Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations

Irreversibility, crucial in both thermodynamics and information theory, is naturally studied by comparing the evolution -- the (forward) channel -- with an associated reverse -- the reverse channel. There are two natural ways to define this reverse channel. Using logical inference, the reverse channel is the Bayesian retrodiction (the Petz recovery map in the quantum formalism) of the original one. Alternatively, we know from physics that every irreversible process can be modeled as an open system: one can then define the corresponding closed system by adding a bath ("dilation"), trivially reverse the global reversible process, and finally remove the bath again. We prove that the two recipes are strictly identical, both in the classical and in the quantum formalism, once one accounts for correlations formed between system and the bath. Having established this, we define and study special classes of maps: product-preserving maps (including generalized thermal maps), for which no such system-bath correlations are formed for some states; and tabletop time-reversible maps, when the reverse channel can be implemented with the same devices as the original one. We establish several general results connecting these classes, and a very detailed characterization when both the system and the bath are one qubit. In particular, we show that when reverse channels are well-defined, product-preservation is a sufficient but not necessary condition for tabletop reversibility; and that the preservation of local energy spectra is a necessary and sufficient condition to generalized thermal operations.


I. INTRODUCTION
Irreversibility is ubiquitous in real life.In science, it was first studied systematically in the context of thermodynamics: this is captured by the Second Law, which stipulates the impossibility of putting all the energy to fruition, leading to the necessary generation of heat-or more generally, entropy [1][2][3][4][5][6][7].Eventually, information theory became the setting in which to study irreversibility: a process is irreversible for an agent when the agent is unable to retrieve from the output all the information about the input.In turn, a theory of optimal retrieval of information was developed, both in classical and in quantum theories [8][9][10][11][12].
Meanwhile, the field of stochastic thermodynamics developed quantitative approaches to irreversibility, based on statistical comparisons between the process under study and its associated reverse process.But how to define the latter?In the case of fully reversible, deterministic processes, the reverse process is obviously the dynamics played backwards.For isothermal evolutions (driven Hamiltonian evolution while the system is in contact with a thermal bath), a possible and very natural reverse process consists in driving the evolution backwards in the presence of the same bath [13,14].Reverse process have also been found for more complex processes, through expert control of the model and its assumptions (see e.g.[15]).A general recipe may be built on the observation that any irreversible process can be seen as a marginal of a global, reversible process involving the system and some environment.The recipe through dilation is then: add a suitable environment (dilation), trivially reverse the reversible global process, and finally remove the environment.
Recently, it was proposed to define the reverse process using the Bayesian recipe for information retrieval, a.k.a.Bayesian inversion or retrodiction [16,17].This recipe through retrodiction requires only choosing a reference state, which plays the role of a Bayesian prior.The connection between reverse processes and Bayesian logic had not been noticed in the context of classical stochastic thermodynamics.In quantum thermodynamics, one the main tools for information recovery had been used, first occasionally [14,18], then systematically [19,20]: the Petz recovery map [11,21,22], which was also proposed early on as a quantum analog of Bayesian inversion [23][24][25][26].
In this work, after a review of known material on reverse processes (Section II), we start by proving that the two recipes by dilation and by retrodiction are identical, both in the classical and the quantum case (Section III).The fact that the two proposed general recipes coincide, combined with the knowledge that all the previously known special cases can be recovered with these recipes, shows that we have the definition of the reverse process under control.Next we bring up the observation that a process and its associated reverse process may be very different.It is indeed well known in the quantum case that implementing the reverse (Petz) of a channel may require very different resources than those needed to implement the channel itself (see e.g.[27]).The cases mentioned above of the reversible and the isothermal processes, whose reverses are "what one would expect" and can be implemented with the same control and the same environment, seem to be the exception.Based on this observation, we set to study which processes have a reverse that can be implemented with the same, or similar, resources (Fig. 1).We shall say that the latter processes possess tabletop reversibility.This is of interest for the structure of the theory of the reverse processes, as well as for possible experimental tests of fluctuation theorems in situations that are not unitary or isothermal.The mathematical definition of tabletop reversibility is defined in Section IV, together with the auxiliary notion of productpreserving channels, which may be of interest in its own right.In Section V, we present both general results valid and a thorough characterization of two-qubit channels.In Section VI, we highlight the implications of these results and scenarios for energetics and reversibility in the quantum regime.Section VII is a conclusion.FIG.1: An illustration of the main goal of this paper.
Any channel E on a system can be viewed as a larger reversible unitary process U involving the system and a bath (top).Applying the recipes by dilation or by retrodiction (proved identical in Section III), one finds that the reverse process must in general be implemented with completely different tools than the original process (bottom left).We set out to characterize the tabletop reversible situations, in which the reverse process can be implemented with the same, or similar, tools as the forward process: namely, by appending an ancilla and inverting the original unitary U.The exact definitions will be given in Section IV, and the results in the following Sections.

