1. Introduction

Conventional wisdom has it that any evolution of a quantum system can be represented by a family of completely positive (CP) maps on its state space. Moreover, there seem to be good arguments that evolutions outside this class must be regarded as unphysical. But orthodoxy is not without dissent; several authors have argued for considering evolutions represented by maps that are not completely positive (NCP).

The debate has implications that have the potential to go deep. The possibility of incorporating NCP maps into our quantum dynamical framework may illuminate much regarding the nature of and relation between quantum entanglement and other types of quantum correlations (Devi et al. Reference Devi, Usha, Rajagopal and Sudha2011). If the use of NCP maps is illegitimate, however, such investigations must be dismissed without further ado.

In the following, we will argue for the proposition that NCP maps should be allowed—but we will add a caveat: one should not regard NCP dynamical maps as descriptions of the ‘not completely positive evolution’ of quantum systems. An ‘NCP map’, properly understood, is a linear extension, to a system’s entire state space, of a CP map that is only defined on a subset of this state space. In fact, as we will see, not much constrains the extension of a partially defined CP map. Depending on the characteristics of the state preparation, such extensions may be not completely positive, inconsistent, or even nonlinear.Footnote ¹

The article will proceed as follows. In section 2 we review the essential aspects of the theory of open quantum systems, and in section 3 we present the standard argument for complete positivity. In section 4 we consider the issues involved in the debate over NCP maps, and in section 5 we present our interpretation of the debate and what we believe to be its resolution.

2. Evolution of a Quantum System

Consider a quantum system S that is initially in a state , represented by a density operator . If the system is isolated, its evolution will be given by a one-parameter family of unitary operators , via

Suppose, now, that the system interacts with another system R, which may include some piece of experimental apparatus. We take R to include everything with which S interacts. Suppose that S is prepared in a state that is uncorrelated with the state of R (although it may be entangled with some other system, with which it does not interact), so that the initial state of the composite system is

The composite system will evolve unitarily:

where now is a family of operators operating on the Hilbert space of the composite system. It is easy to show (see, e.g., Nielsen and Chuang Reference Nielsen and Chuang2000, sec. 8.2.3) that, for each t, there will be a set of operators, which depend on the evolution operators and the initial state of R, such that

This is all in the Schrödinger picture, in which we represent a change of state by a change in the density operator used. We can also use the Heisenberg picture, which represents a state change via a transformation of the algebra of operators used to represent observables:

where

In addition to unitary evolution of an undisturbed system, we also associate state changes with measurements, via the collapse postulate. In the case of a von Neumann measurement, there is a complete set of projections onto the eigenspaces of the observable measured, and the state undergoes one of the state transitions given by

The probability that the state transition will be is . When a measurement has been performed, and we do not yet know the result, the state that represents our state of knowledge of the system is

Note that this, also, has the form (4).

One can also consider selective operations, that is, operations that take as input a state and yield a transformed state, not with certainty, but with some probability less than one, and fail otherwise. One such operation is the procedure of performing a measurement and keeping the result only if the outcome lies in a specified set (e.g., we could do a spin measurement and select only ‘+’ outcomes); the operation fails (does not count as preparing a state at all) if the measurement yields some other result. A selective operation is represented by a transformation of the state space that does not preserve norm. A selective operation , applied to state , produces a final state with probability , and no result otherwise.

Unitary evolution, evolution of a system interacting with an environment with which it is initially correlated, and measurement-induced collapse can all be represented in the form (4). The class of state transformations that can be represented in this form is precisely the class of completely positive transformations of the system’s state space, to be discussed in the next section.

3. Completely Positive Maps

We will want to consider not just transformations of a single system’s state space but also mappings from one state space to another. The operation of forming a reduced state by tracing out the degrees of freedom of a subsystem is one such mapping; as we will see below, assignment maps used in the theory of open systems are another.

We associate with any quantum system a -algebra whose self-adjoint elements represent the observables of the system. For any -algebra , let be its dual space, that is, the set of bounded linear functionals on . The state space of , , is the subset of consisting of positive linear functionals of unit norm.

For any linear mapping , there is a dual map , defined by

If is positive and unital, then maps states on to states on . Similarly, for any mapping of the state space of one algebra into the state space of another, there is a corresponding dual map on the algebras.

