Informational derivation of Quantum Theory

I Introduction

More than eighty years after its formulation, quantum theory is still mysterious. The theory has a solid mathematical foundation, addressed by Hilbert, von Neumann and Nordheim in 1928 HNN28 and brought to completion in the monumental work by von Neumann von32. However, this formulation is based on the abstract framework of Hilbert spaces and self-adjoint operators, which, to say the least, are far from having an intuitive physical meaning. For example, the postulate stating that the pure states of a physical system are represented by unit vectors in a suitable Hilbert space appears as rather artificial: which are the physical laws that lead to this very specific choice of mathematical representation? The problem with the standard textbook formulations of quantum theory is that the postulates therein impose particular mathematical structures without providing any fundamental reason for this choice: the mathematics of Hilbert spaces is adopted without further questioning as a prescription that “works well” when used as a black box to produce experimental predictions. In a satisfactory axiomatization of Quantum Theory, instead, the mathematical structures of Hilbert spaces (or C*-algebras) should emerge as consequences of physically meaningful postulates, that is, postulates formulated exclusively in the language of physics: this language refers to notions like physical system, experiment, or physical process and not to notions like Hilbert space, self-adjoint operator, or unitary operator. Note that any serious axiomatization has to be based on postulates that can be precisely translated in mathematical terms. However, the point with the present status of quantum theory is that there are postulates that have a precise mathematical statement, but cannot be translated back into language of physics. Those are the postulates that one would like to avoid.

The need for a deeper understanding of quantum theory in terms of fundamental principles was clear since the very beginning. Von Neumann himself expressed his dissatisfaction with his mathematical formulation of Quantum Theory with the surprising words “I don’t believe in Hilbert space anymore”, reported by Birkhoff in Birk61. Realizing the physical relevance of the axiomatization problem, Birkhoff and von Neumann made an attempt to understand quantum theory as a new form of logic BirkVN36: the key idea was that propositions about the physical world must be treated in a suitable logical framework, different from classical logics, where the operations AND and OR are no longer distributive. This work inaugurated the tradition of quantum logics, which led to several attempts to axiomatize quantum theory, notably by Mackey Mack63 and Jauch and Piron JauPir63 (see Ref. coeckerev for a review on the more recent progresses of quantum logics). In general, a certain degree of technicality, mainly related to the emphasis on infinite-dimensional systems, makes these results far from providing a clear-cut description of quantum theory in terms of fundamental principles. Later Ludwig initiated an axiomatization program Lud83 adopting an operational approach, where the basic notions are those of preparation devices and measuring devices and the postulates specify how preparations and measurements combine to give the probabilities of experimental outcomes. However, despite the original intent, Ludwig’s axiomatization did not succeed in deriving Hilbert spaces from purely operational notions, as some of the postulates still contained mathematical notions with no operational interpretation.

More recently, the rise of quantum information science moved the emphasis from logics to information processing. The new field clearly showed that the mathematical principles of quantum theory imply an enormous amount of information-theoretic consequences, such as the no-cloning theorem wootterszurek; dieks, the possibility of teleportation tele, secure key distribution wiesner; bb84; e91, or of factoring numbers in polynomial time shor. The natural question is whether the implication can be reversed: is it possible to retrieve quantum theory from a set of purely informational principles? Another contribution of quantum information has been to shift the emphasis to finite dimensional systems, which allow for a simpler treatment but still possess all the remarkable quantum features. In a sense, the study of finite dimensional systems allows one to decouple the conceptual difficulties in our understanding of quantum theory from the technical difficulties of infinite dimensional systems.

In this scenario, Hardy’s 2001 work Har01 re-opened the debate about the axiomatizations of quantum theory with fresh ideas. Hardy’s proposal was based on five main assumptions about the relation between dimension of the state space and the number of perfectly distinguishable states of a given system, about the structure of composite systems, and about the possibility of connecting any two pure states of a physical system through a continuous path of reversible transformations. However, some of these assumptions directly refer to the mathematical properties of the state space (in particular, the “Simplicity Axiom” 2, which is an abstract statement about the functional dependence of the state space dimension on the number of perfectly distinguishable states). Very recently, building on Hardy’s work there have been two new attempts of axiomatization by Dakic and Brukner DakBru09 and Masanes and Müller Mas10. Although these works succeeded in removing the “Simplicity Axiom”, they still contain mathematical assumptions that cannot be understood in elementary physical terms (see e.g. requirement 5 of Ref. Mas10, which assumes that “all mathematically well-defined measurements are allowed by the theory”).

Another approach to the axiomatization of quantum theory was pursued by one of the authors in a series of works maurofirst culminated in Ref. maurolast. These works tackled the problem using operational principles related to tomography and calibration of physical devices, experimental complexity, and to the composition of elementary transformations. In particular this research introduced the concept of dynamically faithful states, namely states that can be used for the complete tomography of physical processes. Although this approach went very close to deriving quantum theory also in this case one mathematical assumption without operational interpretation was needed (see the CJ postulate of Ref. maurolast).

In this paper we provide a complete derivation of finite dimensional quantum theory based of purely operational principles. Our principles do not refer to abstract properties of the mathematical structures that we use to represent states, transformations or measurements, but only to the way in which states, transformations and measurements combine with each other. More specifically, our principles are of informational nature: they assert basic properties of information-processing, such as the possibility or impossibility to carry out certain tasks by manipulating physical systems. In this approach the rules by which information can be processed determine the physical theory, in accordance with Wheeler’s program “it from bit”, for which he argued that “all things physical are information-theoretic in origin” wheeler. Note that, however, our axiomatization of quantum theory is relevant, as a rigorous result, also for those who do not share Wheeler’s ideas on the informational origin of physics. In particular, in the process of deriving quantum theory we provide alternative proofs for many key features of the Hilbert space formalism, such as the spectral decomposition of self-adjoint operators or the existence of projections. The interesting feature of these proofs is that they are obtained by manipulation of the principles, without assuming Hilbert spaces form the start.

The main message of our work is simple: within a standard class of theories of information processing, quantum theory is uniquely identified by a single postulate: purification. The purification postulate, introduced in Ref. purification, expresses a distinctive feature of quantum theory, namely that the ignorance about a part is always compatible with the maximal knowledge of the whole. The key role of this feature was noticed already in 1935 by Schrödinger in his discussion about entanglement Schr35, of which he famously wrote “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought”. In a sense, our work can be viewed as the concrete realization of Schrödinger’s claim: the fact that every physical state can be viewed as the marginal of some pure state of a compound system is indeed the key to single out quantum theory within a standard set of possible theories. It is worth stressing, however, that the purification principle assumed in this paper includes a requirement that was not explicitly mentioned in Schrödinger’s discussion: if two pure states of a composite system ${\rm A}{\rm B}$ have the same marginal on system ${\rm A}$, then they are connected by some reversible transformation on system ${\rm B}$. In other words, we assume that all purifications of a given mixed state are equivalent under local reversible operations anchese….

The purification principle expresses a law of conservation of information, stating that at least in principle, irreversibility can always be reduced to the lack of control over an environment. More precisely, the purification principle is equivalent to the statement that every irreversible process can be simulated in an essentially unique way by a reversible interaction of the system with an environment, which is initially in a pure state purification. This statement can also be extended to include the case of measurement processes, and in that case it implies the possibility of arbitrarily shifting the cut between the observer and the observed system purification. The possibility of such a shift was considered by von Neumann as a “fundamental requirement of the scientific viewpoint” (see p. 418 of von32) and his discussion of the measurement process was exactly aimed to show that quantum theory fulfils this requirement.

Besides Schrödinger’s discussion on entanglement and von Neumann’s discussion of the measurement process, the purification principle is deeply rooted in the structure of quantum theory. At the purely mathematical level, it plays a crucial role in the theory of C*-algebras of operators on separable Hilbert spaces, where the purification principle is equivalent to the Gelfand-Naimark-Segal (GNS) construction arveson and implies the celebrated Stinespring’s theorem stine. On the other hand, purification is a cornerstone of quantum information, lying at the origin of most quantum protocols. As it was shown in Ref. purification, the purification principle directly implies crucial features like no-cloning, teleportation, no-information without disturbance, error correction, the impossibility of bit commitment, and the “no-programming” theorem of Ref. no-prog.

In addition to the purification postulate, our derivation of quantum theory is based on five informational axioms. The reason why we call them “axioms”, as opposed to the the purification “postulate”, is that they are not at all specific of quantum theory. These axioms represent standard features of information-processing that everyone would, more or less implicitly, assume. They define a class of theories of information-processing that includes, for example, classical information theory, quantum information theory, and quantum theory with superselection rules. The question whether there are other theories satisfying our five axioms and, in case of a positive answer, the full classification of these theories is currently an open problem.