II. REVERSE PROCESSES A. Notations
In classical theory, we consider a discrete state space with d distinct states.A generic state is represented by a probability distribution p: p(j) ≥ 0 for j = 1, ..., d; and d j=1 p(j) = 1.It can be represented by a d × 1 probability vector p, whose entries are p j := p(j).A generic channel is a stochastic map φ, defined by d 2 probabilities φ(j ′ |j) of transiting from the input state j to the output state j ′ .These probabilities must satisfy φ(j ′ |j) ≥ 0 for all j, j ′ and d j ′ =1 φ(j ′ |j) = 1 for all j.The channel can be represented by the d × d stochastic matrix φ, whose entries are φ j ′ j := φ(j ′ |j).In this representation, the composition of channels is represented by the standard matrix multiplication: if φ = φ 2 • φ 1 , then φ = φ 2 φ 1 .
An important remark for what follows: even if the matrix φ has an inverse, in general the entries of the matrix φ −1 do not define a valid stochastic map.Analogously, while every matrix can be transposed, the map corresponding to φ T is a valid map if and only if the channel is bistochastic, i.e. satisfies the additional property d j=1 φ(j ′ |j) = 1 for all j ′ .When the inverse or the transpose of the matrix do define valid channels, we shall denote those channels as φ −1 and φ T respectively.
In quantum theory, we consider a finite-dimensional complex vector space of dimension d.A generic state is described by a semidefinite operator ρ ⪰ 0 in this space with Tr(ρ) = 1.Channels are represented by completely positive, trace preserving (CPTP) maps.Given a CPTP map E, the adjoint E † is the unique map for which for operators X and Y .Just as φ T is not a valid stochastic map in general, E † is in general not a valid quantum channel.

B. Reverse Process: Standard Examples
We first review some classes of processes where the reverse-or at least, a candidate for it-is known (see Fig. 2(a) for an illustration).
The most obvious class is that of reversible processes, where the map is a bijection between the space of states.Their reverse processes are naturally defined as the evolution played backwards, i.e. the inverse map.In classical theory, such processes are Hamiltonian evolutions Φ obeying Liouville's theorem: reversal is given by inverting the trajectories in the configuration space.In a discrete state space, the matrix Φ is a permutation matrix, whose inverse and transpose coincide (Φ −1 = Φ T ) and define a valid map.Analogously, in quantum theory, bijective transformations are unitary channels Illustrations for standard examples of reversal.(b) Bayesian inversion or "retrodiction" is a formal recipe that reproduce results of the standard approach while generalizing the definition of reverse processes for any characterized process, and any under setting as captured by a reference state.
U U † = 1: their inverse exists, coincides with the adjoint, and defines a valid map.For a reversible process, the inverse map is the only reasonable candidate of the corresponding reverse process.
Moving to bistochastic/unital processes, their inverse is in general not defined, but their transpose/adjoint is a valid channel and thus provides an immediate, natural candidate for the reverse map.Beyond this class, the transpose/adjoint ceases to define a valid channel.On this basis, it has been argued that only bistochastic/unital processes are fundamental if one wants the theory to be fundamentally reversible [28][29][30].We do not take sides in that discussion: we are not concerned with ultimate constraints on fundamental theories, but with the description of practical irreversibility.
The most obvious irreversible processes describe the dissipation of information in an unmonitored environment, or arise from a coarse-graining over a chaotic dynamics.In both cases, the dynamics is generally not bistochastic/unital.The reverse of some such processes has been constructed on a case-by-case basis by invoking physical arguments.For a system undergoing Hamiltonian evolution while in contact with thermal baths, under the assumption of detailed balance, the recipe is very sinmple: the reverse process consists in playing the Hamiltonian evolution backward, while staying in contact with the same thermal baths [6,13].This recipe was used in the experimental demonstrations [31] of Crooks' fluctuation theorem with biological systems [32,33] and levitating nanospheres [34,35].For quantum channels, the same result holds for the so-called thermal operations T [14,[36][37][38][39][40][41][42][43].

C. Reverse Processes: General Recipe through
Bayesian Retrodiction

Generalities
Having reviewed the prime examples of constructed reverse processes, we now describe a general recipe applicable to every process: Bayesian retrodiction (sometimes called "Bayesian inversion") [16,17].It can be traced back to the works of Watanabe [44,45], ultimately building on the observation that the laws of physics give us the knowledge of the evolution (the channel, in our language) and not the initial state.
In classical theory, given the channel φ, the recipe for Bayesian inversion is standard: one postulates a reference prior γ, then applies Bayes' rule to the joint probability distribution P (a, a ′ ) := φ(a ′ |a)γ(a).The resulting reverse map is where the distribution φ[γ] is given by φ[γ](a ′ ) = a φ(a ′ |a)γ(a), the output obtained by propagating the reference prior γ through the channel φ.In matrix notation, Eq. ( 2) reads where D p is the diagonal matrix with entries corresponding to the distribution p.
The extension of Bayesian formalism to the quantum formalism has been the object of many studies [23-25, 46, 47].Here, we don't need an exhaustive Bayesian toolbox: only a candidate for the reverse map.Our choice is the Petz Recovery map Êα [21,48,49] where α is a state that plays the role of reference prior.The choice of the Petz map as the quantum analog of Bayes' rule is standard [23,25,26].It has recently been shown to fulfill the most crucial intuitions about reversal [20,24,25,49] and to be suited to recover results in stochastic quantum thermodynamics [16,19].
From now onwards, we identify and will use these notations interchangeably when convenient.A property of the Bayes/Petz maps that we shall use later is composability [24]: One may see Appendix C 1 for proofs.This property holds even when the maps are not stochastic.
We note in passing that, due to presence of the term ) is rank-deficient.Of course, one could attempt to side-step this problem several ways: for example, by defining ) only on its support, or by considering a neighbourhood of full-rank states around the rank-deficient output and taking some limit.However, this boils down to a matter of convention, where each approach gives a different retrodiction channel, as we discuss in Appendix F in some detail.We do not make a particular choice, and instead leave the retrodiction channel undefined in this case.

