Weak measurements limit entanglement to area law (with possible log corrections)

Weak measurements limit entanglement to area law (with possible log corrections)

Amos Chan Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom    Rahul M. Nandkishore Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309
   Michael Pretko Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309
   Graeme Smith Department of Physics and Center for Theory of Quantum Matter,
University of Colorado, Boulder, CO 80309
JILA, University of Colorado/NIST, Boulder, CO 80309
July 26, 2019
Abstract

Starting from a state of low quantum entanglement, local unitary time evolution increases the entanglement of a quantum many-body system. In contrast, local projective measurements disentangle degrees of freedom and decrease entanglement. We study the interplay of these competing tendencies by considering time evolution combining both unitary and projective dynamics. We begin by providing a rigorous argument excluding the possibility of volume law Von Neumann entropy at any non-zero measurement rate in generic circuits. We argue for an extension of this analysis which also excludes powers intermediate between area and volume, leaving as the only possibility an area law, with possible logarithmic corrections. We explore these conclusions quantitatively by studying unitary-projective time evolution in various one-dimensional models with analytically tractable dynamics, starting with a toy model of Bell pair dynamics which captures the key features of the problem. Its steady state is dominated by small Bell pairs for any non-zero rate of measurement, leading to an area law for entanglement entropy. We also study entanglement dynamics and the approach to the asymptotic state, and find an ‘overshoot’ phenomenon whereby at intermediate times entanglement entropy exceeds its long time steady state value. Next we study Clifford evolution in qubit systems, as well as Floquet random circuits in the limit of large local Hilbert space dimension. All cases lead to area law saturation. We interpret recent numerical results in terms of a phase transition between area law phases with and without logarithmic corrections.

I Introduction

The notion of quantum entanglement is a unifying theme across numerous areas of modern physics, from the study of solid state systems to the study of black holes. In a condensed matter context, entanglement not only provides a window into the study of quantum ground states, but also is an important tool in characterizing the approach to thermal equilibrium (or the lack thereof). For example, the entanglement entropy of a ground state or a many-body localized state obeys an area law, , where is the linear size of the partition. In contrast, the entanglement entropy of a generic thermalizing state at a non-zero temperature is given by a volume law, .

While such static entanglement signatures are useful, there is also a great deal of information contained in the dynamics of entanglement. Consider preparing a system in a tensor product state, for example by performing a quantum quench. If the system exhibits many-body localization, then the growth of entanglement will be logarithmic in time, , in contrast with the ballistic ( linear) growth which occurs in thermalizing systems. Recently, studies have taken place on the growth of quantum entanglement under generic unitary time evolution, demonstrating in detail the linear growth of mean entanglement entropy, as well as determining the form of fluctuations around the mean.nahum1 () Subsequent analyses have studied both entanglement growth and spreading of local operators under random unitary time evolution, both with and without conservation laws.nahum2 (); khemani (); curt1 (); curt2 (); fracop (); banchi () Similar work has also been done in the context of Floquet and Hamiltonian time evolution. amos1 (); amos2 (); amos3 (); jonay (); nahum3 (); prosen1 (); prosen2 (); prosen3 (); sunderhauf ()

But while unitary dynamics generically leads to the growth of entanglement, there is another more drastic type of time evolution which can decrease the entanglement of a quantum system. Under certain conditions, such as interaction with a macroscopic classical object, a quantum mechanical system can rapidly evolve into an eigenstate of a specific operator, such that the resulting time evolution appears to be a non-unitary projection. Such a process is referred to as a projective measurement. When the system is projected into an eigenstate of a local operator, the corresponding local degree of freedom is disentangled from the rest of the system, resulting in a decrease in overall entanglement. In this way, projective measurements can remove some of the entanglement created by more generic unitary time evolution.

Since unitary time evolution and projective measurements have opposite effects on entanglement, it is natural to ask how a physical system behaves when both types of evolution play a prominent role. For example, a system could be subjected to a continuous series of measurements, as can be accomplished with superconducting qubits.qubit1 (); qubit2 (); qubit3 () As another potential physical realization, it has been proposed by M. Fisher that Posner molecules may play a role in quantum information processing in the brain.fisher (); posner (); halpern () As these molecules bind and unbind, they undergo joint unitary-projective dynamics, generating entanglement between different molecules.