Here we informally illustrate the five axioms, leaving the more detailed description to the remaining part of the paper:

Causality: the probability of a measurement outcome at a certain time does not depend on the choice of measurements that will be performed later.

Perfect distinguishability: if a state is not completely mixed (i.e. if it cannot be obtained as a mixture from any other state), then there exists at least one state that can be perfectly distinguished from it.

Ideal compression: every source of information can be encoded in a suitable physical system in a lossless and maximally efficient fashion. Here lossless means that the information can be decoded without errors and maximally efficient means that every state of the encoding system represents a state in the information source.

Local distinguishability: if two states of a composite system are different, then we can distinguish between them from the statistics of local measurements on the component systems.

Pure conditioning: if a pure state of system ${\rm A}{\rm B}$ undergoes an atomic measurement on system ${\rm A}$, then each outcome of the measurement induces a pure state on system ${\rm B}$. (Here atomic measurement means a measurement that cannot be obtained as a coarse-graining of another measurement).

All these axioms are satisfied by classical information theory. Axiom 5 is even trivial for classical theory, because the only pure states of a composite system ${\rm A}{\rm B}$ are the product of pure states of the component systems ${\rm A}$ and ${\rm B}$, and hence the state of system ${\rm B}$ will be pure irrespectively of what we do on system ${\rm A}$.

A stronger version of axiom 5, introduced in Ref. maurolast, is the following:

Atomicity of composition: the sequential composition of two atomic operations is atomic. (Here atomic transformation means a transformation that cannot be obtained from coarse-graining).

However, it turns out that Axiom 5 is enough for our derivation: thanks to the purification postulate we will be able to show the non-trivial implication: Axiom 5 $\Rightarrow$ Axiom 5’ (see lemma 16).

The paper is organized as follows. In Sec. II we review the framework of operational-probabilistic theories introduced in Ref. purification. This framework will provide the basic notions needed for the formulation of our principles. In Sec. III we introduce the principles from which we will derive Quantum Theory. In Sec. IV we prove some direct consequences of the principles that will be used later in the paper. In Sec. V we discuss the properties of perfectly distinguishable states, while in Sec. VI we prove the existence of a duality between pure states and atomic effects.

The results about distinguishability and duality of pure states and atomic effects allow us to show in Sec. VII that every system has a well defined informational dimension—the operational counterpart of the Hilbert space dimension. Sec. VIII contains the proof that every state can be decomposed as a convex combination of perfectly distinguishable pure states. Similarly, any element of the vector space spanned by the states can be written as a linear combination of perfectly distinguishable states. This result corresponds to the spectral theorem for self-adjoint operators on complex Hilbert spaces. In Sec. IX we prove some results about the maximum teleportation probability, which allow us to derive a functional relation between the dimension of the state space and the number of perfectly distinguishable states of the system. The mathematical representation of systems with two perfectly distinguishable states is derived in Sec. X, where we prove that such systems are indeed two-dimensional quantum systems—a.k.a. qubits. In Sec. XI we construct projections on the faces of the state space of any system and prove their main properties. These results lead to the derivation of the operational analogue of the superposition principle in Sec. XII which allows to prove that systems with the same number of perfectly distinguishable states are operationally equivalent (Subsec. XII.2). The properties of the projections and the superposition principle are then exploited in Sec. XIII—where we extend the density matrix representation from qubits to higher-dimensional systems, thus proving that a system with $d$ perfectly distinguishable states is indeed a quantum system with $d$-dimensional Hilbert space. We conclude the paper with Sec. XIV, where we review our results, discussing future directions for this research.

II The framework

This Section provides a brief summary of the framework of operational-probabilistic theories, which was formulated in Ref. purification. We refer to Ref. purification for an exhaustive presentation of the details of the framework and of the ideas behind it. The operational-probabilistic framework combines the operational language of circuits with the toolbox of probability theory: on the one hand, experiments are described by circuits resulting from the connection of physical devices, on the other hand each device in the circuit can have classical outcomes and the theory provides the probability distribution of outcomes when the devices are connected to form closed circuits (that is, circuits that start with a preparation and end with a measurement).

The notions discussed in this section will allow us to draw a precise distinction between principles with an operational content and exclusively mathematical principles: with the expression ”operational principle” we will mean a principle that can be expressed using only the basic notions of the the operational-probabilistic framework.

A test represents one use of a physical device, like a Stern-Gerlach magnet, a beamsplitter, or a photon counter. The device will have an input system and an output system, labelled by capital letters. The corresponding test can have different classical outcomes, represented by different values of an index $i\in{\rm X}$:

Each outcome $i\in{\rm X}$ corresponds to a possible event, represented as

We denote by ${\mathsf{Transf}}({\rm A},{\rm B})$ the set of all events from ${\rm A}$ to ${\rm B}$. The reason for this notation is that in the next subsection the elements of ${\mathsf{Transf}}({\rm A},{\rm B})$ will be interpreted as transformations with input system ${\rm A}$ and output system ${\rm B}$. If ${\rm A}={\rm B}$ we simply write ${\mathsf{Transf}}({\rm A})$ in place of ${\mathsf{Transf}}({\rm A},{\rm A})$.

A test with a single outcome will be called deterministic. This name is justified by the fact that, if there is a single possible outcome, then this outcome will occur with certainty (cf. the probabilistic structure introduced in the next subsection).

Two devices can be composed in a sequence, as long as the input system of the second device is equal to the output system of the first. The events in the composite test are represented as

and are written in formulas as $\mathscr{D}_{j}\mathscr{C}_{i}$.

For every system ${\rm A}$ one can perform the identity-test (or simply, the identity), that is, a test $\{\mathscr{I}_{\rm A}\}$ with a single outcome, with the property

The subindex ${\rm A}$ will be dropped from $\mathscr{I}_{\rm A}$ where there is no ambiguity.

The letter ${\rm I}$ will be reserved for the trivial system, which simply means “nothing” nothingquantum. A device with input (resp. output) system ${\rm I}$ is a device with no input (resp. no output). The corresponding tests will be called preparation-tests (resp. observation-tests). In this case we replace the input (resp. output) wire with a round portion:

In formulas we will write $\left|\rho_{i}\right)_{\rm B}$ (resp. $\left(a_{j}\right|_{\rm A}$). The sets ${\mathsf{Transf}}({\rm I},{\rm A})$ and ${\mathsf{Transf}}({\rm A},{\rm I})$ will be denoted as ${\mathsf{St}}({\rm A})$ and ${\mathsf{Eff}}({\rm A})$, respectively. The reason for this special notation is that in the next subsection the elements of ${\mathsf{St}}({\rm A})$ (resp. ${\mathsf{Eff}}({\rm A})$) will be interpreted as the states (resp. effects) of system ${\rm A}$.

From every pair of systems ${\rm A}$ and ${\rm B}$ one can form a composite system, denoted by ${\rm A}{\rm B}$. Clearly, composing system ${\rm A}$ with nothing still gives system ${\rm A}$, in formula ${\rm A}{\rm I}={\rm I}{\rm A}={\rm A}$. Two devices can be composed in parallel, thus obtaining a new device with composite input and composite output systems. The events in composite test are represented as

and are written in formulas as $\mathscr{C}_{i}\otimes\mathscr{D}_{j}$. In the special case of states we will often write $\left|\rho_{i}\right)\left|\sigma_{j}\right)$ in place of $\rho_{i}\otimes\sigma_{j}$. Similarly, for effects we will write $\left(a_{i}\right|\left(b_{j}\right|$ in place of $a_{i}\otimes b_{j}$.

Sequential and parallel composition commute: one has $(\mathscr{A}_{i}\otimes\mathscr{B}_{j})(\mathscr{C}_{k}\otimes\mathscr{D}_{l})=\mathscr{A}_{i}\mathscr{C}_{k}\otimes\mathscr{B}_{j}\mathscr{D}_{l}$ for every $\mathscr{A}_{i},\mathscr{B}_{j},\mathscr{C}_{k},\mathscr{D}_{l}$ such that the output of $\mathscr{A}_{i}$ (resp. $\mathscr{B}_{j}$) coincides with the input of $\mathscr{C}_{k}$ (resp. $\mathscr{D}_{l}$).

