Strictly speaking, the distinction we wish to make here is that of dynamic vs kinematic properties, and those we consider “spatial” do not necessarily have to be encoded in spatially distinct degrees of freedom. Since physically relevant setups often involve distributing parts of quantum states to spatially separated parties, we use this nomenclature for its intuitive merit.

The latter are often called “supermaps” throughout the literature.

Similarly to pure quantum states, unitary operators differing only by global phases represent identical physical operations, as verified by the identity $\mathcal{U}(\rho )=U\rho U^{†}=(e^{i\phi }U)\rho (e^{i\phi }U{)}^{†}$ for all real $\phi$.

This poses no inconsistency in notation, as one can always consider the output Hilbert spaces to be trivial.

Note that an operator $L:{\mathscr{H}}_{\mathtt{i}}\to {\mathscr{H}}_{\mathtt{o}}$ between two Hilbert spaces ${\mathscr{H}}_{\mathtt{i}}$ and ${\mathscr{H}}_{\mathtt{o}}$ is linear if it satisfies $L(\alpha |\mathrm{\Psi }⟩+\beta |\mathrm{\Phi }⟩)=\alpha L|\mathrm{\Psi }⟩+\beta L|\mathrm{\Phi }⟩$ for all $\alpha ,\beta \in \text{ℂ}$ and $|\mathrm{\Psi }⟩,|\mathrm{\Phi }⟩\in {\mathscr{H}}_{\mathtt{i}}$.

In this Tutorial, we reserve the term “HOQOs” to refer to linear maps, thus excluding analogous models such as closed time-like curves [108, 109, 110, 111] and multitime states [112, 113, 114], which generally exhibit non-linear features.

Eq. (2.16) ensures that each outcome of an effect is positive and together they sum to unity, i.e., the Born rule leads to a valid probability distribution.

When we say that an operation is CP, we automatically mean that it is also Hermiticity preserving.

Since by definition ${\mathrm{\Phi }}^{+}\in ℒ({\mathscr{H}}_{\mathtt{i}}\otimes {\mathscr{H}}_{{\mathtt{i}}^{\prime }})$, technically, the resulting operator $\mathsf{F}$ is an element of $ℒ({\mathscr{H}}_{\mathtt{o}}\otimes {\mathscr{H}}_{{\mathtt{i}}^{\prime }})$ and an appropriate relabeling of spaces is tacitly assumed; whenever more diligent bookkeeping is required, we will be explicit.

Choi states also possess interesting properties for larger sets of linear maps such as positive but not completely positive maps, which do not necessarily represent quantum channels. For a discussion concerning the CJI for such maps, as well as a more detailed presentation of the CJI for CPTP maps, see, e.g., Refs. [125, 126].

Recall that this additional transpose arises from the definition we take for the CJI; therefore it is entirely conventional (we choose it so that the standard Born rule applies) and has no physical consequence.

While our notation convention suggests that quantum states should technically be denoted by sans-serif letters, i.e., $\mathsf{R}=\mathsf{Choi}(\mathcal{R})$, we maintain the traditional Greek letter notation, i.e., $\rho =\mathsf{Choi}(\mathcal{R})$.

We emphasize that the space the “output” state is defined on is labeled by $1^{\mathtt{i}}$ here, i.e., an “input” ($\mathtt{i}$) label. This is owed to the fact that we label spaces from the perspective of the experimenter, not the superchannel, and the roles of inputs and outputs are exchanged for both. This notational ambiguity is unavoidable and we adhere to the convention that when we denote something as an input/output of an object, we refer to what is inserted into it/comes out of it, independent of the labeling of the respective spaces.

Throughout, we frequently change the labeling of spaces depending on whether we want to emphasize the time—leading to labels $0^{\mathtt{i}},0^{\mathtt{o}},1^{\mathtt{i}},\cdots$—or the laboratory—leading to labels $A^{\mathtt{i}},A^{\mathtt{o}},B^{\mathtt{i}},\cdots$—of the respective spaces.

