Axioms for retrodiction achieving time-reversal symmetry with a prior

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Axioms for retrodiction: achieving time-reversal
symmetry with a prior
Arthur J. Parzygnat and Francesco Buscemi
2023-05-12
We propose a category-theoretic definition of retrodiction and use it to exhibit a
time-reversal symmetry for all quantum channels. We do this by introducing retrod-
iction families and functors, which capture many intuitive properties that retrodiction
should satisfy and are general enough to encompass both classical and quantum theories
alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps de-
fine retrodiction families in our sense. However, averaged rotated Petz recovery maps,
including the universal recovery map of Junge–Renner–Sutter–Wilde–Winter, do not
define retrodiction functors, since they fail to satisfy some compositionality properties.
Among all the examples we found of retrodiction families, the original Petz recov-
ery map is the only one that defines a retrodiction functor. In addition, retrodiction
functors exhibit an inferential time-reversal symmetry consistent with the standard
formulation of quantum theory. The existence of such a retrodiction functor seems to
be in stark contrast to the many no-go results on time-reversal symmetry for quantum
channels. One of the main reasons is because such works defined time-reversal symme-
try on the category of quantum channels alone, whereas we define it on the category
of quantum channels and quantum states. This fact further illustrates the importance
of a prior in time-reversal symmetry.
Contents
1 Retrodiction versus time reversal 2
2 Retrodiction as a monoidal functor 5
3 The Petz recovery maps 9
4 The convexity of certain retrodiction families 11
5 Inverting property for retrodiction families 15
6 Involutive retrodiction 17
7 A summary of recovery maps and their properties 20
A The Hilbert–Schmidt inner product on C∗-algebras 21
B Convex sums of rotated Petz recovery maps 22
C The JRSWW retrodiction family is neither compositional nor tensorial 24
D The SS retrodiction family is not stabilizing 26
Bibliography 27
Key words: Retrodiction; postdiction; Bayes; Jeffrey; probability kinematics; compositionality; category theory;
recovery map; Petz map; inference; prior; time-reversal; monoidal category; dagger; process theory
Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. 1
arXiv:2210.13531v2 [quant-ph] 12 May 2023
1 Retrodiction versus time reversal
As we currently understand them, the laws of physics for closed systems are time-reversal symmetric
in both classical and quantum theory. The associated evolution is reversible in the sense that if
one evolves any state to some time in the future, one can (in theory) apply the reverse evolution,
which is unambiguously defined, to return the initial state.
However, not all systems of interest are closed, and so, not all evolutions are reversible in the
sense described above. For instance, arbitrary stochastic maps and quantum channels are of this
kind. This, together with other phenomena such as the irreversible change due to a measurement [1,
2,3] and the black hole information paradox [4,5,6], has led many to question the nature of time-
reversibility. Indeed, many have provided no-go theorems and occasionally offered proposals for
what time-reversal is or how it can be implemented (physically or by belief propagation) [7,8,9,
10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The following lists some of
the questions considered in these programs.
•In what sense can channels be reversed to construct a time-symmetric formulation of open
dynamics? Namely, what axioms define what we mean by time-reversibility [25,27,28]?
•As opposed to applying classical probabilistic inference associated with experimental out-
comes [29,18,20,24], is it possible to provide a fully quantum formulation of inference that
does not require the classical agent interface?
•Is the apparent directionality of time a consequence of the irreversibility of certain processes
that are more general than deterministic evolution?
•What is a maximal subset of quantum operations that has a reasonable time-reversibility?
For example, it is known that a certain type of reversibility is possible for unital quantum
channels [25,24,27,19,28] and more generally for quantum channels that fix an equilibrium
state [11]. Is this the best we can do without modifying quantum theory [16]?
Although a large body of work has recently focused on quantum theory, a similar problem lies
in the classical theory. More importantly, it has been understood for a long time that complete
reversibility is much too stringent of a constraint to ask for. Instead, one might be more interested
in retrodictability rather than reversibility [7,30], where retrodictability is the ability to infer about
the past and which specializes to reversibility in the case of reversible dynamics. And in the classical
theory, retrodictability is achieved through Bayes’ rule and Jeffrey’s probability kinematics, the
latter of which specializes to Bayes’ rule in the case of definitive evidence, but is strictly more
general in that an arbitrary state can be used as evidence to make inference [31,32,33,14].