When one of the two tests is the identity, we will omit the box and draw only a straight line, as in

The rules summarized in this section define the operational language of circuits, which has been discussed in detail in a series of inspiring works by Coecke (see in particular Refs. kinder; picturalism). The language of circuits allows one to represent the schematic of an experiment, like e.g.

and also to represent a particular outcome of the experiment

In formula, the above circuit is given by

II.2 Probabilistic structure: states, effects and transformations

On top of the language of circuits, we put a probabilistic structure purification: we declare that the composition of a preparation-test $\{\rho_{i}\}_{i\in{\rm X}}$ with an observation-test $\{a_{j}\}_{j\in{\rm Y}}$ gives rise to a joint probability distribution:

with $p(i,j)\geq 0$ and $\sum_{i\in{\rm X}}\sum_{j\in{\rm Y}}p(i,j)=1$. In formula we write $p(i,j)=\left(a_{j}\right|\left.\!\rho_{i}\right)$. Moreover, if two experiments are run in parallel, we assume that the joint probability distribution is given by the product:

where $p(i,k):=\left(a_{k}\right|\left.\!\rho_{i}\right),q(j,l):=\left(b_{l}\right|\left.\!\sigma_{j}\right)$.

II.3 Basic definitions in the operational-probabilistic framework

Here we summarize few elementary definitions that will be used later in the paper. The meaning of the definitions in the case of quantum theory is also discussed.

First, we start from the notions of coarse-graining and refinement. Coarse-graining arises when we join together some outcomes of a test: we say that the test $\{\mathscr{D}_{j}\}_{j\in{\rm Y}}$ is a coarse-graining of the test $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$ if there is a disjoint partition $\{{\rm X}_{j}\}_{j\in{\rm Y}}$ of ${\rm X}$ such that

Conversely, if $\{\mathscr{D}_{j}\}_{j\in{\rm Y}}$ is a coarse-graining of $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$, we say that $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$ is a refinement of $\{\mathscr{D}_{j}\}_{j\in{\rm Y}}$. Intuitively, a test that refines another is a test that extracts information in a more precise way: it is a test with better “resolving power”.

The notion of refinement also applies to a single transformation: a refinement of the transformation $\mathscr{C}$ is given by a test $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$ and a subset ${\rm X}_{0}$ such that

Accordingly, we say that each transformation $\mathscr{C}_{i},i\in{\rm X}_{0}$ is a refinement of $\mathscr{C}$. A transformation $\mathscr{C}$ is atomic if it has only trivial refinements: if $\mathscr{C}_{i}$ refines $\mathscr{C}$, then $\mathscr{C}_{i}=p\mathscr{C}$ for some probability $p\geq 0$. A test that consists of atomic transformations is a test whose “resolving power” cannot be further improved.

When discussing states (i.e. transformations with trivial input) we will use the word pure as a synonym of atomic. A pure state describes a situation of maximal knowledge about the system’s preparation, a knowledge that cannot be further refined.

As usual, a state that is not pure will be called mixed. An important notion is that of completely mixed state:

A state is completely mixed if any other state can refine it: precisely, $\omega\in{\mathsf{St}}({\rm A})$ is completely mixed if for every $\rho\in{\mathsf{St}}({\rm A})$ there is a non-zero probability $p>0$ such that $p\rho$ is a refinement of $\omega$.

Intuitively, a completely mixed state describes a situation of complete ignorance about the system’s preparation: if a system is described by a completely mixed state, then it means that we know so little about its preparation that, in fact, every preparation is possible.

We conclude this paragraph with a couple of definitions that will be used throughout the paper:

A transformation $\mathscr{U}\in{\mathsf{Transf}}({\rm A},{\rm B})$ is reversible if there exists another transformation $\mathscr{U}^{-1}\in{\mathsf{Transf}}({\rm B},{\rm A})$ such that $\mathscr{U}^{-1}\mathscr{U}=\mathscr{I}_{\rm A}$ and $\mathscr{U}\mathscr{U}^{-1}=\mathscr{I}_{\rm B}$. When ${\rm A}={\rm B}$ the reversible transformations form a group, indicated as ${\mathbf{G}}_{\rm A}$.

Two systems ${\rm A}$ and ${\rm B}$ are operationally equivalent if there exists a reversible transformation $\mathscr{U}$ from ${\rm A}$ to ${\rm B}$.

When two systems are operationally equivalent one can convert one into the other in a reversible fashion.

II.3.2 Examples in in quantum theory

Preparation-tests are often called quantum information sources in quantum information theory. A generic state $\rho$ is an unnormalized density matrix. A deterministic state, corresponding to a single-outcome preparation-test is a normalized density matrix $\rho$, with ${\rm Tr}[\rho]=1$.

Diagonalizing $\rho=\sum_{i}\alpha_{i}|\psi_{i}\rangle\langle\psi_{i}|$ we then obtain that each matrix $\alpha_{i}|\psi_{i}\rangle\langle\psi_{i}|$ is a refinement of $\rho$. More generally, every matrix $\sigma$ such that $\sigma\leq\rho$ is a refinement of $\rho$. Up to a positive rescaling, all matrices with support contained in the support of $\rho$ are refinements of $\rho$. A quantum state $\rho$ is atomic (pure) if and only if it is proportional to a rank-one projection. A quantum state is completely mixed if and only if its density matrix has full rank. Note that the quantum state $\chi=\frac{I_{d}}{d}$, where ${I_{d}}$ is the identity $d\times d$ matrix, is a particular example of completely mixed state, but not the only example. Precisely, $\chi=\frac{I_{d}}{d}$ is the unique unitarily invariant state in dimension $d$.

Let us now consider the case of observation-tests: in quantum theory an observation-test is given by a POVM (positive operator-valued measure), namely by a collection $\{P_{j}\}_{j\in{\rm Y}}$ of non-negative $d\times d$ matrices such that

An effect is then a non-negative matrix $P\geq 0$ upper bounded by the identity. In quantum theory there is only one deterministic effect, corresponding to a single-outcome observation test: the unique deterministic effect given by the identity matrix. As we will see in the following section, the fact that the deterministic effect is unique is equivalent to the fact that quantum theory is a causal theory.

An effect $P$ is atomic if and only if $P$ is proportional to a rank-one projector. An observation-test is atomic if it is a POVM with rank-one elements.

is trace-preserving. A general transformation is then given by a trace non-increasing map, called quantum operation, whereas a deterministic transformation, corresponding to a single-outcome test, is given by a trace-preserving map, called quantum channel.

Any quantum operation $\mathscr{C}$ can be written in the Kraus form $\mathscr{C}(\rho)=\sum_{i}C_{i}\rho C_{i}^{\dagger}$, where $C_{i}:{\mathscr{H}}_{1}\to{\mathscr{H}}_{2}$ are the Kraus operators. Up to a positive scaling, every quantum operation $\mathscr{D}$ such that the Kraus operators of $\mathscr{D}$ belong to the linear span of the Kraus operators of $\mathscr{C}$ is a refinement of $\mathscr{C}$. A map $\mathscr{C}$ is atomic if and only if there is only one Kraus operator in its Kraus form. A reversible transformation in quantum theory is a unitary map $\mathscr{U}(\rho)=U\rho U^{\dagger}$, where $U:{\mathscr{H}}_{1}\to{\mathscr{H}}_{2}$ is a unitary operator, that is $U^{\dagger}U=I_{1}$ and $UU^{\dagger}=I_{2}$ where $I_{1}$ $(I_{2})$ is the identity operator on ${\mathscr{H}}_{1}$ (${\mathscr{H}}_{2}$). Two quantum systems are operationally equivalent if and only if the corresponding Hilbert spaces have the same dimension.

II.4 Operational principles

We are now in position to make precise the usage of the expression “operational principle” in the context of this paper. By “operational principle” we mean here a principle that can be stated using only the operational-probablistic language, i.e. using only

the notions of system, test, outcome, probability, state, effect, transformation

their specifications: atomic, pure, mixed, completely mixed

more complex notions constructed from the above terms (e.g. the notion of “reversible transformation”).

The distinction between operational principles and principles referring to abstract mathematical properties, mentioned in the introduction, should now be clear: for example, a statement like “the pure states of a system cannot be cloned” is a valid operational principle, because it can be analyzed in basic operational-probabilistic terms as “for every system ${\rm A}$ there exists no transformation $\mathscr{C}$ with input system ${\rm A}$ and output system ${\rm A}{\rm A}$ such that $\mathscr{C}\left|\varphi\right)=\left|\varphi\right)\left|\varphi\right)$ for every pure state $\varphi$ of ${\rm A}$ ”. On the contrary, a statement like “the state space of a system with two perfectly distinguishable states is a three-dimensional sphere” is not a valid operational principle, because there is no way to express what it means for a state space to be a three-dimensional sphere in terms of basic operational notions. The fact that a state spate is a sphere may be eventually derived from operational principles, but cannot be assumed as a starting point.

