Everettian relative states in the Heisenberg picture

Everett's relative-state construction in quantum theory has never been satisfactorily expressed in the Heisenberg picture. What one might have expected to be a straightforward process was impeded by conceptual and technical problems that we solve here. The result is a construction which, unlike Everett's one in the Schrödinger picture, makes manifest the locality of Everettian multiplicity, its inherently approximative nature and its origin in certain kinds of entanglement and locally inaccessible information. (By Everettian, we are referring not only to Everett's own work, but also to versions of quantum theory that elaborate and refine his. The notion of relative states first appeared in Everett (Everett 1973 In The many worlds interpretation of quantum mechanics (eds BS DeWitt, N Graham)). We are proposing a formalism for relative states that is more detailed and more illuminating than Everett's.) Our construction also allows us to give a more precise definition of an Everett ‘universe’, under which it is fully quantum, not quasi-classical, and we compare the Everettian decomposition of a quantum state with the foliation of a space–time.


Introduction
The dynamical evolution of all quantum systems is unitary at all times except-according to most traditional 'interpretations' of quantum theory-during measurements. The arbitrariness, non-locality, vagueness, inelegance and anthropocentricity of this supposed exception has caused many misconceptions and confusions both in theoretical physics and in philosophy, despite the work of Everett [1], including his relativestate construction, which eliminated the exception. His primary technical innovation was to analyse quantum  measurement processes by treating not only the system being measured but also the measuring apparatus or instrument (or measurer, for short) jointly as a unitarily evolving quantum system. As a consequence of that unitarity, when a measurer has measured an observable of another system, both the system and the measurer then exist in multiple instances such that every possible measurement outcome is observed, but in autonomously evolving parts of reality, often called 'parallel universes'. 1 Everett formulated his construction in the Schrödinger picture, in which the parallel universes are described by autonomously evolving components of the universal state vector. But the construction has never been satisfactorily expressed in the Heisenberg picture. This is potentially problematic for Everettian quantum theory. If, for some reason, relative states could not be formulated in the Heisenberg picture this might indicate that Everettian quantum theory itself is not viable. Or if a successor theory to quantum theory, such as quantum gravity, were to lack a Schrödinger picture, it would not be possible to construct an Everettian interpretation for that theory, and all those traditional problems might reappear. For instance, Dirac [2] cast doubt on the universal validity of the Schrödinger picture, arguing that 'one can obtain a Hamiltonian which gives reasonable field equations in the Heisenberg picture, but which does not allow of solutions of the wave equation to represent physical states in the Schrödinger picture'. Furthermore, the theory of relative states itself, whether by that name or not, is central to all versions of the quantum theory of measurement, for the relative states are where the outcomes of measurements appear explicitly.

Relative states in the Schrödinger picture
In the Schrödinger picture, dynamical evolution is described by the motion of the universal state vector in Hilbert space, and the quantum-mechanical observables are fixed Hermitian operators on that space. So in the Schrödinger-picture relative-state construction, every possible outcome of the measurement of an observable appears, after a measurement, in a component of the universal state vector, and each such component consists of a product of two factors: an eigenstate of the measured observable and a state of the measurer in which there is a record of the measured eigenvalue. When these components do not subsequently undergo quantum interference, the linearity of quantum dynamics ensures that each of them evolves independently, i.e. exactly as it would if the others were not there.