Since ${\mathscr{H}}_{\alpha \cap \beta }$ is traced over, the partial transposition (which acts only on this space) can be applied to either operator $\mathsf{A}$ or $\mathsf{B}$ due to the duality of transposition within said trace; we simply choose $\mathsf{A}$ as a matter of convention.

The terms “quantum comb” and “process tensor” both refer to the same object: causally ordered, sequential deterministic quantum processes with open slots for interventions. The two distinct names are vestiges of the fact that they were derived independently through two different communities. We will use the terms interchangeably throughout to accommodate for the fact that readers will encounter both names throughout the literature.

As mentioned previously, throughout we reserve the term “superchannel” for quantum combs/process tensors with one slot, both for trivial and non-trivial initial input space $0^{\mathtt{o}}$.

In this and the following section, we simplify notation as compared to the previous sections, denoting Hilbert spaces by ${\mathscr{H}}_{\mathtt{i}},{\mathscr{H}}_{\mathtt{o}},\cdots$ instead of ${\mathscr{H}}_{S^{\mathtt{i}}},{\mathscr{H}}_{S^{\mathtt{o}}},\cdots$ and labeling auxiliary spaces by $\mathtt{a},\mathtt{A},\cdots$. Additionally, since it is more intuitive for the channel case, we switch the role of input and input and output and denote them by $\mathtt{i}$ and $\mathtt{o}$ instead of $1^{\mathtt{o}}$ and $2^{\mathtt{i}}$ respectively as would follow from the convention in the previous section. We will return to the latter convention when we discuss genuinely multitime quantum combs.

Since ${\mathsf{C}}_{\mathtt{i}\mathtt{o}}$ is positive semidefinite, the additional dagger on ${\sqrt{\mathsf{C}}}^{†}$ in the definition of the purification of ${\mathsf{C}}_{\mathtt{i}\mathtt{o}}$ is somewhat superfluous. Here, we explicitly include it to adhere to the convention chosen in Refs. [27, 97].

In this section, we often let HOQOs start on an output space ${\mathscr{H}}_{0^{\mathtt{o}}}$. This amounts merely to making a specific choice and has no bearing on the following arguments; the case where combs start on an input space $1^{\mathtt{i}}$ can easily be recovered by setting ${\mathscr{H}}_{0^{\mathtt{o}}}\cong \mathbb{C}$.

We emphasize that—similar to the case of CPTP maps, where trace preservation already implies complete trace preservation (see Sec. 2d1)—here, requiring complete trace preservation preservation does not add any further constraints (see Refs. [89, 140] and below).

Similar to trace non-increasing CP maps—the probabilistic elements corresponding to deterministic quantum channels—we can say that probabilistic HOQOs are those HOQOs $\mathsf{G}$ that satisfy $\mathsf{G}\le \mathsf{T}$ for some deterministic HOQO $\mathsf{T}$ (we will discuss probabilistic HOQOs further in Sec. 2d4).

Here, for simplicity, we assume that ${\mathsf{T}}_{(n+1{)}^{\mathtt{i}}:0^{\mathtt{o}}}$ and ${\breve{\mathsf{T}}}_{(k+1{)}^{\mathtt{o}}:1^{\mathtt{i}}}$ “start” on fixed spaces that are labeled by 0 and 1, respectively, and that they are defined on all spaces between $0(1)$ and $n+1(k+1)$. The arguments can be extended without any added difficulty to the case where these starting times are chosen differently, and where there are “holes” in the set of spaces they are defined on.

When taking the tensor product of two quantum combs, subtleties with respect to the temporal ordering of the resulting comb arise (see Ref. [27]). Here, we do not discuss these issues in detail and assume that the times the resulting comb is defined on are given by the labels of the involved Hilbert spaces.

Alternatively, one can employ the requirement of “compatibility with remote connection” [27], which yields an axiomatic definition of HOQOs that coincides with that of Prop. t30.