For any n, let be an n-state system that does not interact with our system S, although it may be entangled with S. Given a transformation of the state space of S, with associated transformation of S’s algebra, we can extend this transformation to one on the state space of the composite system , by stipulating that the transformation act trivially on observables of .

A mapping is n-positive if is positive, and completely positive if it is n-positive for all n. If S is a k-state system, a transformation of S’s state space is completely positive if it is k-positive.

It can be shown (see, e.g., Nielsen and Chuang Reference Nielsen and Chuang2000, sec. 8.2.4) that, for any completely positive map , there are operators such that

This is equivalent to a transformation of density operators representing the states

The standard argument that any physically realizable operation on the state of a system S must be completely positive goes as follows. We should be able to apply the operation to S regardless of its initial state, and the effect on the state of S will be the same whether or not S is entangled with a “witness” system . Since S does not interact with the witness, applying operation to S is equivalent to applying to the composite system . Thus, we require each mapping to be a positive mapping, and this is equivalent to the requirement that be completely positive.

To see what goes wrong if the transformation applied to S is positive but not completely positive, consider the simplest case, in which S is a qubit. Suppose that we could apply a transformation that left the expectation values of and unchanged, while flipping the sign of the expectation value of .

Suppose that S is initially entangled with another qubit, in, for example, the singlet state, so that

If we could apply the transformation (13) to S when it is initially in a singlet state with W, this would result in a state of satisfying,

This is disastrous. Suppose we do a Bell-state measurement. One of the possible outcomes is the state . The projection onto this state is

A state satisfying (15) would assign an expectation value of to this projection operator, rendering it impossible to interpret this expectation value as the probability of a Bell-state measurement resulting in .

Note that the setup envisaged in the argument is one in which it is presumed that we can prepare the system S in a state that is uncorrelated with the active part of its environment R. This setup includes the typical laboratory setup, in which system and apparatus are prepared independently in initial states; it also includes situations in which we prepare a system in an initial state and then put it into interaction with an environment, such as a heat bath, that has been prepared independently.

4. The Debate concerning Not Completely Positive Dynamical Maps

The early pioneering work of Sudarshan, Mathews, and Rau (Reference Sudarshan, Mathews and Rau1961), and Jordan and Sudarshan (Reference Jordan and Sudarshan1961), did not assume complete positivity but instead characterized the most general dynamical framework for quantum systems in terms of linear maps of density matrices. After the important work of, for instance, Choi (Reference Choi1972) and Kraus (Reference Kraus1983), however, it became increasingly generally accepted that complete positivity should be imposed as an additional requirement. Yet despite the reasonableness of the arguments for complete positivity, the imposition of this additional requirement was not universally accepted. Indeed, the issue of whether the more general or the more restricted framework should be employed remains controversial among physicists. At times, the debate has been quite passionate (see, e.g., Simmons and Park Reference Simmons and Park1981, Reference Simmons and Park1982; Raggio and Primas Reference Raggio and Primas1982). The issues involved in the debate were substantially clarified by an exchange between Pechukas and Alicki that appeared in a series of papers between 1994 and 1995. Pechukas and Alicki analyzed the dynamical map, , for a system into three separate components: an ‘assignment map’, a unitary on the combined state space, and a trace over the environment:

with representing the system of interest and the environment (the ‘reservoir’), respectively, and the assignment map, , given by

Since the unitary and the partial trace map are both CP, whether or not itself is CP is solely determined by the properties of , the assignment map. The map represents an assignment of ‘initial conditions’ to the combined system: it assigns a single state, , to each state . My use of inverted commas here reflects the fact that such a unique assignment cannot be made in general, since in general the state of the reservoir will be unknown. It will make sense to use such a map in some cases, however; for instance if there is a class of possible initial states that is such that, within this class, uniquely determines . Or it might be that, even though there are distinct possible initial states in that yield the same reduced state , the evolution of is (at least approximately) insensitive to which of these initial states is the actual initial conditions.