For example, consider a model quantum measurement in which a measurer, initially in a Schrödinger-picture state |M ∈ H M , measures the computational variableq z of a qubit in the superposition 1/ √ 2(|1 + | − 1 ) ∈ H S of the two eigenstates ofq z . At the initial time t, the composite system H M ⊗ H S is in the state |Ψ (t) = 1/ √ 2|M (|1 + |−1 ), and the measurement, ending at time t + 1 and resulting in the state |Ψ (t + 1) , effects the transformation Here |M 1 and |M −1 (not necessarily distinct from or orthogonal to |M ) are orthonormal eigenstates of a suitable pointer observableM of the measurer, whose eigenvalues M 1 and M −1 are assigned the meanings 'it was a 1' and 'it was a − 1', respectively. The fact that |M 1 and |M −1 are orthogonal, and that the final state has no components |M 1 | − 1 or |M −1 | 1 , is a token of the idealized accuracy of the measurement. Provided there is no subsequent interference between |M 1 |1 and |M −1 | − 1 -for instance, if either the system or the measurer decoheres-the two instances of the combined system described by those two constituents will evolve independently after t + 1 (see, for instance, Wallace [3,4] and Zurek [5] The term 'relative state' can be used in three ways: (i) |M 1 is the relative state of the measurer (relative to the eigenvalue 1 of the observableq z of the qubit), and likewise for |M −1 ; (ii) | 1 is the relative state of the qubit relative to the eigenvalue M 1 ofM), and likewise for | − 1 ; and (iii) the product |M 1 |1 is the relative state of the combined system in which the outcome of the measurement was 1 (relative to the eigenvalues M 1 and 1 of the respective observables), and likewise for M −1 and − 1.
Since the measurer, initially in state |M , measures the computational variableq z , the relative states |M 1 |1 and |M −1 | −1 of (2.1) are eigenstates ofq z . These eigenstates have associated projectorsP where1 is the unit observable of H M ⊗ H S . Each projector in (2.2) has an argument, namely the observableq z , and a subscript, which is an eigenvalue ofq z that specifies which eigenstate the projector projects onto. 2 Note that the sum of projectors (2.2) equals the unit observable1, so (2.2) forms a so-called projection-valued measure (PVM). The relative states are eigenstates ofq z , so we express each relative state of the model measurement (2.1) as a (normalized) component of the state vector |Ψ (t + 1) in the direction specified by a projector of PVM (2.2), e.g. the relative state that contains the 'it was a 1' state where the denominator ensures that (2.3) is normalized. Since the elements of a PVM sum to unity, the state vector, after a quantum measurement, can invariably be expressed as a sum of relative states, each with a complex coefficient. It is a general feature of Everett's construction that the relative states after any perfect measurement of a computational variable can always be defined by a PVM in this manner. 3 The projectors (2.2) have to act on the full state vector in H M ⊗ H S to define the relative states, and similarly, in situations where the multiverse consists not just of a measurer and a measured observable but of other systems, too, the multiverse is again decomposed into relative states through projectors acting on the full state vector. Thus, seemingly, the relativestate construction is non-local: a measurement of one system by another appears to affect the full multiverse instead of only the subsystems that participated in the measurement. Indeed, some authors-for example, Vaidman [6]-have thereby been led to conclude that an Everett universe must be spatially unbounded. In that case, Everettian quantum theory would violate Einstein's criterion of locality that 'the real factual situation of the system S 2 is independent of what is done with the system S 1 , which is spatially separated from the former' (Einstein [7, p. 85]). Some Everettians-including Everett himself 4 -conclude that universes are local by the a priori argument that entanglement spreads through local interactions, but this does not address the point that the Schrödinger-picture relative states are not the products of local states and so apparently violate Einstein's criterion. In other words, Schrödinger-picture relative states are nonlocal because the Schrödinger picture is not a local formalism [8]. Such non-local formalisms are generically admitted by both local and non-local theories of the world. Quantum theory describes a local world because it also admits a local formalism: as we shall show in §4, in the Heisenberg picture, our relative-state construction is manifestly local and, therefore, 'universes' are always spatially bounded.
Formulating the relative-state construction within the Heisenberg picture is important because, for instance, information flow in quantum systems can only be analysed transparently in the Heisenberg picture [9,10]. In the Heisenberg picture, information has a well-defined location in physical systems [8,11]. Related ideas have been explored in the Heisenberg picturefor example, Wallace [4] has used the Heisenberg picture to give an account of decoherent histories, but he did not specify what the Heisenberg-picture relative states are in that picture. A construction similar to ours was presented by Hewitt-Horsman & Vedral [12], in their Section V. But, they do not draw the conclusion that the Heisenberg relative states are states of fully quantum systems, with their own algebras of observables and with the identical structure to those of the original system; nor do they explain that these systems are always local in space and are, therefore, not literally universes.