Examples revisited
The examples of reverse maps of the previous subsection are recovered, and possibly clarified, in the retrodictive approach.Reversible maps are the only ones, for which the Bayes/Petz map does not depend on the reference prior [17,25,45]; and, unsurprisingly, coincides with the inverse: For the case of bistochastic/unital maps, the Bayes/Petz does depend on the reference prior [17,25].The reverse described above is obtained for a very natural choice of reference prior, namely the uniform: γ := u with u(j) = 1/d in the classical case, α := 1/d in the quantum case.Indeed, this prior is preserved by these maps (φ[u] = u, E[1/d] = 1/d), and one can immediately see that Lastly, let us look at thermal operations in the quantum language.Given a non-interacting system-bath Hamiltonian where τ κ (H) = e κH / tr e κH is the thermal (or Gibbs) state with κ = −1/k B T (usually denoted −β, but later in the paper we use β will denote a state of the bath) and where U is constrained to satisfy A channel thus constructed preserves the thermal, or Gibbs, state of the system for the same κ: T [τ κ (H A )] = τ κ (H A ).The Petz map with the Gibbs state as reference prior is found to be ( [14], see also Appendix A for the derivation) which describes indeed a reversal of the unitary dynamics while in contact with the same thermal bath.Notice how, having adopted the retrodictive approach, the usual thermodynamical assumptions called "microreversibility" and "detailed balance" are replaced by the single assumption on the choice of the reference prior.

III. REVERSE PROCESSES AND DILATIONS
Before studying tabletop reversibility, we need to introduce the notion of dilation of a process.The word, common in the language of quantum channels, describes the extension of a process on system A to include an environment (or "bath", or "ancilla") B.
Typically, a dilation is performed with the goal of making the extended system AB a closed one, whose dynamics is therefore reversible.Hence, it is natural to look at defining reverse processes by the following recipe: dilate by adding the environment, reverse the map of the dilation (trivial if reversible), then remove the environment.A priori, this recipe is different from the Bayesian one applied on the system alone: dilations are not unique, and the reverse process obtained by this recipe might depend on the details of the chosen dilation.We proceed to prove that the two recipes actually coincide: given a choice of dilation, only the reference prior chosen on the system determines the reverse process.This holds both for classical and quantum systems.

A. Classical Dilations & Bayes' Rule
Every classical process φ on a system state space A (|A| = d A ) can always be expressed as a marginal of a global process Φ AB on an extended state space AB (|B| = d B ), alongside some potentially-correlated environment β B .This may be expressed by: A tuple (Φ AB , β B ) that fulfills (14) will be called a dilation of φ.We proceed to prove our claim for classical processes: Result 1.Given a classical map φ, the reverse obtained by dilating with an environment, reversing the dilated map, then removing the environment, is the same as that obtained directly through the Bayesian recipe (2) on the system.
Proof.Let us construct the reverse with the dilation.Besides the reference prior on the system A, we have the additional freedom of choosing a dilation (Φ AB , β B ).The total reference prior is then Γ(a, b) = γ(a)β(b|a), and we define the reverse of the dilated map by applying Bayes' rule to Finally, we remove the environment.Writing η := Φ[Γ] for readability, we have ). where on the left-hand side we have used the fact that (Φ AB , β B ) is a dilation of φ, and where φ′ Γ (a|a ′ ) is the reverse map obtained through this recipe.By summing on both sides over a, we see that η(a ′ ) = φ[γ](a ′ ): whence φ′ Γ (a|a ′ ) is identical to Eq. (2).In particular, the only freedom left is indeed that of choosing γ A .
Notice that we did not have to assume that Φ AB is reversible: the proof is valid for any dilation.Also, we did not have to assume that β(b|a) = β(b) carries no initial correlations; of course, one can choose a dilation with this property, if deemed physically important.By contrast, having chosen the dilation, the posterior η :

B. Assignment Maps
Let us now have a more detailed look at the structure of dilations.The first operation (appending an environment) appears as the natural reverse of the last operation (tracing out the environment).We are going to show that this is indeed the case (see Appendix C for supplementary proofs).
Denote the operation of tracing out the environment by Σ B .The map of appending an environment B to the system A (assignment map), with reference state on AB given by Λ, is given by the Bayesian reverse of Σ B : Explicitly, ΣB,Λ [p A ](a, b) = p(a)Λ(b|a) has the form of Jeffrey's update: given a reference joint distribution Λ(a, b), if one gets the information that the distribution of A is actually given by p, the rational way of updating one's knowledge is to update A's marginal while keeping what attains to B unchanged.
In turn, the Bayesian reverse of any classical assignment map, for which Λ(a, b) is product, is the partial trace, for any choice of reference prior: R c ΣB,□⊗β , γ = Σ B for all β and γ.
The generic definition ( 14) of the dilation (Φ AB , β B ) can then be written as with a choice of Λ such that Λ(a,b) a Λ(a,b) = β(b|a).By using the composability property (7), one obtains which is what we proved in Result 1.For relevant proofs see Appendix C 2.
Classically, by choosing Λ(b|a) ̸ = Λ(b), initial systembath correlation are straightforwardly described.The quantum analog, by contrast, took some discussions to be clarified [50][51][52].The Petz map of the partial trace satisfies all the properties of a completely positive assignment map.For a generic reference state Ω of AB, it reads With this definition of the quantum assignment map, we now tackle retrodiction on dilations in the quantum formalism.

C. Quantum Dilations & the Petz Recovery Map
For quantum processes, we focus on unitary dilations with an initially uncorrelated state of the environment.Any quantum channel E can be seen as the marginal of a global unitary U acting on a target input and an ancillary system prepared in a suitable density operator β [22,53]: As in the classical case, it can be seen as the composition of channels The Petz map of E with reference state α can be computed directly, using the fact that the adjoint of ( 21) is where between the first and the second line we inserted two identities , and where we used (19).
Thus, we verified directly the composition that was expected on formal grounds (8).We can then state the claimed: Result 2. Given a quantum map E, the reverse obtained by dilating with an environment, reversing the dilated map (accounting for propagated correlations through the assignment map), then removing the environment, is the same as the Petz map (4) computed directly on the map.In other words, the knowledge of a dilation of E does not add any useful information to define the reverse of E.
Proof.The recipe through the dilation is the composition of Proofs for each individual part of the decomposition are found in Appendix C 3. This composition indeed coincides with the Petz map, as proved in (23).
We summarize the structural symmetries that relate dilation and retrodiction, for classical and quantum theory, in Table I.In both regimes, the role of retrodictive assignment maps Tr B, U [α⊗β] , ΣB,Φ[γ⊗β] ensures consistency in expressing the reverse process.The last line of the table anticipates the definition of tabletop reversibility, the central object of this paper, which we are going to introduce next.