But while the same Bayes’ rule and Jeffrey’s probability kinematics can be used simultaneously
for both spatial and temporal correlations in classical statistics, quantum theory seems to reveal
a subtle distinction between these two forms of inference [13,34,35,36]. The study of the unique
spatial correlations in quantum theory has been a subject of intense studies since at least the work
of Schr¨odinger [37,38,39,40]. Meanwhile, our understanding of the temporal correlations unique
to quantum theory is not as well-understood and is still under investigation [41,42,36]. In order
to contribute towards this important and evolving subject, it is retrodictability, i.e., the inferential
form of time-reversal symmetry, in quantum (and classical) dynamics that will be the focus of this
paper.
However, retrodictability is not in general possible for arbitrary dynamics unless additional
input data are provided. One example of the kind of information that can be used is a prior,
which summarizes the state of knowledge of the retrodictor. Previous proposals that have avoided
the introduction of a prior oftentimes secretly included it. For example, unital quantum channels
are precisely quantum channels that preserve the uniform prior, even though this is not always
phrased in this way because the identity matrix disappears from expressions.
In the present paper, we show that a form of time-reversal symmetry is possible for certain open
system evolutions (classical and quantum channels) together with priors, i.e., we prove the existence
of an inferential time-reversal symmetry for open quantum system dynamics. This bypasses some
of the no-go theorems in the literature, such as in [19,25,27], in at least two ways. First, by
including the prior, the assignment of a time-reversal no longer has input just some channel, but
Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. 2
a channel and a prior. Second, we see no reason to assume that the time-reversal assignment
must be linear in either the channel or the prior, as is done in [25,27] (see axiom 4 on page 2
in [27] or the definition of symmetry in [25]). Indeed, even classical Bayesian inversion is linear in
neither the original process nor the prior, so there is no reason to assume linearity in the quantum
setting, which should specialize to the classical theory. In fact, in the present work, we axiomatize
retrodiction in a way that is agnostic to classical or quantum theory, probability theory, and even
any physical theory as long as it has sufficient structure to describe states and evolution. This is
achieved by formulating a rigorous definition of retrodiction in the language of category theory,
the proper mathematical language for describing processes [43,44,45,46,47,28].
Fortunately, it is possible to summarize our axioms intuitively without requiring familiarity
with category theory.1In order to be agnostic to classical or quantum theory, we try to avoid
using language that is only applicable to either. To describe retrodiction, we should first specify
the collection of allowed systems (denoted A,B,C, . . . ), processes (denoted E,F, . . . ), and states
(denoted α, β, γ, . . . ). Having done this, and as argued above, in order to define the retrodiction of
some process E:A→Bfrom one system Ato another B, it is necessary to have some additional
input. For us, that input is a prior α, which is a state on system A. Hence, we will combine these
data together by writing (α, E). Retrodiction should be some assignment whose input is such a
pair (α, E)and whose output is some map Rα,E:B→Athat only depends on this input (this is
sometimes called universality in the literature).
The definition of retrodiction that we propose consists of the following logical axioms on such
an assignment Rover all systems A,B,C, . . . , evolutions (i.e., processes) E,F, . . . , and states
α, β, γ, . . . (remarks and clarifications on these axioms follow immediately after).
1. Retrodiction should produce valid processes, i.e., the map Rα,Eshould be a valid process.
2. The map Rα,Eshould take the prediction E(α)back to the prior α, i.e., Rα,E(E(α)) = α.
3. The retrodiction of the process that does nothing is also the process that does nothing, i.e.,
Rα,idA= idA.
4. More generally, the retrodiction of a genuinely reversible process E:A→Bis the inverse
E−1of the original process, i.e., Rα,E=E−1, and is therefore independent of the prior.
5. Retrodiction is involutive in the sense that retrodicting on a retrodiction gives back the
original process, i.e., RE(α),Rα,E=E.
6. Retrodiction is compositional in the sense that if one has a prior αon system Aand two
successive processes E:A→Band F:B→C, then the retrodiction Rα,F◦E associated with
the composite process AE
−→ BF
−→ Cis the composite of the retrodictions RE(α),F:C→Band
Rα,E:B→Aassociated with the constituent components, i.e., Rα,F◦E =Rα,E◦RE(α),F.