Quantum computation in the Heisenberg picture
In contrast with the formalism of the Schrödinger picture, that of the Heisenberg picture consists of evolving q-number observables and a fixed state vector (or in the case of subsystems, a fixed density operator). Because of this difference, certain important properties of quantum systems are transparent in the Heisenberg picture but opaque in the Schrödinger picture. In particular, in the Heisenberg picture, the time-varying entities, the Heisenberg observables, are local, and affect each other only locally, just as in classical physics, unlike the time-varying state vector of the Schrödinger picture, which is non-local. This implies that, in the Heisenberg picture, all physical information is assigned a location, even in entangled systems; see Deutsch & Hayden [9].
We shall construct the Heisenberg relative states for a quantum computational network; this technique, as used by Deutsch & Hayden [9], ensures that our results about relative states generalize to all quantum systems. This is not because quantum computational networks are the most general system-a network consists exclusively of qubits-but since such networks are universal in their capacity to simulate with arbitrary accuracy any other physical system, any result obtained about relative states in these networks must be a general feature of quantum theory. 5 Accordingly, consider a quantum computational network N consisting of n interacting qubits Q 1 , Q 2 , . . . , Q n . Following Gottesman [13] and Deutsch & Hayden [9], we represent each qubit at the initial time t as a triple of its observables, which we call the descriptors of the qubit They satisfy the algebraic relationŝ When an index appears twice in a product, once as a superscript and once as a subscript, as k does in (3.2), we implicitly sum it over all its possible values, in accordance with the Einstein summation convention. Here, the values summed over are {x, y, z}.
From (3.2), it follows that, for a network of n qubits, each qubit descriptor can be expressed as a Hermitian 2 n × 2 n matrix, so, for instance, the descriptors of a network consisting of a single qubit can be taken to have the initial matrix representation where1 k is a tensor product of k single-qubit identity operators, andσ = (σ x ,σ y ,σ z ). The initial matrix representation (3.4) of the descriptors will be altered by the subsequent evolution of the network, but since the descriptors of N evolve unitarily the algebra (3.2) is a constant of the motion.
It is convenient and conventional, and loses no generality, to assume that each gate in a network requires exactly 1 unit of time to act, so that the computational state of the network need only be specified at integer values of time. The effect of a general quantum gate G on Q a after one such period isq Here, the unitary U G implements the gate G and is a function of the time-dependent descriptors of N. Note that, in the Heisenberg picture, U G generally varies with time and, therefore, its matrix representation in our basis changes with time as well, but it is a fixed (i.e. characteristic of the gate) function of the time-dependent descriptors. For example, the not gate is a single-qubit gate which, when it acts on qubit Q a , toggles the value of its z-observable when it is sharp; the definition of the not gate in terms of the descriptors is The only gate whose matrix representation in the Heisenberg picture is time-independent is the 'unit wire' I, the effect of which is that it leaves the descriptors of N unchanged, and for a network of n qubits has matrix representation U I =1 n .
Since the unitary of a gate can be expressed as a characteristic function of the descriptors, a gate also has a characteristic effect on those descriptors, which can be specified algebraically. This provides a complementary way of defining a gate in the Heisenberg picture. For example, using (3.5) and (3.6), we find that, after the application of the not gate, the descriptors of Q a at time t + 1 have the following fixed expression in terms of its descriptors at time t: while all other qubits in the network are unaffected. Expressing gates in form (3.7) rather than (3.6) often makes their effect on a network more transparent. For example, it is manifest in (3.7) that the eigenvalues of the z-observable of Q a are toggled by the not gate, asq az . N is fully specified by its time-dependent descriptors and constant Heisenberg state |Ψ . We shall fix the Heisenberg state to be a standard constant by using the basis {|q 1z , . . . ,q nz ; t } of instantaneous eigenstates of the descriptors {q (t) az } so that |Ψ equals |1, . . . ,1;t = 0 . Then the expectation value of an arbitrary time-dependent observableÂ(t) of N is Since the Heisenberg 'state' |Ψ is a fixed constant, it contains no information about the physical state of N. All such information resides instead in the Heisenberg observables of N.
