CCausal structure of the entanglementrenormalization ansatz
C´edric B´eny
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstraße 2, 30167Hannover, Germany
Abstract.
We show that the multiscale entanglement renormalization ansatz(MERA) can be reformulated in terms of a causality constraint on discrete quantumdynamics. This causal structure is that of de Sitter space with a flat spacelike boundary,where the volume of a spacetime region corresponds to the number of variationalparameters it contains. This result clarifies the nature of the ansatz, and suggests ageneralization to quantum field theory. It also constitutes an independent justificationof the connection between MERA and hyperbolic geometry which was proposed as aconcrete implementation of the AdS-CFT correspondence. a r X i v : . [ qu a n t - ph ] M a r ausal structure of the entanglement renormalization ansatz de Sitter space : a solution of Einstein’sequation describing an exponentially expanding universe. This result constitutes aconnection between MERA and hyperbolic geometry which appears complementaryto arguments previously proposed in connection with the AdS-CFT correspondence [12,13] ‡ .In addition, this observation naturally points to quantum field theory (QFT) onde Sitter space as a continuous generalization of MERA. We show that this proposal iscompatible with the cMERA introduced in Ref. [14] as a variational ansatz for quantumfield theories, but more constrained, and amenable to the tools developed in the contextof quantum cosmology. The connection with de Sitter space is interesting for tworeasons. Firstly, QFT on de Sitter background has been extensively studied giventhat it models the early inflationary universe where quantum effects are believed tobe important. Secondly, it is arguably the next simplest spacetime in which to do QFTafter Minkowski space due to its maximum number of symmetries.
1. A quick introduction on quantum channels
A conveniently general type of “process” in quantum mechanics is formalized by quantumchannels , or trace-preserving and completely positive maps. They are defined as themost general maps from density matrices to density matrices which are compatible with ‡ Hyperbolic space is the Euclidean form of both anti de Sitter and de Sitter spacetimes. ausal structure of the entanglement renormalization ansatz E , acting on the (mixed) state ρ A of a system A , can always be written as E ( ρ ) = tr B [ U ( ρ A ⊗ | (cid:105)(cid:104) | B ) U † ]where U is a unitary map describing the joint evolution (interaction) of A togetherwith an auxiliary system B for a fixed amount of time, the state | (cid:105) B is some arbitraryinitial state of B , and tr B is the partial trace over system B . Since a redefinition of | (cid:105) can be easily canceled by a redefinition of U , it is often convenient to simply write E ( ρ ) = tr B V ρ A V † where V : | ψ (cid:105) A (cid:55)→ U ( ψ A ⊗ | (cid:105) B ) is an isometry from system A tosystem AB . More generally, an isometry is any operator V satisfying V † V = (whichimplies that V V † is a projector), which is all we need to produce a quantum channel.Just as for unitary evolutions, quantum channels can be formulated in theHeisenberg picture, defined by the dualitytr( E ( ρ ) X ) = tr( ρ E † ( X ))where ρ is any state and X any observable. If E is defined in terms of the isometry V as above, then E † ( X A ) = V † ( X A ⊗ B ) V, in which we see that we obtain a channel by restricting observations of the state V ρV † of system AB to observables of system A only. This is how channels naturally appearin this work, and generally when evaluating a MERA on a local observable.Most properties of channels can be derived from the Stinespring dilation theoremwhich implies the existence of V for any channel as well as the uniqueness of V up toany partial isometry on the auxiliary system B , i.e., two isometries V and V (cid:48) producingthe same channel must satisfy ( A ⊗ W B ) V = V (cid:48) , where W † B W B , and hence also W B W † B ,are projectors.