In such a system with joint unitary-projective evolution, it is not obvious whether the effects of unitary evolution or projection are more important, or whether there is some transition as a function of measurement rate. However, recent numerical work on free fermion systems with continuous monitoring has indicated that an arbitrarily low rate of measurement is sufficient to keep the system in a state of low entanglement.fermion () A hydrodynamic explanation for this behavior was also advanced. An alternative model (which we will use) has been proposed by Li, Chen, and Fisher pcts (), and consists of time evolution which proceeds via alternating layers of unitary operators and non-unitary projectors, as depicted in Figure 1. If every site were measured during each projective step, then the system would be continually reset to a tensor product state and no entanglement would ever be built up. The more interesting scenario is when the local measurements are sparse. Specifically, consider projective evolution in which each site has probability to be measured at each time step. Equivalently, a randomly distributed fraction of the sites are measured at every step. As , with every site being constantly measured, it is clear that the effects of projection dominate those of the unitary time evolution. But for , there are far more unitary than projective operators in the circuit, so one might naively think that the unitary evolution would proceed largely unaffected by projection, driving the system towards a maximally entangled state with volume law entanglement entropy. However, an analytic understanding of the behavior at remains to be obtained.

Figure 1: We consider time evolution consisting of alternating layers of unitary and projective operators acting on a set of local degrees of freedom. The grey rectangles represent local unitary operators which generate entanglement between adjacent degrees of freedom, while the purple squares represent local projective measurements which remove entanglement from the system.

In this work, we show that the dominance of projection observed in free fermion systems fermion () is a general feature of quantum systems undergoing unitary-projective evolution. We begin in Section II, by providing a general argument that rigorously excludes the possibility of volume law entropy, for non-zero measurement rates and finite onsite Hilbert space dimension. We propose an extension of this argument which also rules out any power law intermediate between area and volume, leaving an area law entanglement as the only possibility. However, this general argument does allow for logarithmic corrections to area laws, and for a phase transition between a ‘high measurement’ phase with a true area law, and a low measurement phase characterized by an area law with a logarithmic correction. While the results in this section are all for Von Neumann entropy, Renyi entropies with are all upper bounded by Von Neumann entropy, and so these results also upper bound all higher Renyi entropies to area law behavior, with potential logarithmic corrections.

Next, we test the predictions of the general argument by analytic computations of entanglement entropy in three tractable models of unitary-projective evolution. These tractable models also yield additional insight into the behavior. We begin in Section III by introducing a toy model of Bell pair dynamics. The model is simple enough to allow an exact calculation of the dynamics of entanglement entropy. Nevertheless, the model captures the crucial piece of physics which determines the interplay between the two types of dynamics: local unitary evolution primarily creates short-range entanglement, while projective measurements can destroy entanglement on any length scale. As we will see, the result is that the system is dominated by small Bell pairs, with an exponentially decaying distribution on Bell pair size in the steady state. Since most of the entanglement is short-ranged, the steady state exhibits an area law for entanglement entropy, in contrast with the volume law expected from more generic unitary evolution. By continually removing long-range entanglement from the system, projective measurements are able to keep the entire system in a state of unusually low entanglement, for any rate of measurement. In addition to the steady state, we also investigate the full dynamics of entanglement by deriving time evolution equations for the Bell pair distribution. We find an overshoot phenomenon, whereby at intermediate times entanglement entropy exceeds its steady state value. This overshoot phenomenon appears to be consistent with the numerics reported in Ref.pcts, .

In Section IV, we consider Clifford evolution in a qubit system, in which the unitary layers of the dynamics have random operators drawn only from the set of Clifford gates. While this is not a universal set of gates, Clifford evolution allows for a convenient description of entanglement spreading in terms of an effective hydrodynamics. We show that, starting from a tensor product state, random Clifford evolution leads to a steady state with area law for both Von Neumann entanglement entropy and all Renyi entropies, consistent with our Bell pair model. Moreover, while the entanglement entropy can fluctuate with the position of the cut, we can upper bound the fluctuations such that the maximum entanglement entropy is an area law with a logarithmic correction. We propose a hydrodynamic description for entanglement growth in this universality class.