When is linear (in the sense that it preserves mixtures)Footnote ²

consistent,

and of product form, one can show that is of necessity CP as well. Pechukas (Reference Pechukas1994) inquired into what follows from the assumption that is linear, consistent, and positive. Pechukas showed that if is defined everywhere on the state space and is linear, consistent, and positive, it must be a product map: , with a fixed density operator on the state space of the reservoir (i.e., all ’s are assigned the same ).Footnote ³ This is undesirable as there are situations in which we would like to describe the open dynamics of systems that do not begin in a product state with their environment. For instance, consider a multipartite entangled state of some number of qubits representing the initial conditions of a quantum computer, with one of the qubits representing a ‘register’ and playing the role of S, and the rest playing the role of the reservoir R. If we are restricted to maps that are CP on the system’s entire state space then it seems we cannot describe the evolution of such a system.

Pechukas went on to show that when one allows correlated initial conditions, , interpreted as a dynamical map defined on the entire state space of S, may be NCP. In order to avoid the ensuing negative probabilities, one can define a ‘compatibility domain’ for this NCP map; that is, one stipulates that is defined only for the subset of states of S for which (or equivalently, ). He writes: “The operator is defined, via reduction from unitary dynamics, only on a subset of all possible ’s. may be extended—trivially, by linearity—to the set of all , but the motions so defined may not be physically realizable. … Forget complete positivity; , extended to all , may not even be positive” (Pechukas Reference Pechukas1994, 1061).

In his response to Pechukas, Alicki (Reference Alicki1995) conceded that the only initial conditions appropriate to an assignment map satisfying all three “natural” requirements—of linearity, consistency, and complete positivity—are product initial conditions. However, he rejected Pechukas’s suggestion that in order to describe the evolution of systems coupled to their environments one must forgo the requirement that be CP on S’s entire state space. Alicki calls this the “fundamental positivity condition.” Regarding Pechukas’s suggestion that one may use an NCP map with a restricted compatibility domain, Alicki writes: “Pechukas proposed to restrict ourselves to such initial density matrices for which . Unfortunately, it is impossible to specify such a domain of positivity for a general case, and moreover there exists no physical motivation in terms of operational prescription which would lead to [an NCP assignment of initial conditions]” (Alicki Reference Alicki1995, 3020).

It is not clear exactly what is meant by Alicki’s assertion that it is impossible to specify the domain of positivity of such a map in general, for does not the condition itself constitute a specification of this domain? Most plausibly, what Alicki intends is that determining the compatibility domain will be exceedingly difficult for the general case. We will return to this question in the next section as well as to the question of the physical motivation for utilizing NCP maps.

In any case, rather than abandoning the fundamental positivity condition, Alicki submits that in situations where the system and environment are initially correlated one should relax either consistency or linearity. Alicki attempts to motivate this by arguing that in certain situations the preparation process may induce an instantaneous perturbation of S. One may then define an inconsistent or nonlinear, but still completely positive, assignment map in which this perturbation is represented.

According to Pechukas (Reference Pechukas1995), however, there is an important sense in which one should not give up the consistency condition. Consider an inconsistent linear assignment map that takes the state space of S to a convex subset of the state space of . Via the partial trace it maps back to the state space of S, but since the map is not necessarily consistent, the traced out state, , will not in general be the same as ; that is,

Now each assignment of initial conditions, , will generate a trajectory in the system’s state space which we can regard as a sequence of CP transformations of the form:

At , however, the trajectory begins from , not ; , in fact, is a fixed point that lies off the trajectory. This may not be completely obvious, prima facie, for is it not the case, the skeptical reader might object, that we can describe the system as evolving from to via the assignment map and then via the unitary transformation to its final state? While this much may be true, it is important to remember that is supposed to represent an assignment of initial conditions to S. On this picture the evolution through time of is a proxy for the evolution of . When is consistent, and there is no issue; however when is inconsistent, , and we can no longer claim to be describing the evolution of through time but only the evolution of the distinct state . And while the evolution described by the dynamical map is completely positive, it has not been shown that the transformation must always be so.

What of Alicki’s suggestion to drop the linearity condition on the assignment map? It is unclear that this can be successfully physically motivated, for it is prima facie unclear just what it would mean to accept nonlinearity as a feature of reduced dynamics. Bluntly put, quantum mechanics is linear in its standard formulation. Commenting on the debate, Rodríguez-Rosario, Modi, and Aspuru-Guzik (Reference Rodríguez-Rosario, Modi and Aspuru-Guzik2010, 1) write: “giving up linearity is not desirable: it would disrupt quantum theory in a way that is not experimentally supported.”