2. MERA from causal order
One step of a circuit defining a MERA takes a quantum state defined on a coarse-grained lattice and isometrically maps it into the larger Hilbert space of a finer lattice.These isometric steps are also required to be implemented through local gates and afixed number of computational steps (see for instance the diagrams in Ref. [1]). Thisimplies a finite speed of information propagation in the circuit. Combined with theexponential nature of the successive coarse-graining operations, this means that theexpectation value of a local observable can be evaluated in a time logarithmic in thelattice size. In order to see this, note that an expectation value can be evaluated byevolving the observable “back in time” in the Heisenberg picture and then computing theexpectation value between the initial fiducial state and the resulting observable. Whentalking about locality it is enlightening to adopt the Heisenberg picture because thereexists an unambiguous concept of a local observable: one which acts nontrivially only ausal structure of the entanglement renormalization ansatz (cid:48) that is in the causal past of Σ [1, 3]. This map is the dualof a local quantum channel mapping states defined on Σ (cid:48) to states on Σ. In performingthis operation, the rest of the isometry can be completely ignored. Furthermore, thecoarse-graining is such that the Hilbert space dimension associated with the causal pastΣ (cid:48) is no larger than that of Σ, and hence the computational load can only decrease ateach step, and is independent of the lattice size.This suggests that, in defining the ansatz, we could replace the ad-hoc requirementthat each isometric step have a particular gate structure, and just require it to pull backlocal observables (on Σ) to local observables (on Σ (cid:48) ). This property is precisely one of causality as it is equivalent to stating that the degrees of freedom outside Σ (cid:48) cannotinfluence those inside Σ through one step of the dynamics [15]. But is such a propertysufficient to obtain an efficient local parameterization of the isometries?It was shown by Arrighi et al. [16] in the context of a unitary dynamical step U thatsuch causal constraints are sufficient and necessary for U to be implementable as a circuitof local operations, or “gates”, with some commutativity constraints between them. Inorder to see that this result is non-trivial, it helps considering the fact that it fails evenwhen the dynamical step is isometric rather than unitary. Indeed, an isometry canalways acausally produce a state correlated over arbitrary distances, something whichcannot be achieved with local gates and a fixed number of computational steps. Considerfor instance the isometry V from two distant systems A and B to the extended systems A = A A and B = B B defined by V ( | ψ (cid:105) A ⊗ | φ (cid:105) B ) = U A ⊗ U B ( | ψ (cid:105) A ⊗ | Ω (cid:105) A B ⊗ | φ (cid:105) B )where Ω is an entangled state and U A and U B are unitary operators acting respectivelyon systems A and B . It is enlightening to rewrite this as the circuit A B V A B = Ω (cid:79) (cid:79) A U A (cid:79) (cid:79) B U BA B where the time flows upward. It is not hard to see that this setup does not allow forany communication at all from A to B nor from B to A . But despite this absence ofcross communication, this isometry cannot be broken down as a product V = V ⊗ V where V maps A to A and V maps B to B . We note that V cannot be unitary inthis example because the output dimension must be larger than the input dimension.Counter examples can become much more intricate if the transformation is implementedby a generic channel rather than an isometry [15], or in fact even by a classical channel(i.e. one mapping diagonal density matrices to diagonal density matrices) [17].For a MERA, the dynamical steps cannot be unitary since the Hilbert spacedimension must increase. In fact, one can show that even for an infinite lattice, the ausal structure of the entanglement renormalization ansatz causal (with respect to this causal relation, or graph), if noinformation is transmitted between pairs which are not in the graph. Hence the graphcontains those pairs which are allowed to communicate.As we have seen, this concept of causality is not strong enough for our purpose.Therefore we will define a new concept, that of pure causality . As explained in theintroduction on quantum channels, a channel can always be written in terms of a unitarymap on a larger space, i.e., with a larger input system and a larger output system. Notethat if the channel is already isometric, i.e., of the form E ( ρ ) = V ρV † for an isometry V ,then we only need to enlarge the input space (so as to make it of the same dimension asthe output space). Since the concept of causality is sufficient to enforce the locality of aunitary map, we will simply say that a