In Section V, we investigate two Floquet random circuits with large on-site Hilbert space dimension. In these circuits, the Renyi- entropies for can be mapped to emergent statistical mechanics problems, which amount to enumerating minimal-length domain wall diagrams. The longer the lengths of the domain walls in these diagrams are, the higher the averaged entanglement entropy of these circuits will be. An area-law saturation of higher Renyi entropies results from the fact that projective measurements can provide effectively “free” segments of domain walls, along which no amount of entanglement entropy is associated.

In Sec.VI we conclude, with a discussion also of two recent papers LiChenFisher (); SkinnerRuhmanNahum () which appeared on the arXiv simultaneously with ours.

Ii A General Argument for Area Law

In this section we present a very general argument excluding the possibility of volume law Von Neumann entropy in generic circuits made out of local unitaries and projectors, when the local Hilbert space dimension is finite and the measurement rate is non-zero, and strongly suggesting that the only possibility is an area law, with possible logarithmic corrections. Insofar as Von Neumann entropy upper bounds Renyi entropies of higher index (e.g. ), this argument also upper bounds higher Renyi entropies to area law, with possible logarithmic corrections. Throughout this section, ‘entropy’ refers to Von Neumann entropy, unless specified otherwise.

We consider a situation where, in alternating time steps, either nearest-neighbor unitaries or projective measurements are applied to each site with probability f. The basic observation is that for any region , the rate of entropy decrease on due to measurement will be roughly proportional to the total entropy of , while the rate of entropy increase is proportional to the size of the boundary . The only way for these rates to balance out is for to satisfy an area law.

ii.1 Excluding volume laws

In this subsection we exclude the possibility of volume law Von Neumann entropy. First, we show that any bipartite unitary acting on dimensions can increase the entanglement entropy between the two parties by no more than . To see this, imagine such a unitary with applied to a state , where Alice holds systems and Bob holds . Then, the increase in entanglement achieved by applying to is

(1)
(2)
(3)

Here we have used the notation of conditional entropy . We have also used the subadditivity of entropy, , and the Araki-Lieb inequality: (Ref.  AL70, ). This is actually a simplified derivation of a bound found in BHLS03, , which studies the general problem of entanglement generation via bipartite unitaries.

Next, assuming is composed of subsystems, we derive an upper bound on the entropy change caused by measuring a constant fraction of those subsystems. Letting be the collection of subsystems that are not measured, and be the classical outcomes of the measurement on the complement, , we see that the average entanglement-entropy change by measurement is given by

(4)
(5)

We can use these two observations to conclude that small-scale volume-law-like scaling must saturate to an area law for sufficiently large sizes in any spatial dimension . For the sake of contradiction, suppose our consists of contiguous spins , and that the entropy of the system scales as

(6)
(7)

where is a correction term. Our goal is to upper bound

(8)

To this end, we apply Eq. 4, but we must handle a slight subtlety. While Eq. 6 posits only the asymptotic behavior of entropies of contiguous sets of spins, the right-hand side of Eq. 4 involves entropies of non-contiguous spins. To see how this works, fix (the spins being measured) and label the contiguous systems between successive points in , . The typical size of will be and there will be such contiguous sets.

We give two arguments. In the first argument, we assume a stronger requirement on our correction term, demanding . Given this, fixing a particular , letting denote a large number of spins to the left of and a large number to the right, strong subadditivity of entropy implies that

(9)

Applying the assumed scaling, this becomes

(10)
(11)
(12)

As a result, we find that

(13)
(14)
(15)
(16)

Substitution into Eq.4 then yields Eq.8 as desired.

In the second argument, we allow a more relaxed scaling, where we do not require that the correction term , but only require that the deviations around area law are independent and random (with mean zero) for different sets and over different realizations of . In this case we find

(17)
(18)
(19)
(20)
(21)

In both cases, we find

(22)

so that the entropy change due to measurement satisfies

(23)
(24)
(25)

The change in entanglement entropy caused by one round of local unitaries satisfies

(26)

where is the number of unitaries that straddle the boundary between and , which is equal to the length of the boundary of . Combining this with the change in the entanglement entropy due to measurement gives us

(27)
(28)

Note that for sufficiently large , this becomes negative since scales more slowly than . As a result, a stable entropy of form Eq. (6) cannot be achieved. In particular, if we hope for volume law scaling of the form , we find positive entropy growth rate can only be sustained for

(29)

which requires

(30)