III The principles

We now state the principles used in our derivation. The first five principles express generic features that are shared by both classical and quantum theory. They could be even included in the definition of the background framework: they define the simple model of information processing in which we try to single out quantum theory. For this reason we will call them axioms. The sixth principle in our derivation has a different status: it expresses the genuinely quantum features. A major message of our work is that, within a broad class of theories of information processing, quantum theory is completely described by the purification principle. To emphasize the special role of the sixth principle we will call it postulate, in analogy with the parallel postulate of Euclidean geometry.

The first axiom of our list, causality purification, is so basic that could be considered as part of the background framework. We decided to explicitly present it as an axiom for two reasons: The first reason is that the framework of operational-probabilistic theories can be developed even without this requirement (see Ref.purification for the general framework and Refs. supermaps; comblong for two explicit examples of non-causal theories). The second reason is that we want to stress that causality is an essential ingredient in our derivation. This observation is important in view of possible extensions of quantum theory to quantum gravity scenarios where the causal structure is not defined from the start (see e.g. Hardy in Ref. causaloid).

The probability of preparations is independent of the choice of observations.

In technical terms: if $\{\rho_{i}\}_{i\in{\rm X}}\subset{\mathsf{St}}({\rm A})$ is a preparation-test, then the conditional probability of the preparation $\rho_{i}$ given the choice of the observation-test $\{a_{j}\}_{j\in{\rm Y}}$ is the marginal

The axiom states that the marginal probability $p\left(i|\{a_{j}\}\right)$ is independent of the choice of the observation-test $\{a_{j}\}$: if $\{a_{j}\}_{j\in{\rm Y}}$ and $\{b_{k}\}_{k\in{\rm Z}}$ are two different observation-tests, then one has $p(i|\{a_{j}\})=p(i|\{b_{k}\})$. Loosely speaking, one may refer to causality as a requirement of no-signalling from the future: indeed, causality is equivalent to the fact that the probability of an outcome at a certain time does not depend on the choice of operations that will be done at later times maurolast.

An operational-probabilistic theory that satisfies the causality axiom 1 will be called causal. As we already mentioned, causality is a very basic requirement and could be considered as part of the framework: it provides the notions used to state the other axioms and it implies several facts that will be used frequently in the paper. In fact, in our derivation we do not use the causality axiom directly, but only through its consequences. In the following we briefly summarize the facts and the notations that characterize the framework of causal operational-probabilistic theories, introduced and discussed in detail in Ref. purification. Similar structures have been subsequently considered in Refs. duotenz; caucat within a formal description of circuits in foliable spacetime regions.

First, causality is equivalent to the existence of an effect $e_{\rm A}$ such that $e_{\rm A}=\sum_{j\in{\rm X}}a_{j}$ for every observation-test $\{a_{j}\}_{j\in{\rm Y}}$. We call the effect $e_{\rm A}$ the deterministic effect for system ${\rm A}$. By definition, the effect $e_{\rm A}$ is unique. The subindex ${\rm A}$ in $e_{\rm A}$ will be dropped when no confusion can arise.

In a causal theory every test $\{\mathscr{C}_{i}\}_{i\in{\rm X}}\subset{\mathsf{Transf}}({\rm A},{\rm B})$ satisfies the condition

As a consequence, a transformation $\mathscr{C}\in{\mathsf{Transf}}({\rm A},{\rm B})$ satisfies the condition

with the equality if and only if $\mathscr{C}$ is a channel (i.e. a deterministic transformation, corresponding to a single-outcome test). In Eq. (4) we used the notation $\left(a\right|\leq\left(a^{\prime}\right|$ to mean $\left(a\right|\left.\!\rho\right)\leq\left(a^{\prime}\right|\left.\!\rho\right)$ for every $\rho\in{\mathsf{St}}({\rm A})$.

In a causal theory the norm of a state $\rho_{i}\in{\mathsf{St}}({\rm A})$ is given by $|\!|\rho_{i}|\!|=\left(e\right|\left.\!\rho_{i}\right)$. Accordingly, one can define the normalized state

In a causal theory one can always allow for rescaled preparations: conditionally to the outcome $i\in{\rm X}$ in the preparation-test $\{\rho_{i}\}_{i\in{\rm X}}$ we can say that we prepared the normalized state $\bar{\rho}_{i}$. For this reason, every state in a causal theory is proportional to a normalized state.

The set of normalized states will be denoted by ${\mathsf{St}}_{1}({\rm A})$. Since the set of all states ${\mathsf{St}}({\rm A})$ is closed in the operational norm, also the set of normalized states ${\mathsf{St}}_{1}({\rm A})$ is closed. Moreover, the set ${\mathsf{St}}_{1}({\rm A})$ is convex purification: this means that for every pair of normalized states $\rho_{1},\rho_{2}\in{\mathsf{St}}_{1}({\rm A})$ and for every probability $p\in$ the convex combination $\rho_{p}=p\rho_{1}+(1-p)\rho_{2}$ is a normalized state. Operationally, the state $\rho_{p}$ is obtained by

performing a binary test with outcomes $\{1,2\}$ and outcome probabilities $p_{1}=p$ and $p_{2}=1-p$

for outcome $i$ preparing $\rho_{i}$, thus realizing the preparation-test $\{p_{i}\rho_{i}\}_{i=1,2}$

coarse-graining over the outcomes, thus obtaining $\rho_{p}=p\rho_{1}+(1-p)\rho_{2}$.

The step 2 (preparation of a state conditionally on the outcome of a previous test) is possible because the theory is causal purification.

The pure normalized states are the extreme points of the convex set ${\mathsf{St}}_{1}({\rm A})$. For a normalized state $\rho\in{\mathsf{St}}_{1}({\rm A})$ we define the face identified by $\rho$ as follows:

The face identified by $\rho\in{\mathsf{St}}_{1}({\rm A})$ is the set $F_{\rho}$ of all normalized states $\sigma\in{\mathsf{St}}_{1}({\rm A})$ such that $\rho=p\sigma+(1-p)\tau$, for some non-zero probability $p>0$ and some normalized state $\tau\in{\mathsf{St}}_{1}({\rm A})$.

In other words, $F_{\rho}$ is the set of all normalized states that show up in the convex decompositions of $\rho$. Clearly, if $\varphi$ is a pure state, then one has $F_{\varphi}=\{\varphi\}$. The opposite situation is that of completely mixed states: by definition 1, a state $\omega\in{\mathsf{St}}_{1}({\rm A})$ is completely mixed if every state $\sigma\in{\mathsf{St}}_{1}({\rm A})$ can stay in its convex decomposition, that is, if $F_{\omega}={\mathsf{St}}_{1}({\rm A})$. An equivalent condition for a state to be completely mixed is the following:

Proof. The condition is clearly necessary. It is also sufficient because for a state $\sigma\in{\mathsf{St}}_{1}({\rm A})$ the relation $\sigma\in\mathsf{Span}(F_{\omega})$ implies $\sigma\in F_{\omega}$ (see Lemma 16 of Ref.purification). $\,\blacksquare$

A completely mixed state can never be distinguished from another state with zero error probability:

Let $\rho\in{\mathsf{St}}_{1}({\rm A})$ be a completely mixed state and $\sigma\in{\mathsf{St}}_{1}({\rm A})$ be an arbitrary state. Then, the probability of error in distinguishing $\rho$ from $\sigma$ is strictly greater than zero.

Proof. By contradiction, suppose that one can distinguish between $\rho$ and $\sigma$ with zero error probability. This means that there exists a binary test $\{a_{\rho},a_{\sigma}\}$ such that $\left(a_{\rho}\right|\left.\!\sigma\right)=\left(a_{\sigma}\right|\left.\!\rho\right)=0$. Since $\rho$ is completely mixed there exists a probability $p>0$ and a state $\tau\in{\mathsf{St}}_{1}({\rm A})$ such that $\rho=p\sigma+(1-p)\tau$. Hence, the condition $\left(a_{\sigma}\right|\left.\!\rho\right)=0$ implies $\left(a_{\sigma}\right|\left.\!\sigma\right)=0$. Therefore, we have $\left(a_{\rho}\right|\left.\!\sigma\right)+\left(a_{\sigma}\right|\left.\!\sigma\right)=0$. This is in contradiction with the normalization of the probabilities in the test $\{a_{\rho},a_{\sigma}\}$, which would require $\left(a_{\rho}\right|\left.\!\sigma\right)+\left(a_{\sigma}\right|\left.\!\sigma\right)=1$. $\,\blacksquare$

III.1.2 Perfect distinguishability

Our second axiom regards the task of state discrimination. As we saw in proposition 1, if a state is completely mixed, then it is impossible to distinguish it perfectly from any other state. Axiom 2 states the converse:

Every state that is not completely mixed can be perfectly distinguished from some other state.

