treatment of geometric quantummechanics—what we refer to as geometric quantum sta-tistical mechanics. Fourth, we propose an experimentalscenario that directly assesses its validity. Fifth, we buildon this to establish two fundamental equations of geomet-ric quantum thermodynamics. The first is a novel versionof the first law of quantum thermodynamics, with its def-inition of quantum heat and quantum work. The secondis a quantum version of Jarzynski’s inequality—one thatdoes not require a two-time measurement scheme. Finally,we expand on the approach’s relevance and discuss futuredirections.

II. GEOMETRIC QUANTUM MECHANICS

Geometric quantum mechanics arose from effortsto exploit differential geometry to probe the often-counterintuitive behaviors of quantum systems. Thissection summarizes the relevant concepts, adapting themto our needs. Detailed expositions are found in the origi-nal literature [2, 3, 5–8, 10–19]. Here, we present the mainideas in a constructive way that builds on the familiarvector-based formalism of quantum mechanics.Any statistical mechanics requires an appropriate, work-able concept of ensemble. To do this, one identifies ensem-bles with coordinate-invariant measures on the manifoldof pure states, a treatment first introduced in Ref. [20].We call these distributions geometric quantum states andin Ref. [1] we give a generic procedure to compute them ina quantum thermodynamic setting of a small system inter-acting with a large environment. Reaching this, though,requires a series of technical steps. The first identifies themanifold of pure states and defines their observables. Thesecond introduces a suitable metric, scalar product, andcoordinate-invariant volume element for the pure-statemanifold. From these, the third step derives the evolutionoperator and equations of motion. Finally, states aredescribed via functionals that map observables to scalarvalues. This is done so that the associated ensembles arecoordinate-invariant measures.Our quantum system of interest has Hilbert space H offinite dimension D. The manifold of its pure states isa complex projective space P (H) = CPD−1 [11]. Givenan arbitrary basis |eα〉D−1

α=0 a pure state is thereforeparametrized by D complex homogeneous coordinates Zαup to normalization and an overall phase:

|ψ〉 =D−1∑α=0

Zα |eα〉 ,

where Z ∈ CD, Z ∼ λZ, and λ ∈ C/ 0.For example, the pure state of a single qubit can be givenby Zqubit = (√p0,

√1− p0e

iν). An observable O is aquadratic real function of the state. It associates to each

point of the pure-state manifold P (H) the expectationvalue 〈ψ| O |ψ〉 of the corresponding operator O on thatstate:

O(Z) =∑α,β

Oα,βZαZβ (1)

and Oβ,α = Oα,β . And so, O(Z) ∈ R.These complex projective spaces are Kahler spaces. Thismeans there is a function K, which in our case is K =logZ · Z, from which one obtains both a metric g:

gαβ = 12∂α∂β logZ · Z ,

with gαβ = gβα, and a symplectic two-form:

Ω = 2igαβdZα ∧ dZβ,

using shorthand ∂α := ∂/∂Zα. It is not too hard to see

that these two structures are parts of the Hermitian formthat defines the scalar product 〈ψ1|ψ2〉 in H. Indeed,using the standard notation, one has [6]:

〈ψ1|ψ2〉 = g(Z1, Z2) + iΩ(Z1, Z2) ,

Each geometric term provides an independent volumeelement.Agreement between these volumes, together with invari-ance under unitary transformations, selects a uniquecoordinate-invariant volume element [20], based on theFubini-Study metric on CPD−1:

dVFS = 1(D − 1)!

(Ω2

)∧(

Ω2

)∧ . . . ∧

(Ω2

)(2a)

=√

det g(Z,Z)dZdZ . (2b)

(See also Ref. [11] for a textbook treatment.) Equippedwith this unique volume element, the total volume of thepure-state manifold CPD−1 is [6, 11]:

Vol (CPn) = πD−1

(D − 1)! .

Since symplectic geometry is the correct environment inwhich to formulate classical mechanics, one can see howthe geometric formalism brings classical and quantummechanics closer together—a point previously raised byStrocchi [2] and made particularly clear by Heslot [5].Indeed, as in classical mechanics, the symplectic two-form Ω is an antisymmetric tensor with two indices thatprovides Poisson brackets, Hamiltonian vector fields, andthe respective dynamical evolution.Given two functions A and B on manifold P(H) we have:

Ω(A,B) = ∂αA∂βBΩαβ

= A,B ,

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where we used Ω = 12ΩαβdZα ∧ dZ

β and Ωαβ = (Ω−1)αβis the inverse: ΩαγΩγβ = δαβ . Using the symplectic two-form one can show that Schrödinger’s unitary evolutionunder operator H is generated by a Killing vector fieldVH as follows:

V αH = Ωαβ∂βh(Z) (3a)dF

dt= F, h (3b)

where h(Z) =∑αβ HαβZ

αZβ and F : P(H) → R is a

real but otherwise arbitrary function. Indeed, it can beshown that Schrödinger’s equation is nothing but Hamil-ton’s equations of motion in disguise [5, 11]:

d |ψt〉dt

= −iH |ψt〉 ⇐⇒ dF

dt= F, h , (4)

for all F . Here, we use units in which ~ = 1.

This framework naturally views a quantum system’s statesas the functional encoding that associates expectationvalues with observables; as done in the C∗-algebra for-mulation of quantum mechanics [28]. Thus, states aredescribed via functionals P [O] from the algebra A ofobservables to the reals:

P [O] =∫P(H)

p(Z)O(Z)dVFS ∈ R ,

for p(Z) ≥ 0 and all O ∈ A. Here, p is the distributionassociated to the functional P . It is important to note herethat dVFS and O(Z) are both invariant under coordinatechanges. Thus, for P [O] to be a scalar, p(Z) must bea scalar itself. A pure state |ψ〉 ∈ H is represented bya Dirac-delta functional concentrated on a single pointof P(H). However, Dirac delta-functions δ(·) are notinvariant under coordinate changes: They transform withthe inverse of the Jacobian: δ → δ/detJ .

To build an invariant quantity, then, we divide it bythe square root √g of the metric’s determinant. Thistransforms in the same way, making their ratio δ = δ/

√g

an invariant quantity. This is a standard rescaling thatturns coordinate-dependent measures, such as Cartesianmeasure, into coordinate-invariant ones. And, this is howthe Fubini-Study measure Eq. (2) is defined from theCartesian product measure. Thus:

Pψ0 [O] =∫P(H)δ[Z − Z0]O(Z)dVFS

= O(Z0)= 〈ψ0| O |ψ0〉 , (5)

where:

δ[Z − Z0] = 1√g

∏α

δ(Zα − Zα0 )

and:

δ(Zα − Zα0 ) = δ(Re[Zα]− Re[Zα0 ])δ(Im[Zα]− Im[Zα0 ]) .