In what follows, to avoid an additional relabeling step in the derivation, we tacitly suppress the terms ${\mathbb{1}}_{1^{\mathtt{o}}\to 1^{\mathtt{i}\prime }}$ and ${\mathbb{1}}_{0^{\mathtt{o}}\to 0^{\mathtt{i}\prime }}$ that respectively occur in the definitions of $V^{(2)}$ and $V^{(1)}$; this is merely done for notational convenience and not a conceptual step.

We again suppress the term ${\mathbb{1}}_{1^{\mathtt{o}}\to 1^{\mathtt{o}\prime }}$ to avoid additional relabeling.

Since in this section we only consider superinstruments—and not the quantum combs that they act on—we omit the “breves” we added to the notation of superinstruments in Sec. 2c3.

Such general superinstruments are, e.g., used to demonstrate the advantage of causally indefinite HOQOs over all causally ordered ones for the task of channel discrimination [75, 76].

Note that for this definition of ${\mathsf{T}}_{\mathtt{i}\mathtt{o}}$, we have ${\mathcal{T}}^{(x)}[{\mathsf{C}}_{\mathtt{i}\mathtt{o}}]={\mathsf{T}}_{\mathtt{i}\mathtt{o}}^{(x)\mathrm{T}}\star {\mathsf{C}}_{\mathtt{i}\mathtt{o}}$, due to the definition of the link product.

Here, again, we employ the breve to signify that the “input” spaces of one comb correspond to the “output” spaces of the other.

Since the output of ${\mathsf{W}}_{AB}$ when acting on non-signaling maps is equal to 1, the projector on the output space is the trivial identity projector $\mathcal{I}$, and we have $\text{tr}[1]=1$ as the (trivial) trace condition on the output space.

In what follows, we assume that $d_{A^{\mathtt{i}}}=d_{A^{\mathtt{o}}}=d_{B^{\mathtt{i}}}=d_{B^{\mathtt{o}}}$ and $d_C=d_{C^{\prime }}=2$.

An alternative option would be to take the adjoint instead of the transpose in this equation [193]. In this case, the resulting channel and the corresponding time flip would depend on the explicit choice of Kraus operators $\{K_B^{(\alpha )}\}$ and we thus do not consider it here.

In more general situations, depending on the sets of objects between which the HOQO of interest maps, the additional demand of “completeness” may be insufficient to guarantee positivity and lead to structurally different sets of HOQOs [140]. Consequently, in all following considerations, we will require positivity of the considered (Choi states of) HOQOs as an explicit assumption.

Since ${\mathsf{W}}_{A^{\mathtt{i}}{A}^{\mathtt{o}}{B}^{\mathtt{i}}{B}^{\mathtt{o}}}\ge 0$, it is already ensured that ${\mathsf{W}}_{A^{\mathtt{i}}{A}^{\mathtt{o}}{B}^{\mathtt{i}}{B}^{\mathtt{o}}}$ yields a positive semidefinite operator when acting on a positive semidefinite object, so that we do not have to demand this property as an additional requirement here.

Similar to the quantum state fidelity, the quantum channel fidelity here respects various properties which are desired for quantifying the similarity between two quantum channels; see, e.g., Refs. [238, 239].

Equations (3.3) and (3.5) differ up to a transpose on ${\mathsf{T}}_k$; however, since the optimization is taken over a class of HOQOs (e.g., parallel, sequential, or causally indefinite), and transposition leaves any such class of HOQOs invariant, it follows that ${\max }_{{\mathsf{T}}_k}\text{\hspace{0.17em}tr}[{\mathrm{\Omega }}_f(d,k){\mathsf{T}}_k^{\mathrm{T}}]={\max }_{{\mathsf{T}}_k}\text{tr}[{\mathrm{\Omega }}_f(d,k){\mathsf{T}}_k]$.

In such cases, these two figures of merit also coincide with the white noise robustness [41].