IV. DEFINITION OF TABLETOP REVERSIBILITY AND RELATED CLASSES OF CHANNELS
In this Section, we introduce the new classes of channels that are the central object of this work.From here onward, we work only in the quantum formalism.When required, we shall highlight whether a result is purely quantum, or is also true for classical processes.

A. Tabletop Reversibility
Our primary concern is the implementation of a Petz map Êα , given the control on the implementation of the FIG.3: Two routes for Bayesian retrodiction illustrated.One can show that these two protocols always give the same reverse process, as long as the propagated correlations formed across the reference prior and the ancillary environment is accounted for.This is captured by the retrodictive assignment map (23).
If no correlations are formed, and reverse map is well-defined ⇒ tabletop reversibility Tabletop Time-Reversibility φTR

TABLE I: Summary table of the relation between retrodiction and dilations.
To stress the comparison between classical and quantum theory, in this table the classical dilation Φ is a reversible channels, and the bath is taken as initially uncorrelated with the system (although these restrictions are not needed, as proved in the text). channel Implementing the Petz map is not straightforward, and approximate realisations have been studied recently [27,54].With what we introduced, we can understand the reason of this difficulty.On the one hand, since the Petz map is a CPTP map, there exist a unitary V and an ancillary state β such that Êα But in general, V ̸ = U † : we may have to build a dedicated unitary.On the other hand, we have just seen in Eq. ( 23) that every Petz map can be written (19): we may have to do something more complicated than appending an ancilla.
We want to identify the special cases, in addition to unitary and isothermal channels, where the reverse channel can be obtained by just appending an ancilla and inverting the unitary: for the same U that enters the dilation of E.
Notice that this definition does not mean that the reverse should be implementable by acting only on the system, a situation studied by Aberg [55] and inspired by dynamical decoupling.Even in the generic case of isothermal processes one may have to invert the systembath interaction, if the latter is not constant.

B. Product-Preserving Maps & Generalized
Thermal Operations Here we introduce two more definitions that will be used below.

Definition 2. A unitary that acts in a joint Hilbert space
Here, α, α ′ ∈ S(H A ) and β, β ′ ∈ S(H B ) are positive semidefinite operators with trace one, and at least one of α, β is not maximally mixed (to exclude the obvious case U (1 ⊗ 1)U † = 1 ⊗ 1).We shall also call any tuple (U, α, β) for which U is product-preserving with respect to α and β a product-preserving tuple.
Contrary to the production of correlations (e.g. in universal entanglers [56,57]) and the preservation of maximal entanglement (Bell-to-Bell maps [57]), productpreservation has not been studied systematically prior to this work.It appears as a natural property of thermal maps, and is at the origin of several results in entropy production, thermalization and reversibility in the quantum regime [14,[36][37][38].By relaxing Eq. ( 12), we enlarge that natural setting in a way that was already used in some other works in the literature [14,18]: where either H A or H B is not proportional to the identity.The corresponding generalized thermal map is given by E that is, (U, τ κ (H A ), τ κ (H B )) are a product-preserving tuple for all generalized thermal operations.
In subsection V A, we prove some general connections between these classes of processes.At a glance: • Theorem 1 fully characterizes the generalized thermal as a subset of product-preserving unitaries.
• Theorem 2 establishes a strong relation between the input pair (α, β) and the output pair (α ′ , β ′ ) of any product-preserving unitary (and Corollary 1 interprets that relation in the thermal case).
• Theorem 3 is almost obvious, but is central to our work: it proves that product preservation leads to tabletop reversibility.
• Theorem 4 proves that the converse is not true, by exhibiting examples of tabletop reversibility that do not arise from product preservation.
In subsection V B, we present a thorough study for twoqubit unitaries.At a glance: • Theorem 5 provides a parametric characterization of two-qubit generalized thermal unitaries.
• Theorem 6 shows that, given any two qubit unitary U and a pure state |β⟩, there always exists |α⟩ such that U (|α⟩ ⊗ |β⟩) is product.This is unique to low-dimensional cases, as the existence of nonproduct-preserving unitaries have been proven in higher dimensions [58].
• Theorems 7 and 8 provide two connections between the generalized thermal character and the productpreservation properties of two-qubit unitaries.