This extends by linearity to ensembles ρ =∑Mk=1 pk |ψk〉 〈ψk| as:

Pρ[O] =M∑h=1

pk

∫P(H)δ[Z − Zk]O(Z)dVFS

=M∑h=1

pkO(Zk)

=M∑h=1

pk 〈ψk| O |ψk〉 .

It is now quite natural to consider generalized ensemblesthat correspond to functionals with a continuous measureon the pure-state manifold.Such ensembles have appeared previously in Refs. [10, 20–22] and elsewhere, where aspects of their properties havebeen investigated extensively. For our purposes, it will beuseful to look at such ensembles from the following pointof view.Consider a probability measure on the natural numbers:pk such that pk ≥ 0 and

∑k pk = 1. Now let Zk be a

countable collection of points in P(H), then δk(dZ) is theDirac measure concentrated on the point Zk. Then, givenpk one can define the measure µ(dZ) on P(H) as:

µ(dZ) =∞∑k=1

pkδk(dZ) , (6)

which gives precise meaning to the notion of a geometricquantum state with support on a countably-infinite num-ber of points. Indeed, with the measure in Eq.(6) andarbitrary observable function O(Z) one has that:

P∞[O] =∫P(H)O(Z)µ(dZ)

=∞∑k=1

pkO(Zk) .

In more general terms, calling B the Borel σalgebra of theopen sets of P(H), then, this procedure defines a measureµ on P(H) such that for a set S ∈ B one has:

µ(S) =∫S

µ(dZ) =∞∑k=1

pkI(Zk ∈ S) ,

where I(Zk ∈ S) is the indicator function which is 1 ifZk ∈ S and zero otherwise.The resulting geometric quantum state has all the prop-erties desired of an appropriately-generalized pure-stateensemble: It preserves normalization and convexity oflinear combinations, each of its elements are invariant

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under coordinate changes, and the entire functional P∞is also invariant under unitary transformations. Withsome abuse of language, we will often refer to both thefunctional P and their underlying measure µ as geometricquantum states.

III. GEOMETRIC QUANTUM STATE ANDTHE THERMODYNAMIC LIMIT

We are now equipped to address how the geometric for-malism arises quite naturally for subsystems of a largersystem in a pure state; in particular, in a quantum ther-modynamic setting.If we have a bipartite system HAB = HA ⊗ HB and|ψAB〉 =

∑α,i ψ

αiAB |aα〉 |bi〉 ∈ HAB , the partial trace over

the subsystem B is:

ρA =dA∑

α,β=1ρAαβ |aα〉〈aβ | ,

where:

ρAαβ =dB∑i=1

ψαiψβi

= (ψψ†)αβ .

dA and dB are A’s and B’s dimensions, respectively.Hence, we can write the partial trace as:

ρA =dB∑j=1|vj〉〈vj | ,

with |vi〉 ∈ HA given as:

|vi〉 :=dA∑α=1

ψαi |aα〉 .

However, |vj〉 is not normalized. To address this, wenotice that:

〈vj |vk〉 = (ψ†ψ)jk= ρBjk

= 〈bj | ρB |bk〉 .

This gives:

pBk = ρBkk

=dA∑α=1

∣∣ψαk∣∣2 .

We see that 〈vj |vk〉 is a Gramian matrix of vectors|vj〉 ∈ HA that conveys the information about the re-duced state ρB on the subspace HA. Though the vectors

|vk〉 are not normalized, we readily define their normalizedcounterpart:

|χk〉 := |vk〉√〈vk|vk〉

=dA∑α=1

ψαk√∑dAβ=1 |ψβk|

2|aα〉 .

And, eventually, we obtain:

ρA =dB∑k=1

pAk∣∣χAk ⟩⟨χAk ∣∣ , (7)

where |χj〉dBj=1 is a set of dB pure states on HA which,usually, are nonorthogonal. This provides the followinggeometric quantum state, at fixed dB :

µAdB (dZ) :=dB∑k=1

pBk δχk (dZ) ,

where δχk is the Dirac measure with support only on thepoint χk ∈ P(HA) corresponding to the ket |χk〉.While it is possible to track all information about

pAkdBk=1

for small dB, in the thermodynamic limit this rapidlybecomes infeasible. A probabilistic description becomesmore appropriate. One could object that this is not aconcern since, at each step in the limit, the spectraldecomposition ρA =

∑dAi=1 λi |λi〉 〈λi|, where the λi are

the Schmidt coefficients of |ψAB〉, is always available.However, this retains only ρA’s matrix elements, erasingthe information contained in the vectors |vj〉 =

√pAj∣∣χAj ⟩.

That is, ρB has been erased from the description.However, this information can be crucial to understandingthe behavior of small system A. The geometric formal-ism resolves this issue as it naturally keeps the “relevant”information by handling measures and probability distri-butions. In the limit of a large “environment” B, despitethe fact that storing all information about the environ-ment’s details is exponential in B’s size, the geometricquantum state’s form (convex sum of Dirac deltas) facili-tates working with smooth approximations of increasingaccuracy. It does so by retaining the information aboutits “purifying environment”.Concretely, one needs to operationally define the thermo-dynamic limit procedure. We do so by confining ourselvesto modular systems and defining an iterative procedure.Modular systems are those made by identical subsystems,each described by a Hilbert space Hd of dimension d.Thus, we imagine our system to containNA such repetitiveunits, while the environment contains NB ≥ NA. Thismeans HA = H⊗NAd and HB = H⊗NBd , so that dA = dNA

and dB = dNB . At the iteration limit, the joint systemwill always be in a pure state |ψAB(NB)〉 ∈ HA ⊗HB .We also imagine the dynamics the whole system has a

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Hamiltonian HAB of fixed functional form. For example,the XXZ model. Starting with NB = NA, at each stepwe add one repetitive unit NB → NB + 1 and choose aseries of pure states |ψAB(NB)〉NB with the requiredproperty that the limit of the average energy has to befinite:

limNB→∞

〈ψAB(NB)|HAB |ψAB(NB)〉NA +NB

= ε .

For example, one can decide to consistently pick theground state of the Hamiltonian HAB . In general, though,there is no unique way of performing the procedure. How-ever, with any specific choice of the series |ψAB(NB)〉NBsatisfying the constraint on average energy, the procedureis well-defined, physical, and meaningful. It provides anoperational way to study the thermodynamic limit of thegeometric quantum state µAdB .

That said, by no means does this guarantee the limitalways exists. However, it does allow exploring it ina physically meaningful way. In particular, given thisoperational implementation of the thermodynamic limit,we say that:

limdB→∞

µAdB = µA∞ ,

This requires a geometric quantum state µA∞ on P(HA)such that, for any ε > 0 arbitrarily small, one can alwaysfind some finite dB such that for any dB ≥ dB one has thatD(µAdB , µ

A∞) ≤ ε. Here, D(µ, ν) is a notion of distance

between geometric quantum states that we take to bethe measure-theoretic counterpart of the total variationdistance: D(µ, ν) := supS∈B |µ(S)− ν(S)|, where B isthe σ-algebra of the Borel sets on P(H).