Alternatively, volume law entanglement must break down around a saturation entropy

(31)

We can also show that a strong volume-law behavior is impossible in with a simpler argument. Given a set , the local-unitary steps will tend to increase the entanglement entropy of , while the projective measurements will tend to decrease it. Our goal is to identify the size at which these competing forces balance out. To understand the rate of entropy reduction due to measurements, we make some assumptions about the structure of the state on . In particular, we consider a situation where the entanglement entropy of is nearly maximal (the state is nearly maximally mixed) and see how large an is consistent with this. In a sense, we are asking how big can be and be consisent with a very strong notion of volume law. Suppose has spins. Then, after one step of measurements, a fraction spins will be measured, and the resulting entropy will be , which is an entropy change of . When a layer of local unitaries is applied, only two of the unitaries will straddle the edges of (one at each end). The unitary step will therefore increase the entanglement entropy of by . So, after one unitary step and one measurement step, the change in entanglement entropy is

(32)
(33)

We therefore find for . This suggests that for , unitary-projective dynamics will increase the entanglement entropy of , but that it will saturate around . This simple argument holds only for near-maximally mixed states on .

ii.2 Excluding power laws intermediate between area and volume

Now we argue that power laws intermediate between area and volume are impossible also in the thermodynamic limit. For the sake of contradiction, suppose our consists of spins , and that the entropy of a system scales as

(34)
(35)

for some . The change in the entanglement entropy due to measurement is

(36)
(37)
(38)
(39)

We have assumed here that the entropy of non-contiguous sets of spins is characterized by a no larger than that for contiguous sets of spins - an assumption that was proven to be safe for volume law scaling in the previous subsection.

This gives us

(40)
(41)
(42)
(43)

Note that for sufficiently large , this becomes negative since scales more slowly than . As a result, a stable entropy of form Eq. (34) cannot be achieved. In particular, a power law can only be sustained for

(44)

ii.3 Logarithmic corrections and phase transitions

Now we argue that our general argument does allow for logarithmic corrections to area laws, and also for phase transitions between area law phases with and without logarithmic corrections. The argument is simple: suppose the entropy scales as

(45)

An argument analogous to that presented in the previous subsection gives

(46)
(47)

For the upper bound on is positive, such that entropy growth of the form Eq.45 can be sustained indefinitely, leading to a correction to area law behavior. For , , and the scaling Eq.45 cannot be sustained, allowing only for a true area law. This also suggests that there should generically be a phase transition at a critical between a high measurement phase with a true area law for entanglement, and a low measurement phase featuring an area law with a logarithmic correction.

Thus, we have shown that any power law scaling with can only be sustained upto a maximum entropy , but an area law with a logarithmic correction can be sustained indefinitely, for small enough measurement rates. This leaves open the possibility also of a system where starts off as volume law at small , then gradually transitions through a range of intermediate power laws with increasing , asymptoting eventually to an area law (with a potential logarithmic correction) in the thermodynamic limit.

Iii A Toy Model for Bell Pair Dynamics

We now test the general understanding from the previous section by explicit computation of entanglement entropy in various analytically tractable models for unitary-projective dynamics. In order to build intuition, we begin with a toy model which captures the important physical features. To this end, we focus on a particularly simple form of entanglement. We consider states which can be fully described in terms of Bell pairs, maximally entangled two qubit states, such as a spin singlet. In other words, we study a system of qubits in which each qubit is either maximally entangled with another qubit or is completely disentangled from the system (see Figure 2). For such a system, we can easily obtain the entanglement entropy by counting the number of Bell pairs which are cut by a given partition. While such Bell pair configurations are a restricted class of states, this model provides important physical intuition for entanglement dynamics under unitary-projective evolution, and we will see later that the behavior of more generic systems is well-described by the results of this toy model. All Renyi entropies are equal for this model.

To build unitary-projective dynamics into the toy model, we must consider the effects of both types of operators on Bell pairs. We first consider applying a layer of local unitary operators, as in Figure 1. Such a layer of operators will result in entanglement between neighboring qubits which were previously disentangled from the rest of the system. Consistent with the restrictions of our toy model, we take this entanglement to be maximal. In other words, local unitary operators can generate Bell pairs between previously unentangled neighboring qubits. When a unitary operator acts on a qubit which was already in a Bell pair, it can move one end of the Bell pair to an adjacent site, which may cause the Bell pair to grow or shrink in size. Bell pairs can move through one another. Starting from a state with mostly small Bell pairs ( a state of low entanglement), generic local unitary time evolution will cause Bell pairs to increase in size, leading to the growth of entanglement for generic spatial partitions.