A. General Results
Theorem 1. U is generalized thermal if and only if it is product-preserving with regard to full rank α, β.
Proof.The "only if" direction has already been established in Eq. ( 28).For the "if" implication, we note that for any full-rank state α, one can always find a corresponding H A for which α = τ κ (H A ) = exp{κ(H A − Z H A )}, and likewise for the bath state β with the same inverse temperature −κ.Then, The same construction can be done for the logarithm of the output product states ln(α ′ ⊗ β ′ ).By invoking the product-preserved behavior of the tuple implies U ln(α ⊗ β)U † = ln(α ′ ⊗ β ′ ), which is Eq.(27).Thus there will always exist for every product-preserved tuple with full rank α, β some H A , H B , H ′ A , H ′ B such that U is a generalized thermal unitary.
Notice that, in the previous theorem, the condition of full rank cannot be relaxed.Indeed, on the one hand, the logarithm of rank-defective states is ill-defined.On the other hand, it is simple to find unitaries that preserve one pure product state, and that are not even close (in any meaningful distance) to a generalized thermal unitary.One such example is the two-qubit unitary Next, in the definition of product-preservation, we have merely required the input states α ⊗ β and output states α ′ ⊗ β ′ to be uncorrelated.Now we show that productpreservation implies a stronger relationship between the two: Theorem 2. If (U, α, β) is a product-preserving tuple with output states α ′ and β ′ , then there exist local unitaries u A and u B such that Proof.We denote σ[ρ] as the eigenspectrum of ρ, and write the spectra of α and β in the following way: As such, σ[α ⊗ β] is given by the set of values m i,j = λ i µ j .This implies that m i,j ≥ m i+1,j and m i,j ≥ m i,j+1 for all i, j.Since the input and output products differ by a unitary transformation, σ[α ′ ⊗ β ′ ] is also given by the same set of m i,j , for which some σ , and likewise for µ ′ j .Now, assume that there is some i for which λ ′ i = cλ i .This means that for every j, µ ′ j = c −1 µ j .But since j µ j = 1 and j µ ′ j = 1, summing over j for both sides of µ ′ j = c −1 µ j gives c = 1.This argument also works for the values of µ j .Therefore, the spectra must always be conserved.
An "up to a swap" is obtained if we begin by assuming that there is some i for which λ ′ i = cµ i .Thus, (U, α, β) is a product-preserving tuple if and only if the spectra of α and β are conserved up to a swap: Note that the second set of conditions can only be fulfilled Since the spectra are conserved, up to a swap, we can always find some local unitary u A that brings α to α ′ , and similarly for u B , which completes the proof.
Notably, Theorem 2 shows that demanding the global unitary to preserve noncorrelation is enough to ensure that the local spectra of the input states are preserved, up to a swap between the input and ancilla.This provides us with a corollary on the level of the Hamiltonian: , where u A and u B are some local unitaries on the system and ancilla respectively.Proof.To prove this, we use Theorem 2 on Eq. ( 28), and then apply Eq. ( 29) to find , up to a swap.Finally, noting that the exponential of a Hermitian operator is full rank, we can take the logarithm to complete the proof.
Let us now state the following theorem Proof.Using (20), it is immediate that which is tabletop reversibility (25).
Next, we prove that the converse of Theorem (3) is not true: Proof.We prove this with two counterexamples.
For the first example, we look at the two-qubit channel Though separable, this is not a product state, so (U 1 , α 1 , β 1 ) is not generalized thermal.
These examples demonstrate that channels can be tabletop time-reversible without being generalized thermal with respect to certain priors.Note that the first example also holds for the classical case by setting b ′ y = 0, while the second example is inherently quantum due to the presence of entanglement.

B. Results for Qubit Channels with Two-Qubit Dilations
We shall now focus our attention to one-qubit generalized thermal channels with two-qubit dilations, and product-preserving unitaries acting on two qubits.It is known that every two-qubit unitary permits the Cartan decomposition [60] where {σ k } 3 k=1 are the usual Pauli operators, while v A , v B , u A , and u B are single-qubit unitaries.Hence, every U would be specified by these local unitaries and angles {t k } 3 k=1 .With reference to this parametrization, we fully characterize all two-qubit generalized thermal unitaries: Theorem 5. A two-qubit unitary U , parameterized as Eq.(33), is generalized thermal if and only if (t j − t k ) mod (π/2) = 0 or (t j + t k ) mod (π/2) = 0 for some j ̸ = k.
Proof.For a given two-qubit unitary U , we prove this by characterizing every possible pair of Hamiltonians H A and H B such that (U, H A , H B ) is a generalized thermal tuple.The proof by direct inspection is done in Appendix E, divided in three lemmas: the main one covers all the U such that t k mod (π/4) = 0 for at most one t k ; the other two settle the remaining special cases.
Due to Theorem 1, the above also fully characterizes all two-qubit unitaries that are product-preserving with respect to full rank states.There are in fact many more product-preserving unitaries for two-qubits.Indeed, every two-qubit unitary is not just product-preserving, but product-preserving with respect to every pure ancilla: Theorem 6.For every two-qubit U and ancilla |β⟩, there exists an |α⟩ such that (U, |α⟩ , |β⟩) is a productpreserving tuple.
, where v A and v B are the same as in Eq. ( 33), and we FIG.4: Schematic of the order relationships between the tuple sets we have introduced.Note that the non-tabletop time-reversible but product-preserved tuples are those for which a Bayesian inversion is not well-defined in general due to the presence of rank-deficient output states: although it is still possible to define some retrodiction channel in this case, it will depend on the chosen convention, and the different approaches are inconsistent with each other (Section VI B, Appendix F).
have assumed for now that a 0 ̸ = 0.Then, where f jk (x) are linear functions of x.The condition that U is product-preserving with respect to |α⟩ ⊗ |β⟩ requires Since this is at most a quadratic equation in Corollary 2. Every two-qubit unitary is productpreserving with regard to some states.
This corollary was already known in the context of universal entanglers, where it has been shown that there is no two-qubit unitary that takes every product state to an entangled state [58].
Finally, we present an characterisation of the generalized thermal two-qubit unitaries, an alternative to Theorem 5, in terms of the product states that they preserve: Theorem 7. A two-qubit unitary U is generalized thermal if and only if it is product-preserving with respect to two pure states |α Proof.The "if" direction: We shall first consider the case where H A and H B are nondegenerate.Let |α ± ⟩ and |β ± ⟩ be the eigenstates of H A and H B respectively, with the ground states labelled |α − ⟩ and |β − ⟩.Taking the low temperature limit lim so U is product-preserving with respect to |α ± ⟩ ⊗ |β ± ⟩.