When the limit exists, we say that the thermodynamiclimit of the geometric quantum state is µA∞ or, equiva-lently, PA∞:

PA∞ [O] =∫P(HA)µA∞(dZ)O(Z)

=∞∑k=1

pAkO(χAk ) .

PA∞ is a functional whose operational meaning is under-stood in terms of ensemble theory, mirroring the way inwhich we interpret the discrete probability distributionpk for a density matrix ρ =

∑k pk |ψk〉 〈ψk|. Geometric

quantum states describe ensembles of independent andnoninteracting instances of the same quantum systemwhose pure states are distributed according to a givenprobability distribution. Loosely speaking, if we pick arandom pure state out of the ensemble described by PA∞,the probability of finding it in a small region around Z isdPZ = µA∞(dZ).

IV. FROM GEOMETRY TO STATISTICS

Several observations serve to motivate defining statisticalmechanics using the geometric formalism. Consider alarge system consisting of a macroscopic number M ofqubits from which we extract, one by one, N qubit states.Describing small subsystems of a macroscopic quantumsystem places us in the realm of quantum statistical me-chanics. It is therefore reasonable to assume that thequbit states are distributed according to Gibbs’ canonicalstate γβ = e−βH/Zβ .

However, one immediately sees that the standard treat-ment of quantum statistical mechanics contains an un-warranted assumption. After we extract the k-th samplefrom the macroscopic system, that sample’s state is sup-posed to be an energy eigenstate

∣∣∣E(k)i

⟩with probability

p(Z(|E(k)i 〉)) ∝ e−βE

(k)i . A priori, however, there is no

reason to assume that the Hamiltonians Hk of all the sam-ples are identical to each other. In fact,

∣∣Ehi ⟩ 6= ∣∣Eki ⟩ andEhi 6= Eki . This point was originally made by Khinchin[29] and Schrödinger [30], who advocated for the use ofensembles of wave-functions.

To address this, a description of the system’s state thatdoes not contain this assumption is provided by the contin-uous counterpart of Gibbs canonical state, first introducedin Ref. [20], written as the following functional:

Pβ [A] = 1Qβ [h]

∫P(H)e−βh(Z)A(Z)dVFS ,

where:

Qβ [h] =∫P(H)e−βh(Z)dVFS ,

with h(Z) =∑αβ HαβZ

βZα. While this distribution

retains a characteristic feature of the canonical Gibbsensemble:

pβ(Z(|En〉)

)pβ(Z(|Em〉)

) = e−β(En−Em) ,

it also extends this “Boltzmann” rule to arbitrary states:

− log[pβ(Z(|ψ〉)

)pβ(Z(|φ〉)

) ] = β [h(Z(ψ))− h(Z(φ))] .

Therefore, formulating the statistical mechanics of quan-tum states via the geometric formalism differs from thestandard development, based on an algebraic formalism.This becomes obvious when we write the Gibbs canonical

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density matrix γβ in the geometric formalism:

pGibbs(Z) =D−1∑k=0

e−βEk

Tr e−βH δ[Z − Z(|Ek〉)]

6= e−βh(Z)

Qβ [h] .

This makes explicit the standard formalism’s assump-tion that the measure is Dirac-like—peaked on energyeigenstates.

Despite quantum statistical mechanics’ undeniable suc-cesses, this assumption is not, in general, justified. Inpoint of fact, it is the origin of the missing environmen-tal information noted above. These arguments motivatean alternative formulation of the statistical mechanicsof quantum systems, first introduced in Ref. [20]—onebased on geometric quantum states rather than on thefamiliar density matrices.

V. STATISTICAL TREATMENT OFGEOMETRIC QUANTUM MECHANICS

Representing a quantum system’s state as a continuousmixed state was first broached, to our knowledge, byBrody and Hughston [20, 21]. Our goal here is to advancethe idea, going from statistical mechanics to thermody-namics. To set the stage for a geometric quantum thermo-dynamics, the following first presents our version of theirresults, derived via the formalism defined in Sec. III, andthen expands on them. We begin with the fundamentalpostulate of classical statistical mechanics and its adap-tation to quantum mechanics—the microcanonical andcanonical ensembles.

A. Classical microcanonical ensemble: A prioriequal probability

At its most basic level, the fundamental postulate of clas-sical statistical mechanics is that, in an isolated system’sphase space, microstates with equal energy have the samechance of being populated. Calling ~q and ~p generalizedvelocities and positions, which provide a coordinate framefor the classical phase-space, the postulate corresponds toassuming that the microcanonical probability distributionPmc of finding the system in a microstate (~p, ~q) is, atequilibrium:

Pmc(~q, ~p) =

1/W (E) if E(~q, ~p) ∈ [E , E + δE ]0 otherwise

.

Here, W (E) is the number of microstates (~q, ~p) belongingto energy shell Imc := [E , E + δE ]:

W (E) =∫E(~q,~p)∈Imc

d~q ∧ d~p

with∫d~q ∧ d~p Pmc(~q, ~p) = 1.

B. Quantum microcanonical ensemble: A prioriequal probability

Quantum statistical mechanics relies on the quantumversion of the Gibbs ensemble. For macroscopic isolatedsystems this is usually interpreted as the quantum systemhaving equal chance pmc to be in any one of the energyeigenstates |En〉, as long as En ∈ Imc:

pmc(En) =

1/Wmc ifEn ∈ [E , E + δE ]0 otherwise

.

Here, Wmc =∑En∈Imc

1 is the number of energy eigen-states that belong to the microcanonical window Imc.Thus, the equal-probability postulate provides the follow-ing definition for the microcanonical density matrix:

ρmc = 1Wmc

∑En∈Imc

|En〉 〈En| .

Geometric quantum mechanics gives an alternative wayto extend equal-probability to quantum systems, whichwe now introduce.

C. Geometric quantum microcanonical ensemble:A priori equal probability

The following summarizes an approach to the statisticalmechanics of quantum systems first presented in Refs.[20, 21, 23]. In geometric quantum mechanics the roleof the Hamiltonian operator as the generator of unitarydynamics is played by the real quadratic function:

h(Z) =∑αβ

HαβZαZ

β,

where Hαβ are the matrix elements of the Hamiltonianoperator in a reference basis; see Eq. (3). As h is thegenerator of Liouville dynamics on the pure-state manifoldP(H), it is easy to see that there is a straightforwardgeometric implementation of the a-priori-equal-probabilitypostulate in the quantum setting:

pmc(Z) =

1/Ω(E) h(Z) ∈ Imc, for all Z ∈ P(H)0 otherwise

.

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Due to normalization, Ω(E) is the volume of the quantum-state manifold enclosed by the microcanonical energy shellImc:

Ω(E) =∫h(Z)∈Imc

dVFS .

where dVFS is the Fubini-Study volume element intro-duced in Sec. II. In probability-and-phase coordinateZα = √pαeiνα the volume element has the explicit form:

dVFS =n∏α=1

dpαdνα2 .