7. Retrodiction is tensorial in the sense that if one has priors αand α0on systems Aand A0,
respectively, as well as two processes E:A→Band E0:A0→B0, then the retrodiction
Rα⊗α0,E⊗E0:B⊗B0→A⊗A0associated with the tensor product of the systems and
processes is equal to the tensor product Rα,E⊗Rα0,E0of the constituent retrodictions, i.e.,
Rα⊗α0,E⊗E0=Rα,E⊗Rα0,E0.
Several clarifications and remarks are in order with regard to these axioms, which we enumerate
in the same order.
1. Part of our axioms assume that we have identified a class of physical systems and processes.
In the setting of quantum theory, and in this work, physical systems A,B,C, . . . will be
mathematically represented by their corresponding algebras, following the conventions of
1In the body of this work, only elementary ideas from category theory will be used to formalize our definitions
and results. If the reader is comfortable with the definitions of a monoidal category, its opposite, and functor, this
should suffice. We will try to ease into this through our setting of retrodiction rather than giving formal definitions.
The reader is encouraged to read [48] for a first impression of category theory and then browse the reference [49,
Chapter 1] for a more detailed, yet friendly, introduction. References that cover advanced topics while maintaining
a close connection to quantum physics include [50,44,51]. Finally, standard in-depth references include [52,53].
Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. 3
prior works on quantum inference [13]. We will take processes to be represented by quantum
channels, i.e., completely positive trace-preserving (CPTP) maps (traces on such algebras are
defined in Appendix A). In the classical theory, physical processes are modeled by stochastic
maps.
2. Note that we are only demanding our prior to be reproduced. If α0is some other state on A,
this axiom is not saying that Rα,E(E(α0)) = α0. Note also that this axiom together with the
first one allows us to think of retrodiction as an assignment that sends a pair (α, E)consisting
of a valid state and process to (E(α),Rα,E), which is another valid state and process. This
simplifies the form of the assignment since it can be viewed as a function that begins in
one class of objects and comes back to that same class of objects. In this work, we will not
consider time-reversals that do not satisfy this crucial state-preserving property, though we
mention that some authors have considered dropping such an axiom [15].
3. Note that retrodicting on the identity process is, in particular, independent of the prior α.
4. In this setting, we define Eto be reversible (i.e., invertible) whenever there exists a process
E−1:B→Asuch that E ◦ E−1= idBand E−1◦ E = idA. In the context of category theory,
Eis also called an isomorphism.
5. This axiom has appeared in many approaches towards time-reversal [15,11,28]. A weakened
form of involutivity was considered in [27].
6. This axiom and the next one describe in what sense retrodiction of a complicated process can
be broken into retrodictions of simpler components. These are some of the essential axioms
where the language of category theory becomes especially important. The compositional
property was considered crucial in [27] and was also emphasized in many other works such
as [11].
7. In the tensorial axiom, we are assuming that our collection of states and processes has a
tensorial structure, sometimes called parallel composition to not conflate it with the series
composition from the previous axiom. This axiom seems to have been emphasized a bit less
in the literature on time-reversal symmetry, but was emphasized in the context of recovery
maps [54,55,56].
In this paper (specifically Theorem 6.5), we prove that among a variety of different proposals
for recovery maps and retrodiction in the quantum setting (including Petz recovery maps, their
rotated variants, and their averaged generalizations, and many other candidates), the only one
that satisfies all these axioms is the Petz recovery map [57,10,58,29,11]. This justifies the
Petz recovery map as a retrodiction map, and hence an extension of time-reversal symmetry to
all quantum channels (see Figure 1). Although we do not characterize the Petz recovery map
(and hence Bayesian inversion in the classical setting) among all possible retrodiction maps, we
propose a precise mathematical problem whether our axioms indeed isolate Petz among all possible
retrodiction maps (and not just the vast examples we have listed).
We emphasize that our axioms are logical as opposed to analytical. It has often been the case
that axioms used to help single out a form of retrodiction (a recovery map to be more precise)
optimized some quantity, such as the difference of relative entropies, more general divergences,
fidelity of recovery, or relative entropy of recovery [57,10,59,60,61,62,54,63]. Rather than
choosing a specific such distance measure and then finding axioms to argue for the necessity of
those, we have preferred to find a logical set of axioms more directly, in spirit of earlier derivations
of classical inference [64]. In addition, the axioms that we identified do not involve concepts
such as measurement, observations, or the need of any intervention, which are concepts that are
mathematically difficult to define outside of specific models. In fact, our axioms can be formulated
without any mention of probability theory. Our axioms are merely those of consistency rather than
properties we might expect from our experiences, which are largely based on classical thinking,
and may therefore skew our understanding of quantum.