channel is purely causal (with respect to a givengraph) if it can be expressed in terms of a unitary map which is causal (with respectto that same graph), where the extra input or output systems are distributed amonginput and output vertices, and the input state on the extra inputs has no correlations.Hence the unitary is defined on the same input and output lattices as the channel, buteach vertices is associated with a possibly larger Hilbert space. Since the operation ofinitializing and tracing out the extra local spaces are local, it is straightforward to seethat a local implementation for the unitary map yields a local implementation for thecorresponding channel.For instance, consider a channel E from systems A and B to systems X and Y , andthe causal relation which only allows for communication from A to X and from B to Y , i.e., defined by the pairs ( A, X ) and (
B, Y ), then E is purely causal with respect tothat causality relation if there exists systems A (cid:48) , B (cid:48) , X (cid:48) and Y (cid:48) , and a unitary operator U from AA (cid:48) BB (cid:48) to XX (cid:48) Y Y (cid:48) causal with respect to the causality relation defined by thepairs ( AA (cid:48) , XX (cid:48) ) and ( BB (cid:48) , Y Y (cid:48) ), and states | (cid:105) A (cid:48) and | (cid:105) B (cid:48) , such that E ( ρ AB ) = tr X (cid:48) Y (cid:48) U ( ρ AB ⊗ | (cid:105)(cid:104) | A (cid:48) ⊗ | (cid:105)(cid:104) | B (cid:48) ) U † . We will now prove that one step of the binary MERA, defined as the set of allpossible isometries which can be implemented with the gate structure of one MERAstep (by picking the right parameters for the gates) is equal to the set of isometries ausal structure of the entanglement renormalization ansatz (cid:63) (cid:63) (cid:95) (cid:95) (cid:84) (cid:84) (cid:74) (cid:74) (cid:63) (cid:63) (cid:95) (cid:95) (cid:84) (cid:84) (cid:74) (cid:74) (cid:63) (cid:63) (cid:95) (cid:95) (cid:84) (cid:84) (cid:74) (cid:74) (cid:63) (cid:63) (cid:95) (cid:95) = · · · (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) · · · (1)where time flows upward. The left-hand side represents the set of isometries betweenthe two one-dimensional lattices whose vertices are represented as dots (with somearbitrarily fixed Hilbert space associated with each dot) which are purely causal withrespect to the causality relation specified by the graph (i.e. communications is onlyallowed between connected dots). The right-hand side of the equation represents theset of isometries which can be implemented by the circuit (or tensor network) obtainedby replacing each box by an isometry. This is precisely one step of the binary MERA[1, 3], except for the fact that in our case there is no constraint on the dimension of theHilbert spaces associated with the intermediate wires. However, the fact that each boxmust be an isometry effectively limits their input of dimension to that of their output.Below we also show that the ternary MERA [3] is equivalent to such a naturalcausality constraint. More generally, our prescription together with the constructivelocalizability result introduced in Ref. [16] allows for the construction of circuitswith equivalent properties on arbitrary lattices, including lattices embedded in higherdimensional spaces.Our approach works just as well if we allow each step to be implemented by aquantum channel rather than just an isometry, hence allowing in principle for thecharacterization of mixed states with long range correlations, such as critical thermalstates. If the state to be described is classical one may furthermore constrain the localquantum channels to be stochastic maps (i.e. to map diagonal matrices to diagonalmatrices).
3. Connection with de Sitter space
Let us call an “event” a lattice site at a given coarse-graining step. Each event isassociated with the Hilbert space dimension of the corresponding lattice site. Thecausal relations between successive coarse-graining generates a partial order betweenany two events, i.e.,
A < B if A is in the causal past of B . Furthermore the originalcausal relation can be recovered uniquely from the partial order by noting that it isdefined by the causal links : pairs of related events with no events “in between”, i.e.,( A, B ) forms a link if
A < B and there is no C such that A < C < B .Therefore, one can recover the MERA simply from the causal order between the setof events, together with the dimensions of their assigned Hilbert spaces. For instance,the usual binary MERA [1] is implied by the partial order shown in Figure 1. Such“causal sets” have been studied before as discrete models of spacetime [18, 19]. The ideafollows from the fact that the geometry of a manifold with Lorentzian signature can berecovered exactly from the partial order between events induced by the metric, together ausal structure of the entanglement renormalization ansatz Figure 1.