Following Heslot [5], we introduce dimensional coordinatesvia:

Zα = Xα + iY α√~

,

where Xα and Y α are real numbers with dimensions[X] =

[√~]

= Length√

Mass/Time and [Y ] =[√

~]

=Momentum

√Time/Mass. The ratio X/Y is a pure num-

ber, while their product XY has the dimension ~ of anaction. Note that dpαdνα/2 = dXαdYα/~. This allows usto write the Fubini-Study measure in a classical fashion:

dVFS =D−1∏α=1

dXαdY α

~

= d ~Xd~Y

~D−1 ,

where Xα play the role of generalized coordinates andY α that of generalized momenta. However, it is worthnoting that the global geometry of the classical phase-space differs substantially from that of P(H).

Given these definitions, it is now possible to calculate thenumber of states Ω(E) ≈ ω(E)δE , where δE is the size ofthe microcanonical energy shell and ω(E) is the densityof states:

ω(E) =∫h(Z)=E

dVFS

= πD−1

(D − 1)!

D−1∑k=0

D−1∏j 6=k,j=0

(E − Ek)+

(Ej − Ek) ,

where (x)+ := max(0, x). Since E ∈ [E0, Emax], thereexists an n such that E ∈]En, En+1[. This means that wecan stop the sum at k = n(E) since for all k > n we have(E − Ek)+ = 0. This gives:

ω(E) = πD−1

(D − 1)!

n(E)∑k=0

(D − 1)(E − Ek)D−2∏D−1j 6=k,j=0(Ej − Ek)

. (8)

This is in agreement with Ref. [21]’s Eq. (5). Appendix

II C provides a detailed proof, using a convenient mathe-matical result by Lasserre [31].

Figure 1. Alternate ensembles in the standard and geometricsettings: Differences are plainly evident. Canonical probabilitydistributions on a qubit’s state manifold CP 1 with coordinatesZ = (Z0, Z1) = (

√1− q,√qeiχ) where q ∈ [0, 1] and χ ∈

[−π, π]. CP 1 discretized using a 100-by-100 grid on the (q, χ)coordinates exploiting the fact that, with these coordinates, theFubini-Study measure is directly proportional to the Cartesianvolume element dVFS = dqdχ/2. The Hamiltonian is H =σx + σy + σz, with ~ = 1 and inverse temperature β = 5(kB = 1). (Right) Gibbs ensemble where circles enclose thepositions around the coordinates of the respective eigenvectors(q(|E0〉), χ(|E0〉)

)= (0.789,−2.356) and

(q(|E1〉), χ(|E1〉)

)=

(0.211, 0.785). (Left) Geometric Canonical Ensemble.

D. Quantum canonical ensemble: Statisticalphysics of quantum states

The geometric approach to microcanonical ensembles ex-tends straightforwardly to the canonical case, definingthe continuous canonical ensemble as:

pβ(Z) = e−βh(Z)

Qβ [h] , (9)

where:

Qβ [h] =∫P(H)e−βh(Z)dVFS .

Reference [20] first proposed the general form of the canon-ical partition function Qβ [h], working it out explicitly inseveral low-dimensional cases. Follow-on work providedan exact formula valid for arbitrary finite-dimensionalHilbert spaces [21]. Appendix II C provides an alternativeproof and explicit examples of:

Qβ [h] =D−1∑k=0

e−βEk∏nj=0,j 6=k(βEk − βEj)

. (10)

This is in full agreement with Ref. [21]’s Eq. (6). Fig-ure 1 plots the standard Gibbs ensemble (Right) and

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its geometric counterpart (Left) for the HamiltonianH = σx + σy + σz at inverse temperature β = 5.With the ensembles laid out we can now highlight thegeometric framework’s experimental consequences. Thenext section presents a proposal to discriminate betweenstandard and geometric ensembles.

VI. EXPERIMENTAL RELEVANCE: QUDITGAS

With its ensemble theory, the standard machinery ofquantum statistical mechanics turns on a system beingdiagonal in the Hamiltonian basis. And, this means thatsuperpositions between different energy eigenstates arenot allowed. While this might be consistent with severalexperimental results, it cannot always be the case. Wepresent an experimentally-concrete scenario that clearlyviolates this assumption, demonstrating the necessity ofstatistical geometric quantum mechanics.Consider a box containing a gas of N weakly interactingquantum systems (qudits) with Hilbert space of dimensionD. Let’s treat the spatial degrees of freedom classicallywhile applying a full quantum treatment to the internaldegrees of freedom. Prepare the system in a productstate |ψ〉 = ⊗Nk=1 |ψk〉 and let it evolve unitarily withHamiltonian H = H0 +HI , where H0 =

∑Nk=1Hk. Here,

H0 is the single-body Hamiltonian and HI is a two-bodyinteraction Hamiltonian. The interaction is weak in thesense that ||HI ||F ||H0||F , where we use the Frobeniusmatrix norm: ||A||F :=

√∑i,j |Aij |2. We also assume

that HI ’s presence removes possible degeneracies presentin H0 so that H’s spectrum is nondegenerate.To simplify the treatment, set D = 2. Let the box havea small hole from which qubits escape. The hole is con-nected to a Stern-Gerlach measurement apparatus thatperforms a projective measurement

Π(θ,φ)±

with:

Π(θ,φ)+ := |ψ(θ, φ)〉 〈ψ(θ, φ)| and (11a)

Π(θ,φ)− := |ψ⊥(θ, φ)〉 〈ψ⊥(θ, φ)| (11b)

where:

|ψ(θ, φ)〉 := cos θ/2 |0〉+ eiφ sin θ/2 |1〉 and|ψ⊥(θ, φ)〉 := sin θ/2 |0〉+ ei(φ+π) cos θ/2 |1〉 .

On the one hand, according to the general precepts ofquantum statistical mechanics the outcome probabilitiesp±(θ, φ) should be, up to experimental uncertainties:

pGibbs± (θ, φ) = e−βE0

Zβ〈E|0 Π(θ,φ)

± |E0〉+

+ e−βE1

Zβ〈E1|Π(θ,φ)

± |E1〉

On the other hand, statistical geometric quantum me-chanics predicts a different answer:

pGeo± (θ, φ) = 1

Qβ [h]

∫P(H)e−βh(Z)P

(θ,φ)± (Z),

where

P θ,φ+ (Z) = 〈ψ(Z)|Π(θ,φ)+ |ψ(Z)〉

= (cos θ/2)2 |Z0|2 + (sin θ/2)2 |Z1|2

+ sin θ eiφZ0Z

1 + e−iφZ0Z1

2 ,

and:

P θ,φ− (Z) = 1− P θ,φ+ (Z) .