Figure 2: We study a qubit system in which each qubit either forms a maximally entangled Bell pair with another qubit (represented by an oval) or is completely disentangled from the rest of the system.

While local unitary operations tend to increase entanglement, via creating small Bell pairs which subsequently grow in size, the projective portion of the time evolution has a radically different effect. Performing a projective measurement on a qubit has the effect of disentangling it from the rest of the system. If that qubit happened to be in a Bell pair with another qubit, that Bell pair is destroyed by the measurement. Notably, this mechanism for Bell pair destruction is equally effective for Bell pairs of any size, since a local measurement on either qubit is sufficient to destroy a Bell pair, regardless of the distance to the other qubit. The model thereby captures the expected interplay between unitary and projective dynamics: creation of short-range entanglement (and its subsequent growth) by unitary operators, coupled with removal of entanglement at all length scales by projective measurements.

Combining these two types of physical processes, we can now very easily write down a set of equations governing the time evolution of the distribution on the spatial size of Bell pairs. At each time step, the local unitaries cause a given Bell pair to either grow (with probability ), shrink (with probability ), or remain the same size (with probability ). We also take a fraction of the Bell pairs to be destroyed, be reset to zero size. Away from , the time evolution equation for takes the form:

(48)

where the time step (defined by one layer of unitaries and one layer of projectors) is taken to be . The probabilities and will not depend on the rate of external measurement, and therefore have no dependence. We also neglect any nonlinearities of this equation, such that the probabilities can be taken to be independent of . The probabilities can, however, generically depend on the size . We can now take the continuum limit of the above time evolution equation to obtain:

(49)

where is the difference in probabilities for growing and shrinking of a Bell pair. The first term on the right, which is the only term present at ( pure unitary evolution), would lead to uni-directional propagation of waves in the distribution, with the direction of propagation depending on the sign of . However, the second term on the right, arising from the projective measurements, causes the distribution to decay, preventing entanglement from propagating very far from . To have a steady state solution, with , the distribution must satisfy:

(50)

If we take to be approximately independent of , then we immediately obtain:

(51)

where . Note that we have not needed to make use of the details of what happens to the distribution at , the details of Bell pair creation at small scales, which only serves to determine the behavior near the origin.

We see that the steady state solution of the joint unitary-projective time evolution is dominated by small Bell pairs, such that the system is mostly short-range entangled. This makes intuitive sense, in that long-range entanglement is being constantly removed from the system by projective measurements, while the entanglement resulting from local unitary evolution is only being created on short scales. We can directly calculate the typical entanglement entropy of the system. For example, consider a one-dimensional system, which we partition into two half-lines. A qubit at a distance from the cut will contribute one bit ( ) to the entanglement entropy if it is a member of a Bell pair of size at least , and if that Bell pair extends in the direction of the cut. Summing contributions from all qubits on one side of the partition, we obtain:

(52)

The entanglement entropy is a constant, independent of the system size , since the exponentially decaying distribution yields a convergent integral. Since the entanglement entropy is constant, we conclude that the asymptotic state of the unitary-projective evolution obeys an area law, as opposed to the behavior of a volume law state. In higher dimensions, we will have the same sort of exponential convergence of the entropy integrals, except with a factor of area arising from integrating over the entire partition. In this way, our Bell pair toy model gives rise to an area law for entanglement entropy in any dimension.

Figure 3: A schematic representation of at various times for an initial tensor product state. The evolution is characterized by a ballistically propagating but decaying peak, as well as the development of an exponential distribution near .