Whenever the expectation value of an observableÂ(t) of N has the property that Â (t) 2 =Â(t) 2 , the Heisenberg state is an eigenstate ofÂ(t) and the observable is then said to be sharp at time t with the value Â (t) (in Everettian terms, it has 'the same value in all universes' at time t), where that value is one of the eigenvalues ofÂ(t).
In the Heisenberg picture, two qubits Q a and Q b are entangled at time t if there exists a pair of descriptorsq [14]. An entangled quantum system is capable of storing locally inaccessible information-information that cannot be recovered through any measurement of that quantum system alone [9]. This fact is transparent in the Heisenberg picture, since it makes the location of all physical information explicit. For instance, consider two (t) bz = 1, so the qubits are in a Bell state. Then, at time t, Q a effects a gate R z (θ ), which rotates Q a by an angle of θ radians about its z-axis. That is to say, the descriptors of Q a evolve as follows:q Manifestly, the information about the angle θ is located in the x-and y-observables ofq (t+1) a , yet none of the expectation values of the qubit's observables reflect the fact that it contains this information since Thus, the angle θ is locally inaccessible-i.e. the results of measurements made on Q a alone, by measurers with no prior knowledge of θ, reveal nothing about θ, even statistically (if the whole experiment preparing Q a is repeated any number of times). As we shall show in §5, owing to entanglement, instances of a system in a relative state are also capable of storing locally inaccessible information.

The Heisenberg-picture relative-state construction
When the relative-state construction is formulated in the Schrödinger picture, the joint state vector of the observed system and the measurer are decomposed into relative-state vectors, each describing a 'universe'. 6 Although translating between the Schrödinger and Heisenberg pictures is usually a straightforward process, doing so for the relative-state construction brings with it the conceptual difficulty of decomposing the Heisenberg descriptors into relative descriptors: q-numbers which, following a measurement, represent individual instances of a combined quantum system, and which should correspond to the Schrödinger-picture relative-state vectors.
Our construction in what follows resolves that difficulty. Consider a measurement performed by a network M that consists of the two qubits Q M and Q S , where Q M represents an idealized measurer that is programmed to measure the z-observable of system Q S . The qubits in M have initial (t = 0) matrix representation (3.4) with n = 2 To reduce notational clutter, we shall abbreviate this aŝ The Heisenberg state |Ψ of the network equals |1, 1;0 , so that, at t = 0, the z-observables of both Q M and Q S are sharp with value 1. The relative-state construction explains why a unitarily evolving measurer appears to observe only one of the eigenvalues of an observable that is nonsharp: the other eigenvalues occurred in different 'universes'. To elaborate this, we apply a Hadamard gate H to Q S before the measurement begins. H is a single-qubit gate which rotates a qubit by π radians about the axis that bisects the angle between the x-and z-axis. That is to say, Thus, at t = 1, the Heisenberg state is not an eigenstate ofq (1) Sz , so this descriptor has no sharp value.
Next, between t = 1 and t = 2, Q M performs an idealized measurement of the z-observable of Q S . This is implemented by the two-qubit 'controlled-not' (or 'perfect measurement') gate cnot. The qubits that this gate acts on are known as the control and target qubits, which in this case are Q S and Q M , respectively. The effect of a cnot gate depends on the value of the control qubit: at a general time t, cnot has no effect on the target ifq (t) Sz of the control is sharp with value 1 and performs a not operation on the target ifq (t) Sz of the control is sharp with value − 1. It follows that the unitary U cnot that implements the cnot operation at a general time t is (4.4) The operatorsP 1 Sz ) are the Heisenberg equivalent of the projectors of (2.2), and their appearance in (4.4) makes explicit that cnot is a conditional not operation.