Furthermore, we illustrate in detail which axioms fail for various other proposals of retrodiction
by providing explicit examples. In particular, we illustrate, to some degree, how independent most
Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. 4
bistochastic channels
adjoint
reversible channels
adjoint
quantum channels
Petz
classical channels
Bayes
Figure 1: The sets (not drawn to scale) and their inclusion structure depict four families of channels (the
inclusion structure is not meant to include the states). The standard time-reversal symmetry is obtained by
taking the Hilbert–Schmidt adjoint of a reversible channel, or, more generally, a bistochastic channel. The states
(drawn schematically as matrices with shaded entries) are irrelevant for reversible channels, but are implicitly the
uniform states for bistochastic channels. For classical channels equipped with arbitrary probability distributions,
standard Bayesian inversion provides a form of time-reversal symmetry that goes beyond the Hilbert–Schmidt
adjoint for bistochastic channels. Finally, the Petz recovery map allows an extension of Bayesian inversion to
all quantum channels and arbitrary states. In brief, this paper isolates axioms for retrodiction and inferential
time-reversal symmetry that are simultaneously satisfied by all of these classes of channels and states.
of our axioms are. For example, the averaged rotated Petz recovery maps that have appeared
recently in the context of strengthening data-processing inequalities via recovery maps [65,55] are
neither compositional, tensorial, nor involutive. Furthermore, we show that the recent proposal
of Surace–Scandi [63] on state-retrieval maps is also not compositional. This emphasizes some
of the key differences between retrodiction and approximate error correction [10,60,63], and
this distinction may have important consequences for quantum information in extreme situations,
such as near the horizons of black holes, where state-dependent approximate error-correction has
recently been used to suggest that information might be stored in Hawking radiation in certain
models of black hole evaporation [66,67,68,69,70,71,72].
2 Retrodiction as a monoidal functor
In the setting of quantum theory, we would like to define retrodiction as an assignment that takes
a prior, defined on some matrix algebra, together with a process involving another matrix algebra,
and produces a map in the opposite direction. However, working with matrix algebras is unneces-
sarily restrictive and it is more appropriate to include classical and hybrid classical/quantum sys-
tems in our analysis. As such, we will model our systems with finite-dimensional unital C∗-algebras,
where commutative algebras correspond to classical systems and general non-commutative algebras
correspond to quantum (or hybrid) systems. We will model our processes with completely positive
trace-preserving (CPTP) maps between such C∗-algebras. Briefly, one benefit of using C∗-algebras,
as opposed to only matrix algebras, is that one can express quantum channels, classical stochastic
maps, measurements, preparations, and instruments all as processes between C∗-algebras (see [36,
Section II.A] for more details). We are expressing processes in the Schr¨odinger picture where the
processes describe the action on states. We emphasize that we do not require Eto preserve any
of the algebraic structure, nor do we require the unit of the algebras to be preserved. The reader
should not be discouraged by our usage of finite-dimensional C∗-algebras because they are equiva-
lent to direct sums of matrix algebras [73, Theorem 5.20 and Proposition 5.26]. We henceforth take
the convention that all C∗-algebras appearing in this work will be finite dimensional and unital
unless specified otherwise.
We now introduce a category that mathematically describes priors and processes.
Definition 2.1. The category of faithful states is the category States whose objects consists
of pairs (A, α), where Ais a finite-dimensional unital C∗-algebra and αis a faithful state on
Accepted in Quantum 2023-04-10, click title to verify. Published under CC-BY 4.0. 5
摘要：

Axiomsforretrodiction:achievingtime-reversalsymmetrywithapriorArthurJ.ParzygnatandFrancescoBuscemi2023-05-12Weproposeacategory-theoreticde nitionofretrodictionanduseittoexhibitatime-reversalsymmetryforallquantumchannels.Wedothisbyintroducingretrod-ictionfamiliesandfunctors,whichcapturemanyintuitivep...

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