In addition to the steady state, it is also easy to obtain the full time evolution of the Bell pair distribution. The generic solution to Equation 49 takes the form:

(53)

where and , while the function is an arbitrary function of . (This form holds only away from , near which the behavior will be modified in a complicated way in order to preserve the overall normalization of . Note also that we should demand that grow no faster than exponentially, such that remains normalizable.) The resulting time evolution takes the form of waves which propagate at velocity , while decaying via the exponential factor . For example, let us consider an initial tensor product state, such that all the weight of is concentrated at and the entanglement entropy is zero. As time evolves, the peak at propagates to the right at speed , just as in the case of pure unitary evolution. For short times (), the entanglement entropy will therefore grow linearly, , with an effective entanglement velocity given by:

(54)

We see that the initial entanglement velocity of this unitary-projective system is smaller than that of a pure unitary system by a factor of . As time evolves, however, the slowdown of entanglement growth becomes more severe, as the weight in the propagating peak decays exponentially and is transferred back to the origin, as depicted in Figure 3. (In a more generic dynamical model, the peak would begin to broaden as time evolves, though this is unimportant for present purposes.) The contribution to the entanglement entropy from the decaying ballistic peak behaves as , which has a maximum value around , after which the entanglement entropy decreases to its area law saturation value, set by the exponential distribution near the origin. The schematic behavior of the entanglement entropy as a function of time is depicted in Figure 4. The ‘overshoot’ phenomenon predicted by our model of Bell pair dynamics is consistent with the numerics reported in pcts, .

Figure 4: For a nonzero projection probability , the entanglement entropy of an initial tensor product state asymptotes to an area law (after an initial overshoot), in contrast with the unitary () system, in which entanglement continues to grow towards a volume law state. For , the initial growth is linear, but with a slope less than that of the pure unitary system.

Iv Clifford Evolution in One Dimension

We now test our understanding with a more generic model of a one-dimensional system evolving via an almost-random set of unitary gates. Specifically, we consider the unitary operators of the time evolution to be randomly drawn from the set of Clifford gates, a form of dynamics referred to as Clifford evolution. While this is not a universal set of gates, Clifford evolution in one dimension provides certain convenient technical simplifications while still capturing most of the qualitative features of truly random unitary evolution. It also has the virtue that all Renyi entropies behave the same way.

Clifford evolution relies on a simple action of Clifford gates on states labeled in terms of stabilizers, operators which leave the state invariant, such that . For a one-dimensional system with sites, such stabilizers will be necessary to fully label the state. For example, if is a tensor product state, then all will be local operators, acting non-identically only on a single site. For an entangled state, the stabilizer operators extend over multiple sites, with the size of stabilizers increasing as the state becomes more entangled. The value of Clifford evolution lies in its simple action on Pauli operators, mapping each Pauli to a product of other Pauli operators. If we begin from a tensor product state labeled by a Pauli stabilizer on each site, then the resulting time evolution is simply described in terms of Pauli strings. For a given stabilizer state, the entanglement across a cut between and is determined as follows. The stabilizer group for a state on can be generated by three subgroups: and consist of stabiilzer elements with support on only or , respectively, while contains stabilzers acting on both and and accounts for correlations between the systems. A set of generators for is called minimal if it contains the minimal number of generators acting on both and , and the total number of such generators is (the size of the minimal generating set of the nonlocal group ). The entanglement between and is then (see Ref.Fattal, for details).

Figure 5: The set of stabilizers intersecting the measured site is supported on a region of size . The individual stabilizers are typically size , leading to a typical mismatch in support of 2 sites between adjacent stabilizers.

iv.1 Simple Crude Argument

We consider the following dynamics. Our system begins in tensor product states, with alternating nearest-neighbour random Clifford gates applied (see Fig. 1) after each such layer, a fraction of the qubits are measured in their local basis. In the absence of measurement, we get the standard random Clifford circuit model, as studied in Ref. Hlow, ; terhal-leung-divincenzo, ; nahum1, . In this case, after steps, the minimal stabilizer generators have weight, the circuit approaches a random Clifford gate, and after O(L) steps we get a volume law: .

We now argue that this picture changes dramatically in the presence of local measurements. We begin with a quick and dirty argument that captures the situation to factors of . A more detailed analysis in the next section tightens this up at the expense of a little more complication. First, let’s start with an arbitrary stabilizer state. After a layer of measurements, a fraction of the stabilizers will be reset to length 1, while the fraction will be updated in a way that potentially increases the length of their supports. After steps of Clifford unitaries followed by measurements, a fraction of stabilizers will not have been touched, while will have been reset to length 1 at some point in the past k steps. This means that a -fraction of the stabilizers have length no more than , since they’ve only been growing for steps. If we let , the untouched fraction is . Since there are only stabilizers, this will be with high probability. In other words, with high probability all of the stabilizers will have had their lengths reset to in these steps, and they will thus all have lengths no more than . Furthermore, because these stabilizers will be roughly evenly distributed spatially, this allows an entropy scaling of , since typically no more than generators will straddle the boundary of . Note that this is independent of the size of , so no volume law is possible. It is, however, consistent with volume-law-like scaling for regions of size no more than . Note also that while the argument was formulated using , it would have worked just as well for for any non-zero , consistent with our general expectation that the most entanglement that is possible in the thermodynamic limit is an area law with a correction.