5. Linearity, Consistency, and Complete Positivity

We saw in the last section that there are good reasons to be skeptical with respect to the legitimacy of violating any of the three natural conditions on assignment maps. We will now argue that there are nevertheless, in many situations, good, physically motivated, reasons to violate these conditions.

Let us begin with the CP requirement. Pace Alicki, one finds a clear physical motivation for violating complete positivity if one notes, as Shaji and Sudarshan (Reference Shaji and Sudarshan2005) do, that if the system S is initially entangled with R, then not all initial states of S are allowed—for instance, cannot be a pure state, since the marginal of an entangled state is always a mixed state. A map defined only on the allowed mixed states of S might lead to negative probabilities if it is extended to pure states of S. On the other hand, the map will be positive for all of the states of S that are possible given this experimental setup; this is the so-called compatibility domain of the map: the subset of states of S that are compatible with .

In light of this we believe it unfortunate that such maps have come to be referred to as NCP maps, for strictly speaking it is not the map but its linear extension to the entire state space of S that is NCP. The map is indeed CP within its compatibility domain. In fact this misuse of terminology is in our view at least partly responsible for the sometimes acrid tone of the debate. From the fact that the linear extension of a partially defined CP map is NCP, it does not follow that “reduced dynamics need not be completely positive.”Footnote ⁴ Alicki and others are right to object to this latter proposition, for given the arguments for complete positivity it is right to demand of a dynamical map that it be CP on the domain within which it is defined. On the other hand it is not appropriate to insist with Alicki that a dynamical map must be CP on the entire state space of the system of interest—come what may—for negative probabilities will only result from states that cannot be the initial state of the system. Thus we believe that ‘NCP maps’—or more appropriately: Partial-CP maps with NCP linear extensions—can and should be allowed within a quantum dynamical framework.

What of Alicki’s charge that the compatibility domain is impossible to “specify” in general? In fact, the determination of the compatibility domain is a well-posed problem (see Jordan, Shaji, and Sudarshan Reference Jordan, Shaji and Sudarshan2004); however, as Alicki suggests, there may be situations in which actually determining the compatibility domain will be computationally exceedingly difficult. But in other cases—when computing the compatibility domain is feasible—we see no reason why one should bar the researcher from using a Partial-CP map whose linear extension is NCP if it is useful for her to do so.Footnote ⁵ Indeed, given the clear physical motivation for it, this seems like the most sensible thing to do in these situations.

Not only are there partial CP maps whose linear extension is NCP; there are also partial-CP maps that admit of no linear extension, CP or otherwise, to the full state space. For instance, consider a correlated bipartite system with the following possible initial states:

The domain of definition of consists of the four states . Suppose we want to extend so that it is defined on all mixtures of these states and is linear. The totally mixed state of S can be written as an equally weighted mixture of and , and also as an equally weighted mixture of and :

If is defined on this state and is required to be a linear function, we must have

from which it follows that

which in turn entails that

so cannot be extended to a linear map on the entire state space of S unless it is a product map.

It would be misleading to say that assignment maps such as these violate linearity, for much the same reason as it would be misleading to say that Partial-CP maps with NCP linear extensions violate complete positivity. It is not that these maps are defined on a convex domain and are nonlinear on that domain; rather, there are mixtures of elements of the domain on which the map is undefined, and there exists no linear extension of the map that would include them. But since we cannot be said to have violated linearity, then pace Rodríguez-Rosario et al., in such situations we see no reason to bar the researcher from utilizing these ‘nonlinear’ maps, for properly understood, they are merely partially defined maps without linear extensions.

Pace Pechukas, there may even be situations in which it is appropriate to use an inconsistent assignment map. Unlike the previous cases, in this case the assignment map will be defined on the system’s entire state space. This will have the disadvantage, of course, that our description of the subsequent evolution will not be a description of the true evolution of the system, but in many situations one can imagine that the description will be “close enough,” that is, that

6. Conclusion

Bohr warned us long ago against extending our concepts, however fundamental, beyond their domain of applicability. The case we have just looked at is an illustration of this important point. The debate over the properties one should ascribe to the extension of a partially defined description is a debate over the properties one should ascribe to a phantom.

Whether or not we should use a map with no linear extension, or a map whose linear extension is NCP, or an inconsistent map, is not a decision that can be made a priori or that can be shown to follow from fundamental physical principles. The decision will depend on the particular situation and on the particular state preparation we are dealing with.