Partial ordered set corresponding to the binary MERA in one dimension,embedded in R such that the speed of light is equal to 1 everywhere. The outputlattice is the top row of dots and time flows upward. The black circles are events andthe lines segments are causal links. with the volume form. The causal order directly makes sense for a discrete spacetime.For the volume form, a natural postulate is that it corresponds to the counting of events.In our case, assuming for simplicity that all events are associated with the same Hilbertspace dimension, the number of events in a given spacetime region is proportional tothe number of variational parameters, thank to the local representability result.In order to see what metric a MERA on a d -dimensional lattice may correspondto, the easiest is to first parameterize its events by coordinates in which the speedof light is constant (and equal to 1), i.e., in a spacetime with metric ds = f ( t, x , . . . , x d )( − dt + (cid:80) i dx i ) . We suppose that each coarse-graining increases thelattice spacing by a factor a , and that sites at the ( k + 1)th coarse-graining step havea causal influence on the sites of the k th step within a radius ra k . Then a constantspeed of light (equal to 1) is achieved by embedding the k th coarse-graining at time t = − ra k / ( a − t = − r/ ( a − f ( t, x , . . . , x d ), we postulate that in coordinates whereour lattices are equally spaced in time, and renormalized, the volume form should beconstant. This makes precise the idea that the number of events in a given region ofspacetime should be proportional to the volume of that region. Such coordinates mustbe of the form τ = − α log[ t/t ] and ζ i = − βx i /t . The constraint is then satisfied bypicking f ( t ) = ( α/t ) . Also, choosing t = − r/ ( a −
1) puts the output boundary k = 0at τ = 0, and β = α normalizes the volume element. In the coordinates ( τ, ζ , . . . , ζ d ) ausal structure of the entanglement renormalization ansatz (cid:45) (cid:45) Ζ(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Τ Figure 2.
The shaded area is the causal past of two disconnected regions of the τ = 0spacelike surface in the static coordinates ( τ, ζ ). The dashed lines indicate the horizonat | ζ | = α = 1. the metric is then ds = (cid:18) ρ α − (cid:19) dτ − ρα dρ dτ + (cid:88) i dζ i . where ρ = (cid:80) i ζ i , and the volume form has component (cid:112) | det g | = 1 . In the conformallyflat coordinates this is ds = (cid:16) αt (cid:17) ( − dt + (cid:88) i dx i ) . This metric is that of de Sitter space. Another common coordinate system is given bythe time coordinate τ together with ξ i = − αx i /t , so that ds = − dτ + e τ/α (cid:88) i dξ i . Basic properties of the MERA can be deduced from considering past lightcones inthe coordinates ( τ, ζ , . . . , ζ d ) with constant volume form. The lightlike worldlines canbe deduced by applying the coordinate change to the Minkowski ones. They are all ofthe form ζ i ( τ ) = − αu i (1 − e τ/α ) + ζ i (0) e τ/α where u i a unit vector. We see that the causal past of any bounded region of size L converges in the past τ /α → −∞ to the ball of radius α (the cosmological horizon),which contains a fixed number of lattice sites at any given time. This is precisely thefeature of a MERA which allows for the efficient computation of local expectation values,since computing a reduced density matrix means simulating the quantum dynamicswithin the horizon which contains a bounded number of sites independent of the latticesize. Figure 2 illustrates this phenomenon. It shows the causal past associated with thecomputation of the correlation function between two local observables for d = 1. The ausal structure of the entanglement renormalization ansatz
4. Continuous MERA