Note that the statement of axiom 2 holds for quantum and for classical information theory. In quantum theory a completely mixed state is a density matrix with full rank. If a density matrix $\rho$ has not full rank, then it must have a kernel: hence, every density matrix $\sigma$ with support in the kernel of $\rho$ will be perfectly distinguishable from $\rho$, as stated in Axiom 2. Applying the same reasoning for density matrices that are diagonal in a given basis, one can easily see that Axiom 2 is satisfied also by classical information theory.

III.1.3 Ideal compression

The third axiom is about information compression. An information source for system ${\rm A}$ is a preparation-test $\{\rho_{i}\}_{i\in{\rm X}}$, where each $\rho_{i}\in{\mathsf{St}}({\rm A})$ is an unnormalized state and $\sum_{i\in{\rm X}}\left(e\right|\left.\!\rho_{i}\right)=1$. A compression scheme is given by an encoding operation $\mathscr{E}$ from ${\rm A}$ to a smaller system ${\rm C}$, that is, to a system ${\rm C}$ such that $D_{\rm C}\leq D_{\rm A}$. The compression scheme is lossless for the source $\{\rho_{i}\}_{i\in{\rm X}}$ if there exists a decoding operation $\mathscr{D}$ from ${\rm C}$ to ${\rm A}$ such that $\mathscr{D}\mathscr{E}\left|\rho_{i}\right)=\left|\rho_{i}\right)$ for every value of the index $i\in{\rm X}$. This means that the decoding allows one to perfectly retrieve the states $\{\rho_{i}\}_{i\in{\rm X}}$. We say that a compression scheme is lossless for the state $\rho$, if it is lossless for every source $\{\rho_{i}\}_{i\in{\rm X}}$ such that $\rho=\sum_{i\in{\rm X}}\rho_{i}$. Equivalently, this means that the restriction of $\mathscr{D}\mathscr{E}$ to the face identified by $\rho$ is equal to the identity channel: $\mathscr{D}\mathscr{E}\left|\sigma\right)=\sigma$ for every $\sigma\in F_{\rho}$.

A lossless compression scheme is maximally efficient if the encoding system ${\rm C}$ has the smallest possible size, that is, if the system ${\rm C}$ has no more states than exactly those needed to compress $\rho$. This happens when every normalized state $\tau\in{\mathsf{St}}_{1}({\rm C})$ comes from the encoding of some normalized state $\sigma\in F_{\rho}$, namely $\left|\tau\right)=\mathscr{E}\left|\sigma\right)$.

We say that a compression scheme that is lossless and maximally efficient is ideal. Our second axiom states that ideal compression is always possible:

For every state there exists an ideal compression scheme.

It is easy to see that this statement holds in quantum theory and in classical probability theory. For example, if $\rho$ is a density matrix on a $d$-dimensional Hilbert space and $\mathsf{rank}(\rho)=r$, then the ideal compression is obtained by just encoding $\rho$ in an $r$-dimensional Hilbert space. As long as we do not tolerate losses, this is the most efficient one-shot compression we can devise in quantum theory. Similar observations hold for classical information theory.

III.1.4 Local distinguishability

The fourth axiom consists in the assumption of local distinguishability, here presented in the formulation of Ref. purification.

(Local distinguishability) If two bipartite states are different, then they give different probabilities for at least one product experiment.

In more technical terms: if $\rho,\sigma\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ are states and $\rho\not=\sigma$, then there are two effects $a\in{\mathsf{Eff}}({\rm A})$ and $b\in{\mathsf{Eff}}({\rm B})$ such that

Local distinguishability is equivalent to the fact that two distant parties, holding systems ${\rm A}$ and ${\rm B}$, respectively, can distinguish between the two states $\rho,\sigma\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ using only local operations and classical communication and achieving an error probability strictly larger than $p_{ran}=1/2$, the probability of error in random guess purification. Again, this statement holds in ordinary quantum theory (on complex Hilbert spaces) and in classical information theory.

Another equivalent condition to local distinguishability is the local tomography axiom, introduced in Refs. maurofirst; barrett. The local tomography axiom state that every bipartite state can be reconstructed from the statistics of local measurements on the component systems. Technically, local tomography is in turn equivalent to the relation $D_{{\rm A}{\rm B}}=D_{\rm A}D_{\rm B}$ Har01 and to the fact that every state $\rho\in{\mathsf{St}}({\rm A}{\rm B})$ can be written as

An important consequence of local distinguishability, observed in Ref. purification, is that a transformation $\mathscr{C}\in{\mathsf{Transf}}({\rm A}{\rm B})$ is completely specified by its action on ${\mathsf{St}}({\rm A})$: thanks to local distinguishability we have the implication

(see Lemma 14 of Ref.purification for the proof). Note that Eq. (5) does not hold for quantum theory on real Hilbert spaces purification.

III.1.5 Pure conditioning

The fourth axiom states how the outcomes of a measurement on one side of a pure bipartite state can induce pure states on the other side. In this case we consider atomic measurements, that is, measurements described by observation-tests $\{a_{i}\}_{i\in{\rm X}}$ where each effect $a_{i}$ is atomic. Intuitively, atomic measurement are those with maximum “resolving power”.

If a bipartite system is in a pure state, then each outcome of an atomic measurement on one side induces a pure state on the other.

The pure conditioning property holds in quantum theory and in classical information theory as well. In fact, the statement is trivial in classical information theory, because the only pure bipartite states are the product of pure states: no matter which measurement is performed on one side, the remaining state on the other side will necessarily be pure.

The pure conditioning property, as formulated above, has been recently introduced in Ref. pcond. A stronger version of axiom 5 is the atomicity of composition introduced in Ref. maurolast:

Atomicity of composition: the sequential composition of two atomic operations is atomic.

Since pure states and atomic effects are a particular case of atomic transformations, Axiom 5’ implies Axiom 5. In our derivation, however, also the converse implication holds: indeed, thanks to the purification postulate we will be able to show that Axiom 5 implies Axiom 5’ (see lemma 16).

III.2 The purification postulate

The last postulate in our list is the purification postulate, which was introduced and explored in detail in Ref. purification. While the previous axioms were also satisfied by classical probability theory, the purification axiom introduces in our derivation the genuinely quantum features. A purification of the state $\rho\in{\mathsf{St}}_{1}({\rm A})$ is a pure state $\Psi_{\rho}$ of some composite system ${\rm A}{\rm B}$, with the property that $\rho$ is the marginal of $\Psi_{\rho}$, that is,

Here we refer to the system ${\rm B}$ as the purifying system. The purification axiom states that every state can be obtained as the marginal of a pure bipartite state in an essentially unique way:

Every state has a purification. For fixed purifying system, every two purifications of the same state are connected by a reversible transformation on the purifying system.

Informally speaking, our postulate states that the ignorance about a part is always compatible with a maximal knowledge of the whole. The existence of pure bipartite states with mixed marginal was already recognized by Schrödinger as the characteristic trait of quantum theory Schr35. Here, however, we also emphasize the importance of the uniqueness of purification up to reversible transformations: this property sets up a relation between pure states and reversible transformations that generates most of the structure of quantum theory. As shown in Ref. purification, an impressive number of quantum features are actually direct consequences of purification. In particular, purification implies the possibility of simulating any irreversible process through a reversible interaction of the system with an environment that is finally discarded.

IV First consequences of the principles

Let $\rho\in{\mathsf{St}}_{1}({\rm A})$ be a state and let $\mathscr{E}\in{\mathsf{Transf}}({\rm A},{\rm C})$ (resp. $\mathscr{D}\in{\mathsf{Transf}}({\rm C},{\rm A})$) be its encoding (resp. decoding) in the ideal compression scheme of Axiom 3.

Essentially, the encoding operation $\mathscr{E}\in{\mathsf{Transf}}({\rm A},{\rm C})$ identifies the face $F_{\rho}$ with the state space ${\mathsf{St}}_{1}({\rm C})$. In the following we provide a list of elementary lemmas showing that all statements about $F_{\rho}$ can be translated into statements about ${\mathsf{St}}_{1}({\rm C})$ and vice-versa.

The composition of decoding and encoding is the identity on ${\rm C}$, namely $\mathscr{E}\mathscr{D}=\mathscr{I}_{\rm C}$.