The "only if" direction: Let |α
Then, where the top-left and bottom-right entries originate from the product-preserving condition, while the rest of the entries are parameterised to impose unitarity.Define From direct computation, U will be found to be generalized thermal with respect to H A and H B for any ω A and ω B if θ = 2nπ for some integer n, and for To handle the degenerate case, we use the full characterization of two-qubit generalized thermal unitaries from Appendix E. Specifically, lemmas 1 ′ & 1 ′′ states that U is generalized thermal with respect to H A ̸ ∝ 1 and H B ∝ 1 if and only if it is generalized thermal with respect to Since HB shares the same spectrum as H A and is therefore nondegenerate, the rest of the proof follows as stated above.
If we have a situation that only one of the states in the product-preserved tuple is pure, then we can also conclude that the unitary is generalized thermal: Theorem 8.For a two-qubit U and full-rank α, if (U, α, |β⟩⟨β|) is a product-preserving tuple, then U is generalized thermal.
Proof.Up to a swap, U is product-preserving with respect to a full-rank α and pure |β⟩⟨β| if where we have used Theorem 2 to conclude that α ′ is full-rank and |β ′ ⟩⟨β ′ | is pure.Consider first the special case of α = 1/2.Let |β + ⟩ := |β⟩ and |β − ⟩ to be its orthogonal state.Then, Hence, β = τ κ (|β Taking the limit n → ∞ with α n / Tr(α n ) leads to Note that the converse does not hold.From the proof, for a U to be product-preserving with respect to some input state of the form α ⊗ |β + ⟩⟨β + |, it must necessarily be generalized thermal with respect to 1 ⊗ τ κ (|β From the proofs in Appendix E, this is only possible when there exists j, k ∈ {1, 2, 3} and j ̸ = k such that either t j , t k mod (π/2) = 0 or t j , t k mod (π/2) = π/4.Therefore, a generalized thermal U is in general not product-preserving with respect to some α ⊗ |β⟩⟨β|.

VI. FURTHER OBSERVATIONS
While the preceding sections put emphasis on the mathematical aspects of product-preserving and generalized thermal unitaries, we now consolidate notable physical insights that the results elucidate.We devote one subsection to each of the notions we introduced: generalized thermal operations, product-preservation, and tabletop reversibility.

A. On Generalized Thermal Maps
Generalized thermal operations are the largest class of operations that can be described by the following procedure [61]: a system (with free Hamiltonian H A ) and a bath (with free Hamiltonian H B ) are brought into contact, allowed to interact for some time with a unitary U , then are decoupled again.As a result, the final Hamiltonian must be without interaction.A scattering experiment, where two distinct collection of particles start far apart, come together to interact in a complicated way, and two (possibly different) collection of particles leave the interaction region, would be an example of such a process.That said, we do not make any assumptions about the interaction unitary U apart from the fact that the initial and final Hamiltonians are decoupled.This procedure indeed describes our definition of generalized thermal operations [Eq.(27)].
Corollary 1 shows that every process described by the procedure above must obey a strong constraint: the local conservation of energy spectra, up to a possible swap (if the system and the bath have the same dimension, we can always choose to redefine which is which).
The requirement [Eq.(12)] that the local Hamiltonians be unchanged, which is the standard definition of thermal FIG.5: All two-qubit unitaries are equivalent up to local unitaries to the Weyl chamber [60], as plotted here with coordinates (t 1 , t 2 , t 3 ).The generalized thermal unitaries are marked out in gray, and occur only on the surfaces (0, 0, 0)- operation [61,62], is strictly stronger.That being said, given a generalized thermal tuple (U, H A , H B ), up to a swap there exists local unitaries v A and v B , such that Indeed, using Corollary 1: which is the claimed result with v X = u † X .In Appendix E, we have characterized all possible generalized thermal tuples (U, H A , H B ) for two qubits; whence all generalized thermal maps that possess a twoqubit dilation can be inferred.This complements the recent characterization of all thermal maps for one qubit coupled to a bosonic bath [63] and similar studies in resource theoretic contexts [43].Such characterizations are necessary for one of the main goals of a resource theory, namely, to identify all states reachable from some initial state using only free operations [64].