The experimental protocol generalizes directly to a Hilbertspace of arbitrary dimension D. As concrete example,fixing the von Neumann measure to lie along the z axis,Fig. 2 compares the temperature-dependent behaviorof σz’s thermal average and standard deviation accord-ing to the Gibbs ensemble versus its geometric canonicalcounterpart. The system Hamiltonian is chosen to bethe same as before: H = σx + σy + σz. This is done forpractical convenience, as it guarantees that the resultsassociated with projective measurements of σx, σy, andσz are the same. The existence of a measurable differ-ence between the canonical ensemble and the geometriccanonical ensemble does not depend on this choice.

Figure 2. Gibbs and geometric canonical ensemble comparison:β-dependent behavior of the average (left panel) and standarddeviation (right panel) of a projective measurement alongthe z axis. While the qualitative dependence on β for theGibbs ensemble and the geometric canonical ensemble appearssimilar, quantitative differences are clear, both for the average(SGibbsz versus SGeo

z ) and for the fluctuations (∆SGibbsz versus

∆SGeoz ).

9

VII. GEOMETRIC QUANTUMTHERMODYNAMICS

With a consistent statistical geometric quantum mechan-ics in hand, we can now reformulate the thermodynamicsof quantum systems. Thermodynamic behavior is mod-eled via the geometric canonical state Eq. (9). Noticethat, in this setting, an appropriate entropy definition hasyet to be given. Paralleling early work by Gibbs, considerthe functional:

Hq [p] = −kB∫P(H)p(Z) log p(Z)dVFS .

A detailed information-theoretic analysis of this functionalwas done in Ref. [32].Let’s consider its role, though, for the quantum founda-tions of thermodynamics. In particular, assuming thatthe probability distribution of quantum states is thermal,we explore if this functional provides a thermodynamicentropy that is an actionable alternative to von Neumannentropy. For the geometric canonical ensemble of Eq. (9),this gives:

Hq = β(U − F ) ,

where:

U :=∫P(H)pβ(Z)h(Z)dVFS and

F := − 1β

logQβ

are, respectively, the average energy and the free energyarising from the geometric partition function Qβ .This means that we can directly import a series of funda-mental results from classical thermodynamics and statisti-cal mechanics into the quantum setting, fully amortizingthe effort invested to develop the geometric formalism.

A. First Law

The first result is a straightforward derivation of the FirstLaw:

dU =∫P(H)dVFSp(Z)dh(Z) +

∫P(H)dVFSdp(Z)h(Z)

= dW + dQ , (12)

We call the contribution dW work, since it arises from achange in the Hamiltonian h(Z) generated by an externalcontrol operating on the system. We call the contributiondQ heat, as it is associated with a change in entropy.Indeed, by direct computation one sees that:

dHq = βdQ and dF = dW .

This gives the standard form of the First Law for isother-mal, quasi-static processes:

dU = TdHq + dF ,

where T := (kBβ)−1.

B. Second Law

The Second Law follows from the Crooks [33] and Jarzyn-ski [34] fluctuation theorems [27, 35, 36]. Their treatmentcan be straightforwardly exploited, thanks to the Hamil-tonian nature of Schrödinger’s equation when written onthe quantum-state manifold P(H).

As summarized in Eq. (3), given a Hamiltonian h(Z, λ) onP(H) that depends on an externally-controlled parameterλ = λ(t), the unitary evolution is given by the Liouvilleequation Eq. (3) as in classical mechanics:

∂p(Z)∂t

= p(Z), h(Z, λ) .

One can now apply Jarzynski’s original argument [37]to driven quantum systems, without the need to exploitthe two-times measurement scheme [27]. The setup isstandard. The ensemble of quantum systems starts in ageometric canonical state defined by Eq. (9) and is thendriven with a Hamiltonian that depends on a parameterλ following the time-dependent protocol λ = λ(t) witht ∈ [0, 1]. An ensemble of several protocol realizations isrealized. And, we can define the single-trajectory workas:

W =∫ 1

0λ(t)∂h

∂λ

(Z(ψt), λ(t)

)dt ,

where λ = dλdt , Z(ψt) are the homogeneous coordinates

on CPD−1 for |ψt〉 and, therefore, are the solutions of Eq.(4).

With these premises, Jarzynski’s original argument appliesmutatis mutandis to give:

⟨e−βW

⟩ens = Qβ [h(λf )]

Qβ [h(λi)]= e−∆F , (13)

where λ(0) = λi and λ(1) = λf and 〈x〉ens denotes theensemble average over many protocol realizations. Fromthis, one directly applies Jensen’s inequality:⟨

e−βW⟩

ens ≥ e−β〈W 〉ens

to obtain the Second Law’s familiar form:

〈W 〉ens ≥ F . (14)

10

VIII. DISCUSSION AND CONCLUSION

While standard quantum mechanics is firmly rooted inan algebraic formalism, a geometric alternative based onthe differential geometry of the quantum-state manifoldP(H) = CPD−1 is readily available.As previous works repeatedly emphasized [2, 5, 11], thegeometric approach brings quantum and classical mechan-ics much closer. In both cases the manifold of states is aKähler space, with two intertwined notions of geometry—Riemannian and symplectic. It also sports a preferrednotion of measure, selected by invariance under unitarytransformations—the Fubini-Study measure. One canexploit this rich geometric structure to define continuousprobability distributions on the quantum-state manifold.The resulting formalism is equivalent to the standard onein familiar cases. However, it allows working with a newkind of quantum state, dubbed the geometric quantumstate, that generalizes the familiar density matrix andprovides more information about a quantum system’sphysical configuration.Leveraging parallels between the geometric formalism andclassical mechanics, the statistical treatment of geometricquantum mechanics provides a continuous counterpart ofGibbs ensembles. In particular, we looked at the micro-canonical and canonical continuous ensembles. Remark-ably, predictions from standard quantum statistical me-chanics and its geometric counterpart differ. This poseda challenge: Which theory should one use? To addressthis, Sec. VI proposed a concrete experiment with whichto directly probe, by means of simple projective mea-surements, the different predictions. Adopting a moregeneral point of view, thanks to the concentrationof measure phenomenon we expect the geomet-ric approach to provide the same physical predic-tions as quantum statistical mechanics, for large(macroscopic) systems. However, we expect thegeometric approach, both to statistical mechanicsand to thermodynamics, to be more appropriateto describe the phenomenology of small systemsat the nanoscale. In this sense, future work in thisdirection will be devoted to building a stochasticthermodynamics of quantum systems.Rounding out the development, Sec. VII laid out howto establish quantum thermodynamics on the basis ofthe geometric formalism. Building on Sec. V’s statisticaltreatment of geometric quantum mechanics, it derivedthe First and Second Laws of Geometric Quantum Ther-modynamics. Despite the two results appearing identicalto existing ones, derived within standard quantum statis-tical mechanics, they involve quantities that are genuinelydifferent. Understanding how Eqs. (12), (13), and (14)connect to their standard counterparts [27] is a chal-lenge that we must leave for the future. We note that asimilar result was obtained in Ref. [38] which, withoutthe geometric perspective, considered microcanonical andcanonical ensembles of pure states, as first advocated byKhinchin [29] and Schrödinger [30].