Proof. Since the compression is maximally efficient, for every state $\tau\in{\mathsf{St}}_{1}({\rm C})$ there is a state $\sigma\in F_{\rho}$ such that $\mathscr{E}\sigma=\tau$. Using the fact that $\mathscr{D}\mathscr{E}\sigma=\sigma$ (the compression is lossless) we then obtain $\mathscr{E}\mathscr{D}\tau=\mathscr{E}\mathscr{D}\mathscr{E}\sigma=\mathscr{E}\sigma=\tau$. By local distinguishability [see Eq. (5)], this implies $\mathscr{E}\mathscr{D}=\mathscr{I}_{\rm C}$. $\,\blacksquare$

The image of ${\mathsf{St}}_{1}({\rm C})$ under the decoding operation $\mathscr{D}$ is $F_{\rho}$.

Proof. Since the compression is maximally efficient, for all $\tau\in{\mathsf{St}}_{1}({\rm C})$ there exists $\sigma\in F_{\rho}$ such that $\tau=\mathscr{E}\sigma$. Then, $\mathscr{D}\tau=\mathscr{D}\mathscr{E}\sigma=\sigma$. This implies that $\mathscr{D}({\mathsf{St}}_{1}({\rm C}))\subseteq F_{\rho}$. On the other hand, since the compression is lossless, for every state $\sigma\in F_{\rho}$ one has $\mathscr{D}\mathscr{E}\sigma=\sigma$. This implies the inclusion $F_{\rho}\subseteq\mathscr{D}({\mathsf{St}}_{1}({\rm C}))$. $\,\blacksquare$

If the state $\varphi\in F_{\rho}$ is pure, then the state $\mathscr{E}\varphi\in{\mathsf{St}}_{1}({\rm C})$ is pure. If the state $\psi\in{\mathsf{St}}_{1}({\rm C})$ is pure, then the state $\mathscr{D}\psi\in F_{\rho}$ is pure.

Proof. Suppose that $\varphi\in F_{\rho}$ is pure and that $\mathscr{E}\varphi$ can be written as $\mathscr{E}\varphi=p\sigma+(1-p)\tau$ for some $p>0$ and some $\sigma,\tau\in{\mathsf{St}}_{1}({\rm C})$. Applying $\mathscr{D}$ on both sides we obtain $\varphi=p\mathscr{D}\sigma+(1-p)\mathscr{D}\tau$. Since $\varphi$ is pure we must have $\mathscr{D}\sigma=\mathscr{D}\tau=\varphi$. Now, applying $\mathscr{E}$ on all terms of the equality and using lemma 2 we obtain $\sigma=\tau=\mathscr{E}\varphi$. This proves that $\mathscr{E}\varphi$ is pure. Conversely, suppose that $\psi\in{\mathsf{St}}_{1}({\rm C})$ is pure and $\mathscr{D}\psi=p\sigma+(1-p)\tau$ for some $p>0$ and some $\sigma,\tau\in{\mathsf{St}}_{1}({\rm A})$. Since $\mathscr{D}\psi$ is in the face $F_{\rho}$ (lemma 3), also $\sigma$ and $\tau$ are in the same face. Applying $\mathscr{E}$ on both sides of the equality $\mathscr{D}\psi=p\sigma+(1-p)\tau$ and using lemma 2 we obtain $\psi=\mathscr{E}\mathscr{D}\psi=p\mathscr{E}\sigma+(1-p)\mathscr{E}\tau$. Since $\psi$ is pure we must have $\mathscr{E}\sigma=\mathscr{E}\tau=\psi$. Applying $\mathscr{D}$ on all terms of the equality we then have $\sigma=\tau=\mathscr{D}\psi$, thus proving that $\mathscr{D}\psi$ is pure. $\,\blacksquare$

We say that a state $\sigma\in F_{\rho}$ is completely mixed relative to the face $F_{\rho}$ if every state $\tau\in F_{\rho}$ can stay in the convex decomposition of $\sigma$. In other words, $\sigma$ is completely mixed relative to $F_{\rho}$ if one has $F_{\sigma}=F_{\rho}$. Note that in general $\sigma\in F_{\rho}$ implies $F_{\sigma}\subseteq F_{\rho}$.

If the state $\omega\in F_{\rho}$ is completely mixed relative to $F_{\rho}$, then the state $\mathscr{E}\omega\in{\mathsf{St}}_{1}({\rm C})$ is completely mixed. If the state $\upsilon\in{\mathsf{St}}_{1}({\rm C})$ is completely mixed, then the state $\mathscr{D}\upsilon\in F_{\rho}$ is completely mixed relative to $F_{\rho}$.

Proof. Suppose that $\omega$ is completely mixed relative to $F_{\rho}$. Then every state $\sigma\in F_{\rho}$ can stay in its convex decomposition, say $\omega=p\sigma+(1-p)\sigma^{\prime}$ with $p>0$ and $\sigma^{\prime}\in F_{\rho}$. Applying $\mathscr{E}$ we have

Since the compression is maximally efficient, for every state $\tau\in{\mathsf{St}}_{1}({\rm C})$ there exists a state $\sigma\in F_{\rho}$ such that $\tau=\mathscr{E}\sigma$. Choosing the suitable $\sigma\in F_{\rho}$ and substituting $\tau$ to $\mathscr{E}\sigma$ in Eq. (6) we then obtain that for every state $\tau\in{\mathsf{St}}_{1}({\rm C})$ there exists probability $p>0$ and a state $\sigma^{\prime}\in F_{\rho}$ such that

This implies that $\mathscr{E}\omega$ is completely mixed. Suppose now that $\upsilon\in{\mathsf{St}}_{1}({\rm C})$ is completely mixed. Then every state $\tau\in{\mathsf{St}}_{1}({\rm C})$ can stay in its convex decomposition, say $\upsilon=p\tau+(1-p)\tau^{\prime}$. with $p>0$ and $\tau^{\prime}\in{\mathsf{St}}_{1}({\rm C})$. Applying $\mathscr{D}$ on both sides we have

Now, using lemma 3 we have that every state $\sigma\in F_{\rho}$ can be written as $\sigma=\mathscr{D}\tau$ for some $\tau\in{\mathsf{St}}_{1}({\rm C})$. Choosing the suitable $\tau\in{\mathsf{St}}_{1}({\rm C})$ and substituting $\sigma$ to $\mathscr{D}\tau$ in Eq. (7) we then obtain that for evert state $\sigma\in F_{\rho}$ there exists a probability $p>0$ and a state $\tau^{\prime}\in{\mathsf{St}}_{1}({\rm C})$ such that $\mathscr{D}\upsilon=p\sigma+(1-p)\mathscr{D}\tau^{\prime}$. Therefore, $\mathscr{D}\upsilon$ is completely mixed relative to $F_{\rho}$. $\,\blacksquare$

We now show that the system ${\rm C}$ used for ideal compression of the state $\rho$ is unique up to operational equivalence:

If two systems ${\rm C}$ and ${\rm C}^{\prime}$ allow for ideal compression of a state $\rho\in{\mathsf{St}}_{1}({\rm A})$, then ${\rm C}$ and ${\rm C}^{\prime}$ are operationally equivalent.

Proof. Let $\mathscr{E},\mathscr{D}$ and $\mathscr{E}^{\prime},\mathscr{D}^{\prime}$ denote the encoding/decoding schemes for systems ${\rm C}$ and ${\rm C}^{\prime}$, respectively. Define the transformations $\mathscr{U}:=\mathscr{E}^{\prime}\mathscr{D}\in{\mathsf{Transf}}({\rm C},{\rm C}^{\prime})$ and $\mathscr{V}=\mathscr{E}\mathscr{D}^{\prime}\in{\mathsf{Transf}}({\rm C}^{\prime},{\rm C})$. It is easy to see that $\mathscr{U}$ is reversible and $\mathscr{U}^{-1}=\mathscr{V}$. Indeed, since the restriction of $\mathscr{D}^{\prime}\mathscr{E}^{\prime}$ and $\mathscr{D}\mathscr{E}$ to the face $F_{\rho}$ is the identity, using Lemma 3 one has $\mathscr{D}^{\prime}\mathscr{E}^{\prime}\mathscr{D}=\mathscr{D}$ and similarly $\mathscr{D}\mathscr{E}\mathscr{D}^{\prime}=\mathscr{D}^{\prime}$. Hence, we have $\mathscr{U}\mathscr{V}=\mathscr{E}^{\prime}\mathscr{D}\mathscr{E}\mathscr{D}^{\prime}=\mathscr{E}^{\prime}\mathscr{D}^{\prime}=\mathscr{I}_{{\rm C}^{\prime}}$ and $\mathscr{V}\mathscr{U}=\mathscr{E}\mathscr{D}^{\prime}\mathscr{E}^{\prime}\mathscr{D}=\mathscr{E}\mathscr{D}=\mathscr{I}_{{\rm C}}$.$\,\blacksquare$

It is useful to introduce the notion of equality upon input of $\rho$. We say that two transformations $\mathscr{A},\mathscr{A}^{\prime}\in{\mathsf{Transf}}({\rm A},{\rm B})$ are equal upon input of $\rho\in{\mathsf{St}}({\rm A})$ if their restrictions to the face identified by $\rho$ are equal, that is, if $\mathscr{A}\sigma=\mathscr{A}^{\prime}\sigma$ for every $\sigma\in F_{\rho}$. If $\mathscr{A}$ and $\mathscr{A}^{\prime}$ are equal upon input of $\rho$ we write $\mathscr{A}=_{\rho}\mathscr{A}^{\prime}$.