B. Product-preservation Involving Rank-Deficient States
The discrepancy between product-preserving and generalized thermal unitaries boils down to the states they are acting upon.In the preceding sections, we have given explicit examples of unitaries that are productpreserving, but not generalized thermal, with respect to pure states.
For all C d channels For any α, β U, HA, HB is a generalized thermal tuple to U, α, β is product preserved to α ′ , β ′ ⇒ E(U, β) is tabletop reversible with regard to α Eq.( 31) For any full-rank α, β U, α, β is product preserved to α ′ , β ′ ⇔ U, HA, HB is a generalized thermal tuple to TABLE II: Summary of key results for general channels.For relevant results, For all two-qubits channels  However, Gibbs states are full rank for finite inverse temperature.So, pure states, or more generally rankdeficient states, are zero-temperature states in the thermodynamic picture, and are not physically feasible except in the limiting sense.Even outside the field of thermodynamics, since real-world experiments are always susceptible to noise and uncertainty, pure states are really idealizations that can never be actually prepared in the lab.
Whichever the motivation, a product-preserving unitary with respect to a rank-deficient state is pragmatically useful only when there is a neighborhood of full-rank states around it whose product structure is also preserved by the same unitary.If so, one can choose a full-rank approximation of the target state whose product structure is also preserved by the same unitary.
For two-qubit unitaries, which are all productpreserving, Theorem 7 therefore provides a simple check for when that unitary has a product-preserving property.Its proof also offers an exact construction of the input and ancilla Hamiltonians for which they are a generalized thermal tuple together with interaction unitary.Meanwhile, the Stinespring dilation theorem asserts the uniqueness of the dilation unitary for every quantum channel when the ancilla state is pure, up to an isomorphism on the ancilla [65].Therefore, for channels with dilations whose input and ancilla are both a single qubit, Theorem 8 connects the product-preserving property of the Stinespring dilation and the generalizedthermal property of the dilation unitary.
It must be emphasized here that considering full-rank product-preserving states in the neighborhood of a rankdeficient state, and taking the former to be full-rank approximations of the latter, is a convenient choice: one that is motivated by thermodynamic arguments, but an otherwise arbitrary one.Bayes' rule and the Petz alike in-volve the inverse of the propagated reference.When this state is rank-deficient, an inverse does not exist, rendering such channels undefined.One might consider a neighborhood of full-rank states whose inverses exists, and define the retrodiction channel with some limiting process.However, our prevening discussion and an explicit example in appendix F 1 c shows that defining retrodiction this way depends on the chosen neighborhood of full-rank states.
Where the composition of many copies of the same channel involves the same β as the ancilla in each step, it can be desirable for the reversal of the composite channel to involve the same β ′ as the ancilla in each step in the opposite direction.We call channels that satisfy Eq. ( 42) as composable tabletop time-reversible channels.
Apart from the reduction in reversal complexity of a tabletop-time-reversible channel, one might also desire for this behavior to apply for compositions of the same channel.For example, when the prior and ancilla are thermal states, the reverse channel of the composition of a thermal operation is the composition of the reverse channel of a single thermal operation [14].
More generally, it would be convenient if the reverse channel of a composition of many copies of the same quantum operation can be implemented as illustrated in Fig. 6.Formally, for a tabletop-time-reversible channel E with unitary dilation (U , β) and positive integer L, one desires the reverse channel of (42) For brevity, we shall use TR c (α, β ′ |U, β) to denote such composable tabletop-time-reversible channels-TR(α, β ′ |U, β) that also satisfy Eq. (42).Let us provide a few examples of such channels.
Unitary channels.The action of both the unitary and its inverse on the input is independent of the state of the ancilla, so unitary channels are composably tabletop time-reversible.While this is a trivial case, it aligns with the intended definition of composable tabletop timereversible channels as illustrated in Fig. 6.
Reverse channel with N -steady state prior.Consider Then, Êαn [•] = Tr U † (• ⊗ β ′ )U for all n, hence TR c (α n , β ′ |U, β).We call this the "N -steady state" as the set {α n } N −1 n=0 is unchanged under the channel.The N = 1 case is the usual steady state of a channel, which includes the class of thermal operations as previously studied [14].An example of such a channel for any N is an N -dit channel E given by the ancilla β = N −1 k=0 b k |k⟩⟨k| and the dilation unitary Here, σ is a permutation of order N , {|ψ k ⟩} N −1 k=0 and {|k⟩} N −1 k=0 are orthonormal bases, and with u † k u k ′ ̸ ∝ 1 for all k ̸ = k ′ , which ensures that E is not a unitary channel.Meanwhile, the priors are defined as ) This satisfies Eq. ( 43), and hence this channel is composably tabletop time-reversible with respect to α n .
Channels with idempotent reverse channels.These are tabletop time-reversible channels whose reverse channels are idempotent, both with themselves and subsequent priors: that is, they have the property Êα • ( ÊL−1 ) E[α] = Êα • Êα = Êα .If so, Eq. ( 42) is also satisfied.An example of such a channel is when the dilation unitary is a swap, as for every α.
, so the reverse channel with respect to the prior α is idempotent.
These are the classes of composable tabletop timereversible channels that we have thus far identified.In our study of generalized thermal channels with two-qubit dilations, all TR c channels we have found belong to one of the above classes.It is not known if this is an exhaustive list, as there might be richer families of TR c channels in higher dimensions.

VII. CONCLUSION
Using a recipe from Bayesian inference, one can associate any physical channel, however irreversible, to a family of reverse channels indexed by the choice of a reference prior.We proved that an apparently different recipe, based on dilating the system to include the bath into which information is dissipated, leads exactly to the same family of reverse channels.
For thermal channels, it was known that a natural reverse channel consists of reversing the unitary evolution while in contact with the same thermal bath (in our framework, this is obtained by choosing the Gibbs state as reference prior).We ask for which other channels such phenomenon happens: that the reverse channel can be implemented with the same devices and similar baths as the original channel ("tabletop reversibility").These questions further inspire the related definition of "product-preserving" maps that contains generalized thermal channels as an important subclass.We then proved several relations between these classes, both in general and with more detail for two-qubit unitaries.In particular, we show that when the reverse channel is well-defined (Section VI B), product-preservation leads to tabletop reversibility (Theorem 3).As such, with fullrank states, the latter is strictly more general (Theorem 4).As a by-product of this work, we found that the preservation of local energy spectra is a necessary and sufficient characterization of generalized thermal operations.Characterizing tabletop reversibility in a necessary and sufficient way remains an open problem.