See also Refs. [292, 293, 294] for results on the task of “Hamiltonian learning,” which aims to learn an unknown Hamiltonian given only access to its dynamics. However, these works do not explicitly employ the framework of HOQOs.

The number $d^4$ is the dimension of the space spanned by the set of CP maps acting on $ℒ({\mathscr{H}}_d)$, which can be seen by noting that the Choi states of CP maps are $d^2\times d^2$ operators that span a $d^4$-dimensional vector space.

Interestingly, similar phenomena can arise in classical stochastic processes where measurements are noisy and cannot be resolved to a sufficient level of granularity [388, 389, 390]; although these can be “argued away” on grounds of fundamentality in the classical realm, it remains an important issue from a practical standpoint.

Such causal breaks are a generalization of “do-operations” that are used in the field of classical causal modeling [396] to discover causal relations by actively resetting the state of the system to a known one.

This definition closely resembles the causal Markov condition employed for classical stochastic processes where interventions are allowed, such as in causal modeling [396]. There, too, one requires a notion such as a causal break, since active interventions can be unsharp, as opposed to the assumption of standard classical probability theory where the measurements are assumed to be sharp. See Refs. [52, 86, 95, 96] for related discussion regarding quantum causal modeling.

We emphasize that the notion of classicality discussed in this section is not the only way of conceptualizing classical processes (see, e.g., Ref. [410] for an overview). It is, however, the one that is most amenable to being phrased in terms of HOQOs, which is why we opt to present it here.

Since this approach does not explicitly rely on the formalism of HOQOs, we will not discuss it in detail here. For an overview, see, e.g., Refs. [420, 421].

In Refs. [435, 436], the possibility to model external quantum noise a classical noise field was employed as an explicit notion of classicality based on quantum processes. Here, concerning classical memory effects, we instead focus on structural properties of HOQOs.

Here, a pure operation is any rank-one operation, which could be proportional to a unitary operation, projective measurement, or projective measurement followed by a pure state preparation (see Sec. 2c3 for the general framework of superinstruments).

Throughout this section, we take the process tensor to be normalized such that entropic quantities are well-behaved.

See also the early work in Ref. [551] for discussion on the thermodynamics of irreversible quantum processes.

The consequences of quantum mechanics holding locally, i.e., in Alice’s and Bob’s laboratory, combined with an additional non-signaling constraint between them (which we do not impose here) has been investigated in Ref. [572], which demonstrated that correlations obtained in such non-signaling scenarios can always be explained by means of a global quantum state that produces standard quantum correlations in a Bell non-locality scenario.

A notable exception is the quantum time flip [193], whose action is only well-defined on a subset of all CPTP maps (see Sec. 2e4).

Due to differences in the definition of the CJI, our definition here may differ by a transposition from that found in the literature (see, e.g., Refs. [30, 49]); this difference is merely notational.

In addition to advantages for such information theoretic games, causally non-separable process matrices have also been shown to be resourceful for implementing quantum functions, as well as for metrological and channel discrimination tasks; see Sec. 3a.

We emphasize again that the definition of causal non-separability is more involved in the multipartite case; nonetheless, the distinction between causally non-separable processes and those that violate a causal inequality remains the same conceptually.

While the possibility of superposition distinguishes causal relations in quantum mechanics from those in classical physics, the admissible signaling structures, i.e., the sets of directed graphs depicting how events can influence each other, are likely the same in both theories [602].

A discussion of whether experiments implement or rather simulate a causally non-separable HOQOs, and what implementation means in the first place, can be found, e.g., in Refs. [90, 191, 610, 611, 612, 613]. Here, we shall stay agnostic with respect to this question.

“Switch-like” processes refer to the quantum switch and simple generalizations thereof, where one can perform arbitrary unitary operations both before and after the input operations, and before and after the global past and global future.

Note that a similar construction works in the continuous variable case, with the sum replaced by an integral, probability distribution replaced by a probability density function, etc. (see, e.g., Ref. [77]).