Using the notion of equality upon input of $\rho$ we can rephrase the fact that the compression is lossless for $\rho$ as $\mathscr{D}\mathscr{E}=_{\rho}\mathscr{I}_{\rm A}$. Similarly, we can state the following:

The encoding $\mathscr{E}$ is deterministic upon input of $\rho$, that is $\left(e_{\rm C}\right|\mathscr{E}=_{\rho}\left(e_{\rm A}\right|$.

Proof. For every $\sigma\in F_{\rho}$ we have $\left(e_{\rm C}\right|\mathscr{E}\left|\sigma\right)\geq\left(e_{\rm A}\right|\mathscr{D}\mathscr{E}\left|\sigma\right)=\left(e_{\rm A}\right|\left.\!\sigma\right)=1$, having used Eq. (4) and the fact that the compression is lossless. Since probabilities are bounded by 1, this implies $\left(e_{\rm C}\right|\mathscr{E}\left|\sigma\right)=\left(e_{\rm A}\right|\left.\!\sigma\right)$ for every $\sigma\in F_{\rho}$, that is, $\left(e_{\rm C}\right|\mathscr{E}=_{\rho}\left(e_{\rm A}\right|$. $\,\blacksquare$

The decoding $\mathscr{D}$ is deterministic, that is $\left(e_{\rm A}\right|\mathscr{D}=\left(e_{\rm C}\right|$.

Proof. For every $\tau\in{\mathsf{St}}_{1}({\rm A})$ we have $\left(e_{\rm A}\right|\mathscr{D}\left|\tau\right)\geq\left(e_{\rm C}\right|\mathscr{E}\mathscr{D}\left|\tau\right)=\left(e_{\rm C}\right|\left.\!\tau\right)$, having used Eq. (4) and lemma 2. Hence, $\left(e_{\rm A}\right|\mathscr{D}=\left(e_{\rm C}\right|$. $\,\blacksquare$

IV.2 Results about purification

The purification postulate 1 implies a large number of quantum features, as it was shown in Ref. purification. Here we review only the facts that are useful for our derivation, referring to Ref. purification for the proofs.

An elementary consequence of the uniqueness of purification is that the group ${\mathbf{G}}_{\rm A}$ of reversible transformations on ${\rm A}$ acts transitively on the set of pure states:

For every couple of pure states $\varphi,\varphi^{\prime}\in{\mathsf{St}}_{1}({\rm A})$ there is a reversible transformation $\mathscr{U}\in{\mathbf{G}}_{\rm A}$ such that $\varphi^{\prime}=\mathscr{U}\varphi$.

Proof. See Lemma 20 of Ref. purification. $\,\blacksquare$

Transitivity implies that for every system ${\rm A}$ there is a unique state $\chi_{\rm A}\in{\mathsf{St}}_{1}({\rm A})$ that is invariant under reversible transformations, that is, a unique state such that $\mathscr{U}\chi_{\rm A}=\chi_{\rm A}$ for every $\mathscr{U}\in{\mathbf{G}}_{\rm A}$:

For every system ${\rm A}$, there is a unique state $\chi_{\rm A}$ invariant under all reversible transformations in ${\mathbf{G}}_{\rm A}$. The invariant state has the following properties:

$\chi_{{\rm A}{\rm B}}=\chi_{\rm A}\otimes\chi_{\rm B}$.

Proof. See Corollary 34 and Theorem 4 of Ref. purification. The proof of item 2 uses the local distinguishability axiom. $\,\blacksquare$

When there is no ambiguity we will drop the subindex ${\rm A}$ and simply write $\chi$.

The uniqueness of purification in postulate 1 requires that if $\Psi_{\rho},\Psi_{\rho}^{\prime}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ are two purifications of $\rho\in{\mathsf{St}}_{1}({\rm A})$, then there exists a reversible transformation $\mathscr{U}\in{\mathbf{G}}_{\rm B}$ such that $\Psi_{\rho}^{\prime}=(\mathscr{I}_{\rm A}\otimes\mathscr{U})\Psi_{\rho}$. The following lemma extends the uniqueness property to purifications with different purifying systems:

(Uniqueness of the purification up to channels on the purifying systems) Let $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ and $\Psi^{\prime}\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ be two purifications of $\rho\in{\mathsf{St}}_{1}({\rm A})$. Then there exists a channel $\mathscr{C}\in{\mathsf{Transf}}({\rm B},{\rm C})$ such that

Proof. See Lemma 21 of Ref. purification. $\,\blacksquare$

Another consequence of the uniqueness of purification is the fact that any ensemble decomposition of a given mixed state can be obtained by performing a measurement on the purifying system:

(Purification of preparation-tests) Let $\rho\in{\mathsf{St}}_{1}({\rm A})$ be a state and $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ be a purification of $\rho$. If $\{\rho_{i}\}_{i\in{\rm X}}$ be a preparation-test such that $\sum_{i\in{\rm X}}\rho_{i}=\rho$, then there exists an observation-test $\{a_{i}\}_{i\in{\rm X}}$ on the purifying system such that

Proof. See lemma 8 of Ref. purification. $\,\blacksquare$

If $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ is a purification of $\rho\in{\mathsf{St}}_{1}({\rm A})$ and $\sigma$ belongs to the face $F_{\rho}$, then there exists an effect $b$ and a non-zero probability $p>0$ such that

An important consequence of purification and local distinguishability is the relation between equality upon input of $\rho$ and equality on the purifications of $\rho$:

(Equality upon input of $\rho$ vs equality on purifications of $\rho$) Let $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ be a purification of $\rho\in{\mathsf{St}}_{1}({\rm A})$, and let $\mathscr{A},\mathscr{A}^{\prime}\in{\mathsf{Transf}}({\rm A},{\rm B})$ be two transformations. Then one has

Proof. See theorem 1 of Ref. purification. The proof of the direction $\Longleftarrow$ uses the local distinguishability axiom. $\,\blacksquare$

As a consequence, the purification of a completely mixed state allows for the tomography of transformations:

Let $\omega\in{\mathsf{St}}_{1}({\rm A})$ be completely mixed and $\Psi_{\omega}\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ is a purification of $\omega$. Then, for all transformations $\mathscr{A},\mathscr{A}^{\prime}\in{\mathsf{Transf}}({\rm A},{\rm B})$ one has

Proof. By theorem 1 the first condition is equivalent to $\mathscr{A}=_{\omega}\mathscr{A}^{\prime}$. Since $\omega$ is completely mixed, this means $\mathscr{A}\sigma=\mathscr{A}^{\prime}\sigma$ for every $\sigma\in{\mathsf{St}}_{1}({\rm A})$. By local distinguishability [see Eq. (5)] this implies $\mathscr{A}=\mathscr{A}^{\prime}$. $\,\blacksquare$

Corollary 2 shows that the state $(\mathscr{A}\otimes\mathscr{I}_{\rm C})\Psi_{\omega}$ characterizes the transformation $\mathscr{A}$ completely. We will express this fact by saying that the state $\Psi_{\omega}$ is dynamically faithful maurolast, or just faithful, for short. Using this notion we can rephrase corollary 2 as:

If $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ is pure and its marginal on system ${\rm A}$ is completely mixed, then $\Psi$ is dynamically faithful for system ${\rm A}$.

Let us choose a fixed faithful state for system ${\rm A}$, say $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$. Then for every transformation $\mathscr{C}\in{\mathsf{Transf}}({\rm A},{\rm B})$ we can define the Choi state $R_{\mathscr{C}}\in{\mathsf{St}}({\rm B}{\rm C})$ as

(Choi isomorphism) For a given faithful state $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ the map $\mathscr{C}\mapsto R_{\mathscr{C}}:=(\mathscr{C}\otimes\mathscr{I}_{\rm C})\Psi$ has the following properties:

it defines a bijective correspondence between tests $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$ from ${\rm A}$ to ${\rm B}$ and collections of states $\{R_{i}\}_{i\in{\rm X}}$ for ${\rm B}{\rm C}$ satisfying

the transformation $\mathscr{C}$ is atomic if and only if the corresponding state $R_{\mathscr{C}}$ is pure.