We conclude by highlighting an important feature that,so far, we did not make explicit. The geometric formalismsuggests a new concept of Boltzmann entropy for quantumstates—one markedly closer to that in classical statisticalmechanics:

SqB := kB logW ,

where W is the volume of accessible microstates in thecomplex projective manifold CPD−1.Along similar lines, we considered extending Shannon’sinformation functional to a nonmicrocanonical continuousensemble:

Hq [p] = −kB∫P(H)p(Z) log p(Z)dVFS .

When the probability distribution over the manifold of thestates is discrete (convex combinations of Dirac-deltas)this becomes formally equivalent to the von Neumannentropy functional. Moreover, these two notions of ther-modynamic entropy (SqB and Hq) are conceptually iden-tical to their classical counterpart: They evaluate thevolume of microstates compatible with certain macro-scopic conditions. This is a major conceptual differencewith the standard treatment of quantum statistical me-chanics, which is founded on the concept of von Neumannentropy. However, from the perspective of dynamicalsystems theory—arguably the umbrella under which bothclassical and quantum mechanics live—there is no con-ceptual difference. We believe the new concept of entropydeserves further attention, for example, by connectingto experiment via the maximum entropy estimation ofgeometric quantum states [39].

ACKNOWLEDGMENTS

F.A. thanks Marina Radulaski, Davide Pastorello, andDavide Girolami for discussions on the geometric for-malism of quantum mechanics. F.A. and J.P.C. thankDavid Gier, Dhruva Karkada, Samuel Loomis, and Ari-adna Venegas-Li for helpful discussions and the TellurideScience Research Center for its hospitality during visits.This material is based upon work supported by, or in partby, a Templeton World Charity Foundation Power of In-formation Fellowship, FQXi Grant FQXi-RFP-IPW-1902,and U.S. Army Research Laboratory and the U. S. ArmyResearch Office under contracts W911NF-13-1-0390 andW911NF-18-1-0028.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.

11

[1] F. Anza and J. P. Crutchfield. Beyond density matrices:Geometric quantum states. arXiv:2008.08682, 2020. 1, 2

[2] F. Strocchi. Complex Coordinates and Quantum Mechan-ics. Rev. Mod. Physics, 38(1):36–40, 1966. 1, 2, 10

[3] T. W. B. Kibble. Geometrization of Quantum Mechanics.Comm. Math. Phys., 65(2):189–201, 1979. 1, 2

[4] P. R. Chernoff and J.E. Marsden. Some remarks on hamil-tonian systems and quantum mechanics. In Foundationsof Probability Theory, Statistical Inference, and StatisticalTheories of Science, volume III, pages 35–53. Springer,1976. 1

[5] A. Heslot. Quantum mechanics as a classical theory. Phys.Rev. D, 31(6):1341–1348, 1985. 1, 2, 3, 7, 10

[6] G. W. Gibbons. Typical states and density matrices. J.Geom. Physics, 8(1-4):147–162, 1992. 1, 2

[7] A. Ashtekar and T. A. Schilling. Geometry of quantummechanics. In AIP Conference Proceedings, volume 342,pages 471–478. AIP, 1995. 1

[8] A. Ashtekar and T. A. Schilling. Geometrical Formulationof Quantum Mechanics. In On Einstein’s Path, pages 23–65. Springer New York, New York, NY, 1999. 1, 2

[9] L. P. Hughston. Geometric aspects of quantum mechanics.Ch. 6 in “Twistor Theory” (Stephen Huggett, ed.) - LectureNotes in Pure and Applied Mathematics, Vol. 169. MarcelDekker, 1995. 1

[10] D. C. Brody and L. P. Hughston. Geometric quantummechanics. J. Geom. Physics, 38(1):19–53, 2001. 2, 3

[11] I. Bengtsson and K. Zyczkowski. Geometry of QuantumStates. Cambridge University Press, Cambridge, 2017. 2,3, 10

[12] J. F. Cariñena, J. Clemente-Gallardo, and G. Marmo.Geometrization of Quantum Mechanics. Theor. Math.Physics, 152(1):894–903, 2007.

[13] D. Chruściński. Geometric Aspects of Quantum Mechan-ics and Quantum Entanglement. J. Physics: Conf. Ser.,30:9–16, 2006.

[14] G. Marmo and G. F. Volkert. Geometrical descriptionof quantum mechanics—transformations and dynamics.Physica Scripta, 82(3):038117, 2010.

[15] J. Avron and O. Kenneth. An elementary introductionto the geometry of quantum states with pictures. Rev.Math. Physics, 32(02):2030001, 2020.

[16] D. Pastorello. A geometric Hamiltonian description ofcomposite quantum systems and quantum entanglement.Intl. J. Geom. Meth. Mod. Physics, 12(07):1550069, 2015.

[17] D. Pastorello. Geometric Hamiltonian formulation ofquantum mechanics in complex projective spaces. Intl. J.Geom. Meth. Mod. Physics, 12(08):1560015, 2015.

[18] D. Pastorello. Geometric Hamiltonian quantum mechanicsand applications. Intl. J. Geom. Meth. Mod. Physics,13(Supp. 1):1630017, 2016.

[19] J. Clemente-Gallardo and G. Marmo. The EhrenfestPicture and the Geometry of Quantum Mechanics. IlNuovo Cimento C, 3:35–52, 2013. 1, 2

[20] D. C. Brody and L. P. Hughston. The quantum canonicalensemble. J. Math. Physics, 39(12):6502–6508, 1998. 1,

2, 3, 5, 6, 7[21] D. C. Brody, D. W. Hook, and L. P. Hughston. On

quantum microcanonical equilibrium. J. Physics: Conf.Ser., 67:012025, 2007. 6, 7

[22] D. C. Brody and L. P Hughston. Thermodynamicsof quantum heat bath. J. Physics A: Math. Theo.,49(42):425302, 2016. 1, 3

[23] D. C. Brody, D. W. Hook, and L. P Hughston. Quantumphase transitions without thermodynamic limits. Proc.Roy. Soc. A: Math. Phys. Engin. Sci., 463(2084):2021–2030, 2007. 1, 6

[24] U Seifert. Stochastic thermodynamics: Principles andperspectives. Euro. Physic. J. B, 64(3-4):423–431, 2008.1

[25] U. Seifert. Stochastic thermodynamics, fluctuation the-orems and molecular machines. Rep. Prog. Physics,75(12):126001, dec 2012. 1

[26] J. Gemmer, M. Michel, and G. Mahler. Quantum Ther-modynamics, volume 657 of Lecture Notes in Physics.Springer Berlin Heidelberg, Berlin, Heidelberg, 2004. 1

[27] S. Deffner and S. Campbell. Quantum Thermodynamics.IOP Publishing, 2019. 1, 9, 10

[28] F. Strocchi. An introduction to the mathematical structureof quantum mechanics: A short course for mathematicians.World Scientific, Singapore, 2008. 3