This analysis yields a possible approach to building a continuous MERA, to serve asan ansatz for the state of quantum field theories. The field state to be representedlives on the spacelike boundary at τ = 0, and is the state produced by the evolution of some matter field inside de Sitter space. The boundary state is a function of the actualmatter Hamiltonian, which plays the same role as the gate parameters in the discretecase. Indeed, the fact that the field “lives” in de Sitter space simply means that itsdynamics respects the corresponding causality conditions, which is usually formalizedby saying that the field operators evaluated at spacelike-separated events must commute(or anticommute if they are fermions). Just as in the discrete case, one must also choosean initial state inside the horizon at a sufficiently early time. For instance, starting at τ = − α log(( L + 2 α ) / α ) allows for correlations length up to L . However, this initialvalue problem is not the only way of using the ansatz. For instance, one may alsochoose a stationary matter Hamiltonian, i.e., symmetric under translation in τ with ζ constant—which mirrors the strategy of choosing the gates to be the same at each stepin the discrete case—and then define the boundary state to be a stationary state. Thisis expected to yield long range correlations.Of course, the discrete MERA can also be used to attempt to approximate thestate of quantum fields provided the QFT is first discretized. However, working directlyin the continuum can present many advantages, such as the ability to deal with moresymmetries. A naive re-discretization would yield back a MERA, but one could conceiveother ways of producing a numerical algorithm such as, for instance, by applying acovariant cutoff which preserves the continuous symmetries such as in Ref. [20].A continuous MERA, or cMERA, was proposed recently by Haegeman et al. [14].We will see that our proposal constitutes in some sense a subset of theirs. In orderto compare them, we consider the simplest example of a quantum scalar field: withLagrangian density L = 12 (cid:112) | det g | [ − g µν φ ,µ φ ,ν − V ( φ )] . To be clear, it is to be understood that this is not the boundary theory thatthe cMERA is meant to help us solve, but instead a particular choice of the ansatz’svariational parameters (with some freedom left in the choice of V , which could dependon both space and time). Hence we may say that this Lagrangian defines the bulk theory.The ansatz itself is reflected in the fact that this field, once properly quantized, respectsde Sitter causality.The proposal in Ref. [14] does not explicitly require causality constraints. Instead,emphasis is put on imposing an ultraviolet cutoff. Given a local Hamiltonian, this wouldautomatically create approximate causality constraints due to the Lieb-Robinson bounds ausal structure of the entanglement renormalization ansatz τ . That is, if a cutoff wereto be applied, it would be measured relative to our metric. For instance, the binaryMERA corresponds to a cutoff ∆ ζ (cid:39) α/
3. This tells us that the coordinate systemused in Ref. [14] must be compared to our static coordinate system ( τ, ζ , . . . ), giventhat they use the same cutoff at each time step. The Lagrangian L can be quantized bystandard methods. If ˆ π ( ζ , . . . , ζ d ) are the canonical conjugates to the field operatorsˆ φ ( ζ , . . . , ζ d ), then we obtain the Hamiltonian H = K + L , where K = 12 (cid:90) d d ζ (cid:104) ˆ π + (cid:88) i ( ˆ φ ,i ) + V ( ˆ φ ) (cid:105) is the Hamiltonian for the field on flat spacetime and L = − α (cid:90) d d ζ (cid:88) i (cid:104) ˆ πζ i ˆ φ ,i + ζ i ˆ φ ,i ˆ π (cid:105) . generates the expansion of space. One can easily check that, with canonical commutationrelations between ˆ π and ˆ φ , this yields the right Heisenberg equations of motion. Thisstructure for H is compatible with that proposed by Haegeman et al .