Proof. See Theorem 17 of Ref. purification.

A simple consequence of the Choi isomorphism is the following:

Let $\{\mathscr{C}_{i}\}_{i\in{\rm X}}\subset{\mathsf{Transf}}({\rm A},{\rm B})$ be a collection of transformations. Then, $\{\mathscr{C}_{i}\}_{i\in{\rm X}}$ is a test if and only if

In particular, let $\{a_{i}\}_{i\in{\rm X}}\subset{\mathsf{Eff}}({\rm A})$ be a collection of effects. Then, $\{a_{i}\}_{i\in{\rm X}}$ is an observation-test if and only if

Proof. Apply item 1 of theorem 2 to the collection of states $\{R_{i}\}_{i\in{\rm X}}$ defined by $R_{i}:=(\mathscr{C}_{i}\otimes\mathscr{I}_{\rm C}){\Psi}$. $\,\blacksquare$

A much deeper consequence of the Choi isomorphism is the following theorem:

(States specify the theory) Let $\Theta,\Theta^{\prime}$ be two theories satisfying the purification postulate. If $\Theta$ and $\Theta^{\prime}$ have the same sets of normalized states, then $\Theta^{\prime}=\Theta$.

Proof. See Theorem 19 of Ref.purification.$\,\blacksquare$

Thanks to theorem 3 to derive quantum theory we will only need to prove that our principles imply that for every system ${\rm A}$ the normalized states ${\mathsf{St}}_{1}({\rm A})$ can be described as positive Hermitian matrices with unit trace. Once this is proved, theorem 3 automatically ensures that all the dynamics and all the measurements allowed by the theory are exactly the dynamics and the measurements allowed in quantum theory.

Note that in the definition of the Choi state we left the freedom to choose the faithful state $\Psi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$. Among many possibilities, one convenient choice is to take a faithful state $\Phi\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ obtained as a purification of the invariant state $\chi\in{\mathsf{St}}_{1}({\rm A})$. Moreover, as we will see in the next paragraph, we can always choose the purifying system ${\rm C}$ in such a way that the marginal on ${\rm C}$ is completely mixed.

IV.3 Results about the combination of compression and purification

An important consequence of the combination of the purification postulate with the compression axiom is the fact that one can always choose a purification of $\rho$ such that the marginal state on the purifying system is completely mixed. To prove this result we need the following lemma:

Let $\rho\in{\mathsf{St}}_{1}({\rm A})$ be a state and let $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ be a purification of $\rho$. If $\mathscr{E}\in{\mathsf{Transf}}({\rm A},{\rm C})$ is the encoding operation in the compression scheme of axiom 3, then the state $\Psi_{\rho}^{\prime}:=(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ is pure.

Proof. Let $\mathscr{D}\in{\mathsf{Transf}}({\rm C},{\rm A})$ be the decoding operation. Since the compression is lossless for $\rho$ we know that $\mathscr{D}\mathscr{E}=_{\rho}\mathscr{I}_{\rm A}$. By theorem 1 this is equivalent to the condition $(\mathscr{D}\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}=\Psi_{\rho}$. Now, suppose that $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}=\sum_{i\in{\rm X}}\Gamma_{i}$. Applying $\mathscr{D}$ on both sides we then obtain $\Psi_{\rho}=\sum_{i\in{\rm X}}(\mathscr{D}\otimes\mathscr{I}_{\rm B})\Gamma_{i}$, and, since $\Psi_{\rho}$ is pure, for every $i\in{\rm X}$ we must have $(\mathscr{D}\otimes\mathscr{I}_{\rm B})\Gamma_{i}=p_{i}\Psi_{\rho}$, where $p_{i}\geq 0$ is some probability. Finally, since $\mathscr{E}\mathscr{D}=\mathscr{I}_{{\rm C}}$ (lemma 2), one has $\Gamma_{i}=p_{i}(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$. Hence, $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ admits only decompositions with $\Gamma_{i}=p_{i}(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$, that is, $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ is pure.$\,\blacksquare$

We are now in position to prove the desired result:

For every state $\rho\in{\mathsf{St}}_{1}({\rm A})$ there exists a system ${\rm C}$ and a purification $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm C})$ of $\rho$ such that the marginal state on system ${\rm C}$ is completely mixed. Moreover, the system ${\rm C}$ is unique up to operational equivalence.

Let $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ be a purification of $\rho\in{\mathsf{St}}_{1}({\rm A})$ and let $\mathscr{E}\in{\mathsf{Transf}}({\rm A},{\rm C})$ be the encoding for $\rho$. Then, the state $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm C}{\rm B})$ is dynamically faithful for ${\rm C}$.

Proof. The marginal of $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ on system ${\rm C}$ is $\mathscr{E}\rho$, which is completely mixed by lemma 5. Hence, $(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ is dynamically faithful by corollary 3. $\,\blacksquare$

The decoding transformation $\mathscr{D}\in{\mathsf{Transf}}({\rm C},{\rm A})$ in the ideal compression for $\rho\in{\mathsf{St}}_{1}({\rm A})$ is atomic.

Proof. Let $\Psi_{\rho}\in{\mathsf{St}}_{1}({\rm A}{\rm B})$ be a purification of $\rho$, for some purifying system ${\rm B}$. Since $\mathscr{D}\mathscr{E}=_{\rho}\mathscr{I}_{\rm A}$ (the compression is lossless), we have $(\mathscr{D}\mathscr{E}\otimes\mathscr{I}_{{\rm B}})|\Psi_{\rho})=|\Psi_{\rho})$ (theorem 1). Now, by corollary 5 $(\mathscr{E}\otimes\mathscr{I}_{\rm B})|\Psi_{\rho})$ is faithful for ${\rm C}$ and by lemma 13 $(\mathscr{E}\otimes\mathscr{I}_{\rm B})|\Psi_{\rho})$ is pure. Using the Choi isomorphism with the faithful state $\Psi:=(\mathscr{E}\otimes\mathscr{I}_{\rm B})\Psi_{\rho}$ we then obtain that $\mathscr{D}$ is atomic. $\,\blacksquare$

IV.4 Teleportation and the link product

Proof. See Corollary 19 of Ref. purification. $\,\blacksquare$

Let us choose $\Psi^{({\rm A})}$ to be the faithful state in the definition of the Choi isomorphism. Then the sequential composition of transformation induces a composition of Choi states in following way:

For two transformations $\mathscr{C}\in{\mathsf{Transf}}({\rm A},{\rm B})$ and $\mathscr{D}\in{\mathsf{Transf}}({\rm B},{\rm C})$ the Choi state of $\mathscr{D}\mathscr{C}\in{\mathsf{Transf}}({\rm A},{\rm C})$ is given by the link product

Proof. See Corollary 22 of Ref. purification. $\,\blacksquare$

We conclude this paragraph with an important result that follows from the combination of the link product structure with the pure conditioning axiom:

The composition of two atomic transformations is atomic.

Proof. Let $\mathscr{C}\in{\mathsf{Transf}}({\rm A},{\rm B})$ and $\mathscr{D}\in{\mathsf{Transf}}({\rm B},{\rm C})$ be two atomic transformations. By the Choi isomorphism, the (unnormalized) states $R_{\mathscr{C}}$ and $R_{\mathscr{D}}$ are pure. Since the teleportation effect $E^{({\rm B})}$ in Eq. (9) is atomic (lemma 15), the pure conditioning axiom 5 implies the state $R_{\mathscr{D}\mathscr{C}}$ is pure. By the Choi isomorphism this means that $\mathscr{D}\mathscr{C}$ is atomic. $\,\blacksquare$

IV.5 No information without disturbance

We say that a test $\{\mathscr{C}_{i}\}_{i\in{\rm X}}\subset{\mathsf{Transf}}({\rm A})$ is non-disturbing upon input of $\rho$ if $\sum_{i\in{\rm X}}\mathscr{C}_{i}=_{\rho}\mathscr{I}_{\rm A}$. If $\rho$ is completely mixed, we simply say that the test is non-disturbing.

A consequence of the purification postulate is the following “no-information without disturbance” result:

A test $\{\mathscr{C}_{i}\}_{i\in{\rm X}}\subset{\mathsf{Transf}}({\rm A})$ is non-disturbing upon input of $\rho$ if and only if there is a set of probabilities $\{p_{i}\}_{i\in{\rm X}}$ such that $\mathscr{C}_{i}=_{\rho}p_{i}\mathscr{I}_{\rm A}$ for every $i\in{\rm X}$.