Bayes' Rule on Classical Decomposition
In this Appendix, we lay out an explicit proof for Result 1 on the level of matrices.This elucidates how retrodiction occurs on every step of the decomposition and how Bayesian inversion applies on physically valid channels, including those that are not stochastic (that is, even channels that do not correspond to a Markov matrix, such as the marginalization of a subspace and the assigning of an environment).Note that here, for the sake of symmetry with the quantum case, we will assume that the global process Φ is bijective and that the environment β is uncorrelated with the input.Nevertheless, general insights also apply when these assumptions are removed.
Firstly, we write Eq. ( 14) in terms of matrices: Here, v 1 and v 0 are d B -dimensional column vectors of ones and zeros respectively, and v(Λ aB , which comes from Eq. ( 16).Now, □ indicates any choice of state in A. The matrix ΣB,□⊗β is independent of this choice, since . . .
Thus, it follows that ΣB,□⊗β [• A ] = • A ⊗β B .Hence, each matrix performs the role of its corresponding channel.Turning now to retrodiction, we recall that Bayes' rule is composable [24].This simply captures the time-reverse ordering and propagation of the reference prior, expected of Bayesian inversion.When composability ( 7) is applied to Eq. (C4), insights are yielded: where the d B -length vector v a (p(ã)) Thus it is shown that, given the propagated reference priors, the Bayesian inversion of the assignment map gives a marginalizing channel and vice versa: Thus, every step of Eq. ( 24) is proven, thus giving an alternative route to Eq. ( 23).These derivations highlight that Bayesian inference can be applied in a logically consistent and physically insightful way to any valid channel, including that of marginalization and the assignment of environments and so on.
Appendix D: Special Case of Pure Product-preserving Tuple for Two-qubit Unitaries For a 0 ̸ = 0, the explicit form of Eq. ( 34) is The condition for (U, |α⟩ , |β⟩) to be product-preserving is given in Eq. ( 35), which can be expressed as a quadratic equation ax 2 + bx + c = 0, with ) Unless a = b = 0 and c ̸ = 0, a solution always exists for x, and the corresponding |α⟩ for the given U and |β⟩ to make the tuple product-preserving.However, if a = b = 0 and c ̸ = 0, the condition simplifies to c = 0, which is a contradiction.In that case, ib 0 e −it3 sin (t 1 + t 2 ) b 0 e −it3 cos (t 1 + t 2 ) b 1 e iφ e it3 cos (t and it can be verified that Eq. ( 35) is satisfied with a = 0.
In summary, for a given U and |β⟩, if a as defined in Eq. (D2) is nonzero, ax 2 + bx + c = 0 is solved for x with the coefficients given in Eq. (D2), and we set |α⟩ = Since one of 1 + (−1) n k or 1 − (−1) n k must be zero, it is either the case that both right columns are zero, or both left columns are zero.As such, h A,j = h B,j = 0.
Similarly letting 2t j = n j π/2, repeating the above steps give h A,k = h B,k = 0.
Meanwhile, for the h A,l and h B,l case, (E10) When the parities of n j and n k do not match, h A,l = h B,l = 0.In other words, if t j mod (π/2) ̸ = t k mod (π/2), then U is not generalized thermal.
Otherwise, the equations are trivially satisfied.Then, Eq. (E6) gives with analogous expressions for h ′ B,l .Placing these expressions back into the definitions of the Hamiltonians end the proof.Lemma 1 ′′ .For U such that t k mod (π/4) = 0 for all k ∈ {1, 2, 3}, U is generalized thermal with respect to H A , H B , H ′ A , and H ′ B if and only if • t 1 mod π 2 = t 2 mod π 2 = t 3 mod π 2 = 0, H A and H B are arbitrary, and or; • t 1 mod π 2 = t 2 mod π 2 = t 3 mod π 2 = π 4 , H A and H B are arbitrary, and (E13) Proof.The proof is almost identical to that of Lemma 1 ′ , so we shall only provide a sketch.Substituting 2t k = n k π/2 for all k ∈ {1, 2, 3} into Eq.(E5), we will find that h A,k = h B,k = 0 for all k if the parities of n 1 , n 2 , and n 3 do not all match.This means that U is not generalized thermal if t j mod π 2 ̸ = t k mod π 2 for any two j ̸ = k.Otherwise, Eq. (E6) gives, for all k, with similar expressions for h B,k .Substituting these back into the Hamiltonians concludes the proof.
Appendix F: Retrodiction with Rank-deficient Outputs is Ill-defined in General The Petz recovery map in Eq. ( 4) involves the operator (E[α]) − 1 2 , which does not exist when E[α] is rankdeficient.One might attempt to circumvent this problem in in several ways: (1) Define Êα,1 using the pseudoinverse.The inverse can be defined only on the support of E[α], resulting in the so-called pseudoinverse of E[α], as is the convention in quantum information [66].
(3) Naïvely define Êα,3 with product-preserving tuples.As Eq. ( 31) gives a simple form for the retrodiction channel when (U, α, β) is a product-preserving tuple for full rank α and β, one might extend it to the rank-deficient case by imposing that the retrodiction channel should also take the same form even when α ′ is not full rank.
In the following section, we show that the retrodiction channels as defined by the above approaches do not agree in general.As such, the retrodiction channel for a rankdeficient output state depends on the convention chosen, and cannot be consistently defined when the output state is rank-deficient. (F2) (1) Define Ûα,1 using the pseudoinverse.
2. Product-preserved tuples are almost tabletop time-reversible with the pseudoinverse convention Although we have not given a preference to any particular approach for retrodicting with rank-deficient outputs, it is worth pointing out that the product-preserved tuples are almost tabletop time-reversible with the pseudoinverse convention.
Given a product-preserving tuple (U, α, β) with U (α ⊗ β)U † = α ′ ⊗ β ′ where α ′ is not necessarily full-rank, Eq. ( 19) with the pseudoinverse reads which in turn gives a trace-non-increasing retrodiction channel We say that it is almost tabletop time-reversible as Eq.(F31) is the same as Eq. ( 25) up to a projection onto the support of E[α] upon the input state.

TABLE III :
Summary of key results for channels with two-qubit dilations