By using (3.5) and (4.4), we find that the effect of the cnot gate on the descriptors of the two qubits at a general time t is If both descriptorsq Mz is sharp with value 1, then a cnot operation still 'copies' the value of a superposed control ontoq (t) Mz by evolving the target into a mixed state, and the combined system is then entangled. Note in passing, by inspection of (4.5), that the cnot gate also effects a perfect measurement of the x-observable of Q M , storing the result in the x-observable of Q S ; in that interpretation of what the gate does, the roles of 'target' and 'control' are interchanged. In general, whether a system is being measured or performs a measurement depends not only on the interaction between them but also on how the information in each system is subsequently used, and, in general, it cannot be freely chosen which information stored in the pair of qubits to use. Decoherence would, for example, determine which of the qubits in (4.5) can act as a measurer [5,17].
Using (4.3) and (4.5), we find that the descriptors at t = 2, after the application of the cnot gate, areq The Schrödinger state vector of the qubits at t = 2 is Because of relations (3.2), the projectorsP 1 (q Sz ) commute withq (2) M . This ensures that the order of the factors in (4.9) makes no difference and thatq So, in each Everettian 'universe', quantum theory holds in full, but the matrix representation required to represent separately the sets of relative descriptors is of a smaller dimension than that of the full descriptors of Q M .
Because Q S also contains a copy ofq (2) Mz , there is a similar relative-state description of Q S relative to the values 1 and − 1 ofq (2) Mz . These instances of Q S are Q S,1 :q The order of the factors in (4.11) is again irrelevant since the observables of Q M and Q S commute. And the instances Q S,1 and Q S,−1 again obey the appropriate form of the Pauli algebra, but now with relative unit observables1 S,1 Mz ). Analogously to the Schrödinger-picture relative-state construction, the projectorsP 1 (q (2) Sz ) and P −1 (q (2) Sz ) of the measured observable form a PVM, and the relative descriptors (4.9) of Q M are defined as products ofq (2) M and the projectors of this PVM. Since quantum computational networks are universal, it must be a general feature of the Heisenberg relative-state construction that the relative descriptors of any measurer, after a perfect measurement, are the absolute descriptors projected via a PVM in the above manner. This resolves the conceptual difficulty of expressing the Heisenberg descriptors in terms of relative descriptors, and, importantly, our solution is also in agreement with Hewitt-Horsman & Vedral [12].
Borrowing a term from differential geometry, we say that entangled systems, such as Q M and Q S , can be foliated into relative states. In the Heisenberg picture, such a foliation is always spatially local 7 since only those systems that are entangled as a result of a measurement are foliated into relative states, whereas no other systems are affected. This is evident from the fact that the foliations of Q M and Q S are individually specified, the latter in (4.11) and the former in (4.9). By contrast, the Schrödinger picture, which is not a local description of quantum systems, does not allow systems to be foliated individually: a composite system can only be foliated jointly by a projector operating on the full Schrödinger state vector. So, in the Heisenberg picture, unentangled systems cannot be foliated into physically meaningful relative states. Foliating a quantum system changes the expectation value in relative states from those in the absolute network M. For example, at time t = 2, the conditional expectation value for an arbitrary descriptorq (2) Mi of Q M given that the z-observable of Q S was measured to be a 1 is . (4.12) Thus, so long as no further entangling or unentangling interactions take place, the expectation values of Q M,1 (and any other system in a relative state, relative to the result 1 of our measurement) are the same as they would be if the Heisenberg state had changed during the measurement, (4.13) We call |Ψ M,1 the relative Heisenberg state, relative to the result 1 of the measurement ofq (2) Sz . Note that the normalization factor P 1 (q (2) Sz ) is necessarily non-zero, asq (2) Sz is non-sharp. Since the expectation values of the relative descriptors differ from those of the observables of the absolute (multiversal) network M, it is possible for an observable, such asq (2) Mz , not to be sharp with respect to the absolute network (i.e. q (2) Mz = 0) but forq (2) (M,1)z to be sharp relative toq (2) Sz having been measured with value 1 (i.e. q (2) (M,1)z M,1 = 1). We can describe this informally asq (2) Mz having a sharp value in each universe (or in each relative state) but not in the multiverse at large.