5. Continuum limit of MERA
At this point, it is worth making some observations concerning the continuum limit of adiscrete MERA. One possibility (which corresponds to what is done in Ref. [21]) wouldimply keeping the same infinite network all along, but virtually rescaling the lengthand time coordinates x and t as if we were looking at the network from far away. Thehorizon length is sent to zero in this scheme which, therefore, cannot lead to a properde Sitter limit. This makes sense if this is to lead to a scale-invariant boundary state inthe limit. Indeed, the presence of a finite horizon introduces a characteristic lengthscalewhich would be of no use for describing a scale invariant state.Note that the horizon size should not be thought of as a correlation length, whichwould be better encoded in a time-dependent (i.e. scale-dependent) bulk Hamiltonian.Instead, the state inside the horizon would look like it has been produced by somedynamics on a flat spacetime, and may exhibit strong entanglement that scales as thevolume of a region.In order to obtain de Sitter space from a discrete MERA, one may imagine anetwork which is not defined exactly by de Sitter causality but instead by a causalstructure resembling our universe: with an initial period of exponential expansion (i.e.,de Sitter geometry) which transitions to a smaller (polynomial) rate of expansion untilit outputs the present state of the universe. If the polynomial period is long enough, thede Sitter horizon can be macroscopically large compared to the lattice spacing, so that ausal structure of the entanglement renormalization ansatz
6. AdS-CFT correspondence
Previous works have compared MERA to a different continuum theory, namely anti deSitter (AdS) space. The idea being that MERA may be a concrete embodiment of theAdS-CFT correspondence for its ability to represent critical scale-invariant states andthe similarity in which entanglement entropy is calculated using a minimal surface inthe bulk [12, 13]. This suggests an interpretation where the MERA circuit representsa field theory on AdS spacetime, whereas the state it describes is a CFT on a timelike boundary. The AdS metric is ds = (cid:16) αx (cid:17) ( − dt + (cid:88) i dx i )and the boundary considered corresponds to a fixed value of the coordinate x . If wemake the signature Euclidean by changing the sign of dt —a common trick used inQFT—then both the AdS metric and the de Sitter metric become the same hyperbolicmetric. Furthermore, the boundary that we have been considering matches the oneconsidered in the AdS-CFT correspondence.This shows that the Euclidean form of our field theory lives precisely on the samespacetime, and with the same boundary, as the Euclidean form of AdS spacetime. Itis interesting that the argument we have used to relate MERA to hyperbolic geometryappears to be distinct from previous arguments in the context of AdS-CFT [12], whichare based essentially on an analogy between the way entanglement entropy is calculatedin AdS-CFT and MERA.Going back to a Lorentzian metric signature, we see that the AdS boundary beingtimelike, it naturally contains a time direction in which the boundary theory is to evolve.In this sense the AdS time is the physical time, whereas the role of scale is played bythe space coordinate x . In our de Sitter picture, however, the Lorentzian time t is thescale parameter and the boundary is spacelike. This does not mean that the ansatzcannot be used to describe a time evolving state. Typically, conformal field theoriesare described in the Euclidean form, and hence would naturally live on our spacelikeboundary. This is in fact the standard approach used with the discrete MERA [21]. ausal structure of the entanglement renormalization ansatz
7. Causality and locality
We now sketch the proof of the statement represented by Equ. 1. Namely, that the setof isometries which are purely causal with respect to the causality relation representedon the left-hand side is equal to the set of isometries which can be formed by replacingeach box of the right hand side by an isometry (without constraint on the dimensionof the Hilbert spaces associated with the middle wires). First we consider a unitarymap U whose inputs are grouped into systems A and B , and outputs are grouped intosystems A (cid:48) and B (cid:48) , with the constraint that B cannot influence A (cid:48) . This means that inthe Heisenberg picture, any operator X acting on system A (cid:48) is mapped to an operator Y acting only on system A , i.e. U † ( X A (cid:48) ⊗ B (cid:48) ) U = Y A ⊗ B , which can be rewritten as( X A (cid:48) ⊗ B (cid:48) ) U = U ( Y A ⊗ B ) . (2)This implies that for any pure state | x (cid:105) of B , Y = ( ⊗ (cid:104) x | ) U † ( X ⊗ ) U ( ⊗ | x (cid:105) ) . The trick, inspired by Ref. [16], is to replace the local operator X by a swap between A (cid:48) and a new system C in Equation 2. If we initialize the system C to an arbitrarystate | y (cid:105) and trace it out after the action of U and the swap, the left hand side becomessimply U . This yields the expansion AB U A (cid:48) B (cid:48) = x (cid:47) (cid:47) A U y (cid:47) (cid:47) U † x (cid:47) (cid:47) (cid:47) (cid:47) B U (cid:47) (cid:47) A (cid:48) (cid:47) (cid:47) B (cid:48) (3)where the states | x (cid:105) and | y (cid:105) can be chosen arbitrarily. This is the only algebraic propertythat we will need. The vertical bar ending the fourth wire means that this systemis traced out: hence both sides of this equation represent channels rather than justoperators. The channel on the left-hand side is just ρ (cid:55)→ U ρU † : being unitary, it is aminimal Stinespring dilation of the channel on the right hand side. From the uniquenessof the Stinespring dilation of a channel, the right-hand side has also only one Krausoperator. To find its precise form, first note that the operator A V AA (cid:48) := x (cid:47) (cid:47) A U y (cid:47) (cid:47) U † x (cid:47) (cid:47) A (cid:47) (cid:47) A (cid:48) is an isometry as can be checked by tracing out A (cid:48) and B (cid:48) on both sides of Equation 3.Then the Stinespring dilation theorem tells us that there is an isometry (here just a ket) | ψ (cid:105) embedding C into the Hilbert space of the system A (cid:48) such that (cid:47) (cid:47) A (cid:47) (cid:47) B U ψ (cid:47) (cid:47) A (cid:48) A (cid:48) (cid:47) (cid:47) B (cid:48) = (cid:47) (cid:47) V (cid:47) (cid:47) B U (cid:47) (cid:47) A (cid:48) B (cid:48) A (cid:48) ausal structure of the entanglement renormalization ansatz N ( ρ ) := tr A (cid:48) U ρU † in Equ. 3 by ρ (cid:55)→ XρX † ,with X := ( B (cid:48) ⊗ (cid:104) ψ | A (cid:48) ) U . Furthermore, since the whole expression must be unitary,and hence trace-preserving, the operator X is isometric when restricted to its possibleinputs in the circuit, and can therefore be replaced by an isometry.This can be used to parameterize the classes of unitary maps causal with respectto a relation like that of Equ. 1 as follows: we start by grouping all the inputs (resp.outputs) which have the set of children (resp. parents) to obtain a new causal relationon the grouped systems. If the resulting graph is such that removing one particularinput A breaks it into two independent parts, then the remaining inputs and outputscan be grouped so as to satisfy the causality relation A This represents two causality constraints (i.e. missing links). By applying the instanceof Equ. 3 allowed by one of the constraint, and then again on the first instance of U inthe circuit for the other constraint, we obtain that (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) U (cid:79) (cid:79) (cid:79) (cid:79) = V (cid:79) (cid:79) U (cid:79) (cid:79) U (cid:79) (cid:79) (cid:79) (cid:79) for some isometry V . This scheme can be applied recursively on the remaining copiesof U , until the circuit respects all the causality constraints.If we lift the restriction that our computational step be unitary, and assume insteadthat it is an isometry (as is required for a MERA) or more generally a quantumchannel, then we demand that it can be represented by a unitary interaction with alocal environment, such that the unitary map respects the same causal relation. Wealso require that the environment’s initial state is separable. We can then apply ourprocedure to this unitary map to show that it has a local representation. In this way,one obtains the result express in Equ. 1. This method also works for the ternary MERA,showing that · · · · · · = · · · (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) · · · , with the same disclaimer about the fact that the dimensionality of intermediate wiresare not constrained, but limited by the fact that the boxes must represent isometries.As mentioned in the introduction, for more general causality relations, in particularas applied to higher-dimensional lattices, one must use the general prescriptionsintroduced in Ref. [16]. ausal structure of the entanglement renormalization ansatz Acknowledgment
The author is grateful to Tobias Osborne and Guifre Vidal for discussions aboutthis work. This work was supported by the cluster of excellence EXC 201 QuantumEngineering and Space-Time Research.
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