[29] Khinchin. Mathematical Foundations of Quantum Statis-tics. Dover: New York, 1951. 5, 10

[30] Schroedinger. Statistical Thermodynamics. CambridgeUniversity Press, 1952. 5, 10

[31] J. B. Lasserre. Volume of slices and sections of the simplexin closed form. Optimization Letters, 9(7):1263–1269, 2015.7, 1, 4

[32] D. C. Brody and L. P. Hughston. Information content forquantum states. J. Math. Physics, 41(5):2586–2592, 2000.9

[33] G. E. Crooks. Entropy production fluctuation theoremand the nonequilibrium work relation for free energy dif-ferences. Phys. Rev. E, 60(3):2721–2726, 1999. 9

[34] C. Jarzynski. Equalities and Inequalities: Irreversibilityand the Second Law of Thermodynamics at the Nanoscale.Ann. Rev. Cond. Matt. Physics, 2(1):329–351, 2011. 9

[35] M. Campisi, P. Hänggi, and P. Talkner. Colloquium:Quantum fluctuation relations: Foundations and applica-tions. Rev. Mod. Phys., 83(3):771–791, 2011. 9

[36] R. Klages, W. Just, and C. Jarzynski. Nonequilibriumstatistical physics of small systems: Fluctuation relationsand beyond. Wiley-VCH, 2013. 9

[37] C Jarzynski. Nonequilibrium Equality for Free EnergyDifferences. Phys. Rev. Lett., 78(14):2690–2693, 1997. 9

[38] M. Campisi. Quantum fluctuation relations for ensemblesof wave functions. New J. Phys., 15, 2013. 10

[39] F. Anza and J. P. Crutchfield. Geometric quantum stateestimation. arXiv:2008.08679, 2020. 10

1

Supplementary Materials

Geometric Quantum Thermodynamics

Fabio Anza and James P. Crutchfield

The following presents proofs and detailed calculations supporting the main text’s claims.

I. INDEPENDENT RESULT

For completeness, the following summarizes Ref. [31]’s result called on in calculating the density of states. Giventhe n-simplex ∆n :

~x ∈ Rn+ : ~e · ~x ≤ 1

, where ~e is the vector of ones in Rn, a section of the simplex is defined by a

vector ~a ∈ Sn and we want to compute the n-dimensional and (n− 1)-dimensional volume of the following sets:

Θ(~a, t) := ∆n ∩~x ∈ Rn : ~aT · ~x ≤ t

and

S(~a, t) := ∆n ∩~x ∈ Rn : ~aT · ~x = t

,

where ~aT is the transpose of ~a. The result assumes flat geometry, which is obtained from the volume elementdp1dp2 . . . dpn. Letting (x)+ := max(0, x) and a0 = 0, then:

Vol (Θ(~a, t)) = 1n!

n∑k=0

(t− ak)n+∏nj 6=k , j=0(aj − ak)

= 1n!

tn∏nk=1 ak

+ 1n!

n∑k=1

(t− aj)n+∏nj 6=k , j=0(aj − ak)

and:

Vol (S(~a, t)) = 1(n− 1)!

n∑k=0

(t− ak)n−1+∏n

j 6=k , j=0(aj − ak)

= 1(n− 1)!

tn−1∏nk=1 ak

+ 1(n− 1)!

n∑k=1

(t− aj)n−1+∏n

j 6=k , j=0(aj − ak).

II. GEOMETRIC QUANTUM DENSITY OF STATES AND CANONICAL ENSEMBLE

Again for completeness, we first recall the basic definitions, given in the main text, used in the two sections that followto calculate the density of states and statistical physics of quantum states in the geometric formalism.

A. Setup and notation

Consider a Hilbert space H of finite-dimension D. The manifold P(H) of states is the complex projective spaceCPD−1. A point Z on the manifold is a set of D homogeneous and complex coordinates Zα. A point correspondsto a pure state with the identification Z ↔ |ψ〉 =

∑D−1α=0 Z

α |eα〉, where |eα〉α is an arbitrary but fixed basis of H.This parametrization underlies the choice of a reference basis that, however, is ultimately irrelevant. While concretecalculations of experimentally measurable quantities can be made easier or harder by an appropriate coordinate system,the overall result is independent on such choices. The quantum mechanical expectation value is a quadratic and realfunction on the manifold of the quantum states:

a(Z) := 〈ψ(Z)|A |ψ(Z)〉

=D−1∑α,β=0

Aα,βZαZ

β.

2

When A = H is the system’s Hamiltonian, the function a(Z) = h(Z) generates the vector field VH on CPD−1. Theassociated Hamiltonian equations of motion become the Schrödinger equation (and its complex conjugate) when usingthe standard formalism with Hilbert spaces. In the geometric formalism, states are functionals from the algebra ofobservables to the real numbers. Effectively, they are probability distributions, both discrete and continuous, on thequantum-state manifold CPD−1.

B. Microcanonical density of states: Proof of Eq. (8)

We start with the a priori equal probability postulate and build the microcanonical shell as follows:

pmc(Z) =

1/W (E) if h(Z) ∈ [E , E + δE ]0 otherwise

.

Due to normalization we have:

W (E) =∫h(z)∈Imc

dVFS ,

where dVFS is the volume element of the Fubini-Study metric:

dVFS = 12n dp1dp2 . . . dpndν1 . . . dνn .

This gives the manifold volume:

Vol(CPn) = πn

n! .

For concrete calculations, normalize the measure so that CPD−1’s total volume is unity, using:

dµn = dVFSVol(CPn)

= n!(2π)n

n∏k=1

dpk

n∏k=1

dνk .

This does not alter results in the main text. On the one hand, calculations of measurable quantities are independent ofthis value. On the other, here, at the calculation’s end, we reintroduce the appropriate normalization.

We can now computeW (E) for a generic quantum system. Assuming that δE |Emax−Emin|, we haveW (E) = Ω(E)δEand Ω(E) is the area of the surface Σ defined by h(Z) = E :

Ω(E) =∫

Σdσ ,

where dσ is the area element resulting from projecting both the symplectic two-form and the metric tensor onto thesurface Σ. To compute this we choose an appropriate coordinate system:

Zα = 〈Eα|ψ(Z)〉= nαe

iνα

3

adapted to the surface Σ:

h(Z) = 〈ψ(Z)|H |ψ(Z)〉

=n∑k=0

Ek| 〈ψ|Ek〉 |2

=n∑k=0

Ekn2k

= E .

On both sides we subtract the ground state energy E0 and divide by Emax − E0 to obtain the following definingequation for Σ ⊂ CPn:

F (n0, n1, . . . , nn, ν1, . . . , νn) =n∑k=0

εkn2k − ε

= 0 ,

with:

εk = Ek − E0

Emax − E0∈ [0, 1] and

ε = E − E0

Emax − E0∈ [0, 1] .