Everettian 'universes'
The relative-state construction, in any picture, is a calculus for expressing a quantum system in terms of the behaviour of its constituent instances in situations where some or all of those instances evolve autonomously. In such situations, the system consists of parallel 'histories' or 'universes', each represented by a relative state. However, in generic situations, a foliation into such universes does not exist. Whether it does depends on the dynamical evolution of the entangled system.
To study the types of dynamical evolution that permit an entangled system to be foliated into parallel 'universes', let us once more consider the quantum computational network M at t = 2, when the network is foliated into Q S,1 with Q M,1 , and Q S,−1 with Q M,−1 . This foliation is possible because, immediately after the measurement, Q M contains a copy of the descriptorq (2) Sz . 8 We say that a qubit like Q M contains a copy of a descriptor of another qubit, such asq (2) Sz , if the product of an observable of Q M andq (2) Sz is sharp (e.g.q (2) Mzq (2) Sz is sharp with value 1) and all descriptors of Q M commute withq (2) Sz (this second condition ensures thatq (2) Sz is not one of the descriptors of Q M , in which case it would trivially contain a copy of that descriptor). If at a more general time t ≥ 2, Q M still contains a copy ofq (2) Sz , the qubit decomposes into the instances Likewise, by applying our definition to Q S , we find that, at t = 2, it contains a copy ofq (2) Mz , and if Q S retains this copy at a more general time t ≥ 2, then the qubit can be foliated into the relative states The gates that cause these relative states of M to evolve autonomously will, owing to the universality of quantum computational networks, also model when a general quantum system can be foliated into 'universes'. A qubit Q a evolves autonomously between t and t + 1 if its descriptorsq (t+1) a depend solely onq (t) a , and similarly a relative state evolves autonomously if its relative descriptors at time t + 1 depend only on those same relative descriptors at time t. Using this definition, which class of gates, acting between t = 2 and t = 3, would ensure that the relative states of M evolve autonomously between those times? Evidently, not all gates do since, for instance, a second application of cnot would erase the information about Q S contained in Q M , so this gate would destroy the relative-state structure. On the other hand, any gate that does not affectq (2) Sz andq (2) Mz will ensure that the relative states evolve autonomously as it will leave the information aboutq (2) Sz contained inq (2) Mz unaffected. As such, consider a gate F, which is implemented by the unitary U F , and which commutes withq (2) Sz andq (2) Mz so as to leave these descriptors unaltered. The most general expression of U F is U F (q (2) S ,q (2) M ) = α1 + βq (2) Mz + γq (2) Sz + δq Because of the form of F, both qubits still store the same information about the other qubit as before. For instance, Q S still stores a copy ofq (2) Mz after the application of the gate, so, by using (5.2), we find that the effect of F on that qubit's relative descriptors iŝ where the unitaries are defined as U F (q (2) S,1 ) def = ((α + β)1 S,1 + (γ + δ)q (2) (S,1)z ) and U F (q (2) S,−1 ) def = ((α − β)1 S,−1 + (γ − δ)q (2) (S,−1)z ). As can be readily deduced from (5.5), Q S,1 and Q S,−1 evolve independently between t = 2 and t = 3, as the instances perform the unconditional single-qubit rotations F and F , respectively. It is similarly possible to express U F in terms of the relative descriptors of Q M,1 and Q M,−1 from which it follows, in like manner, that those relative descriptors evolve autonomously under gate F too. Moreover, if the network enacts a single-qubit rotation of Q M or Q S or both (separately), then the relative states evolve independently as well, as such gates do not effect interactions between the qubits and so cannot unentangle them. Thus, the most general gate that ensures the relative states of M evolve autonomously between t = 2 and t = 3 are products of a gate of the form F followed by a single-qubit rotation of either or both qubits. This also makes evident the inherently approximate nature of the relative-state construction: in practice, descriptors will only ever be approximately aligned and the system's dynamics will only approximately preserve this alignment, so a foliation into Everettian universes is never exact.
As we mentioned in §3, the relative-state instances of both Q M and Q S can store locally inaccessible information. Consider, for instance, the case in which the gate R z (θ) acts on Q M between t = 2 and t = 3, after which the relative descriptors of Q M,1 and Q M,−1 depend on the angle θq