We use octant coordinates for CPn:

(Z0, Z1, . . . , Zn) =(n0, n1e

iν1 , n2eiν2 . . . , nne

iνn),

where nk ∈ [0, 1] and νk ∈ [0, 2π[. With the transformation pk = n2k the equation for Σ becomes:

n∑k=0

pkεk − ε = 0 .

1. Qubit Case

The state space of a single qubit is CP 1. The latter’s parametrization:

pε0 + (1− p)ε1 = 1− p

means that h(Z) ≤ E is equivalent to 1− p ≤ ε. The volume is therefore given by:

Voln=1(E) = 1π

∫h(φ)≤E

dVFS

= 12π

∫ 1

1−εdp

∫ 2π

0dν

= ε

= E − E0

E1 − E0.

In turn, this gives;

Wn=1(E) = Voln=1(E + δE)−Voln=1(E)

= 1E1 − E0

δE .

4

In other words:

Ωn=1(E) = 1E1 − E0

,

which is a constant density of states.

2. Qutrit Case

The state space of qutrits is CP 2, with parametrization Z = (Z0, Z1, Z2) = (1 − p − q, peiν1 , qeiν2). With thesecoordinates, the equations defining the constant-energy hypersurface is:

(1− p− q)ε0 + pε1 + qε2 = pε1 + q ≤ ε .

And, it has volume:

Voln=2(E) = 2(2π)2

∫∫dqdq

∫∫dν1dν2

= 2∫∫

S

dpdq .

In this, we have the surface S :=

(p, q) ∈ R2 : p, q ≥ 0, p+ q ≤ 1, q ≤ ε− pε1. Examining the geometry we directly

see that the region’s area is:

A(S) =

12 −

12

(1−ε)2

1−ε1when ε ≥ ε1

ε2

2ε1when ε < ε1

.

Or:

A(S) =

12 −

12

(E2−E)2

(E2−E1)(E2−E0) when E ≥ E112

(E−E0)2

(E1−E0)(E2−E0) when E < E1.

One can check that the function A(S)[E ] and its first derivative are continuous. Eventually, we have:

Wn=2(E) = Voln=2(E + δE)−Voln=2(E)

= 2(E2−E)

(E2−E1)(E2−E0)δE when E ≥ E12(E−E0)

(E2−E0)(E1−E0)δE when E < E1.

3. Generic Qudit Case: CPn

To use Ref. [31]’s result, summarized in App. I, we must change coordinates. Again, using “probability + phase”coordinates:

n∑k=0

pkEk = E

5

means that:n∑k=1

pkak = t(E)

ak = a(Ek)

= Ek − E0

R,

R =

√√√√ n∑k=1

(Ek − E0)2, and

t(E) = E − E0

R.

In this way, we can apply the result, finding:

Voln (E) =n∑k=0

(t− ak)n+∏nj 6=k , j=0(aj − ak)

=n∑k=0

(E − Ek)n+∏nj 6=k,j=0(Ej − Ek)

.

Since E ∈ [E0, Emax], there exists an n such that E ∈]En, En+1[. This means that the sum in the second term stopsat k = n because after that (E − Ek)+ = 0. Hence, there exists n(E) such that for all k > n we have (E − Ek)+ = 0.This, in turns, shows that:

Voln (E) =n(E)∑k=0

(E − Ek)n∏nj 6=k,j=0(Ej − Ek)

.

This leads to the desired fraction of CPn microstates in a microcanonical energy shell [E , E + dE ]:

Wn(E) = Ωn(E)dE

=

n(E)∑k=0

n(E − Ek)n−1∏nj 6=k,j=0(Ej − Ek)

dE .

This allows defining the statistical entropy S(E) of a quantum system with finite-dimensional Hilbert space of dimensionD = n+ 1 as:

S(E) = logWD−1(E) .

C. Statistical physics of quantum states: Canonical ensemble

The continuous canonical ensemble is defined as:

ρβ(ψ) = e−βh(ψ)

Qβ [h] ,

where:

Qβ [h] =∫CPD−1

e−βh(ψ)dVFS .

The following analyzes the simple qubit case and then moves to the general treatment of a finite-dimensional Hilbertspace H.

6

1. Single Qubit

The Hilbert space here is H while the pure-state manifold is CP 1. And so, we have:

Qβ [h] = 14

∫ π

0dθ sin θ

∫ 2π

0dφ e−βh(θ,φ) ,

with h(θ, φ) = ~γ · 〈~σ〉 = ~γ ·~b(θ, φ).

Since we consider a single qubit, whose state space is S2 embedded in R3, we can write ~γ ·~b(ψ) = ||~γ|| cos θ, where θ isthe angle between ~γ and ~b(ψ). Thus, we can use an appropriate coordinate h(φ, θ) = ||~γ|| cos θ aligned with ~γ to find:

Qβ [h] = πsinh β||~γ||β||~γ||

.

Or, using “probability + phase” coordinates (p, ν) we can also write:

12

∫ 1

0dp

∫ 2π

0dν e−β[(1−p)E0+pE1] = π

e−βE0 − e−βE1

β(E1 − E0) .

The change in coordinates is given by the result of diagonalization: E0 = −||~γ|| and E1 = ||~γ||. This yields the expectedresult:

Qβ [h] = πe−βE0 − e−βE1

β(E1 − E0)

= πsinh β||~γ||β||~γ||

.

2. Generic Treatment of CPn

We are now ready to address the general case of qudits:

Qβ [h] =∫CPn

e−βh(Z)dVFS

= 12n

∫ n∏k=0

e−βpkEkn∏k=1

dpkdνk

= πn∫

∆n

n∏k=0

e−βpkEkδ

(n∑k=0

pk − 1)dp1 . . . dpn .

To evaluate the integral we first take the Laplace transform:

In(r) :=∫

∆n

n∏k=0

e−βpkEkδ

(n∑k=0

pk − r

)dp1 . . . dpn

to get:

In(z) :=∫ ∞

0e−zrI(r)dr .

7

Calculating, we find:

In(z) =n∏k=0

(−1)k

(βEk + z)

= (−1)n(n+1)

2

n∏k=0

1z − zk

.

with zk = −βEk ∈ R.The function In(z) has n + 1 real and distinct poles: z = zk = −βEk. Hence, we can exploit the partial fractiondecomposition of In(z), which is:

(−1)n(n+1)

2

n∏k=0

1z − zk

= (−1)n(n+1)

2

n∑k=0

Rkz − zk

,

where:

Rk =[(z − zk)In(z)

]z=zk

=n∏

j=0, j 6=k

(−1)n(n+1)

2

zk − zj.

The inverse Laplace transform’s linearity, coupled with the basic result:

L−1[

1s+ a

](t) = e−atΘ(t) ,

where:

Θ(t) =

1 t ≥ 00 t < 0

,

gives:

In(r) = L−1[In(z)](r)

= Θ(r)n∑k=0

Rkezkr .

And so, we finally see that:

Qβ [h] = In(1)

=n∑k=0

e−βEk∏nj=0, j 6=k(βEk − βEj)

.