iv.2 Stabilizer Size Distribution

We now provide a more detailed argument that eliminates the dependence found above. Suppose a random Clifford circuit has been run for some time , so that the typical weight of a minimal stabilizer generator is . We call its stabilizer group . The stabilizer generators of will have support that is spatially localized to a region of width , and their supports will be distributed uniformly in space. What is the effect of a single measurement at site ? Stabilizers whose support doesn’t include site will remain generators of the stabilizer group of the new state. There are approximately of these. Around half of the remaining stabilizers will have a or at location and can also remain in the stabilizer. However, there are around stabilizer generators that have either an or at site and these must be updated to reflect the measurement. The stabilizer group of the new state will be the group generated by the subgroup of that commutes with , which we call , and . To find the generators of this group, we must focus on the generators with an or at site . These can be ordered from left to right, , with the leftmost generating having an towards its right end, and the rightmost having an towards its left end. For these generators, the typical mismatch in support with an adjacent generator is sites, since they are size and spread out over a region of width . We can now find a set of generators for our updated stabilizer group. In particular, letting

(55)
(56)
(57)
(58)
(59)

and adding these to the unchanged generators of the original stabilizer gives our new stabilizer generators. Note that due to the mismatch in support, of the new generators will have their support increased to . Thus, our new stabilizer generators have a similar weight distribution to the original generators, with one important difference: For a given generator, the measurement has a large chance of either increasing its length by 2 or keeping it the same, and a small chance of reducing the weight to 1. This has the same basic structure as the model of Bell pair dynamics - stabilizers (previously Bell pairs) either grow by a constant finite amount, or collapse to unit length. We thus expect similar entanglement dynamics.

In particular, we can incorporate the measurement process into a master equation describing the evolution of the probability of a given stabilizer generator having weight . Without any measurement, we have

(60)

where is the probability of a generator having weight at time , and is the probability that a stabilizer grows in length by one unit, and is the probability that it shrinks by one unit. Taken to the continuum limit, this gives an evolution equation of

(61)

where is the net increase in stabilizer supports. Adding in the measurement process on a fraction sites leads to a similar master equation of the form

(62)

where is a new growth rate which takes into account the (constant) additional growth of generators during the measurement process.

The steady-state solution of Eq. (63) satisfies

(63)

so that

(64)

Thus, we see that at late times typical stabilizer weights will be . The entanglement across a cut scales as the minimal number of stabilizers straddling that cut which, due to the generators’ localized supports of scales like . Since scales like (and in particular is independent of ), for length scales less than we find a volume-law growth, which saturates to area-law for length scales greater than .

Of course, since there is a distribution of stabilizer lengths, the entanglement entropy will fluctuate somewhat from point to point. We can however bound these fluctuations. Recall that there are stabilizers total. The largest size stabilizers present in the system can then be estimated from , which yields a maximum stabilizer size . This then bounds the maximum entanglement entropy to .

iv.3 Quasiparticle Picture and Hydrodynamics

This reasoning can also be understood using a representation of Clifford evolution in terms of a set of fictitious “particles,” as developed in Reference nahum1, , which also allows a slightly more refined analysis. We briefly recap the central idea behind Clifford evolution and its particle representation, referring the reader to Reference nahum1, for further details.

Figure 6: Clifford evolution can be represented in terms of a set of fictitious particles (blue circles) representing the endpoints of stabilizers. The system contains as many particles as sites, and no more than two particles can occupy any site. A tensor product state corresponds to a state with uniform density, while a maximally entangled state corresponds to all particles shifted to one side. Figure adapted from Reference nahum1, .

For many purposes, it is sufficient to only keep track of the endpoints of the stabilizer, which encode information about the length of the Pauli strings. To this level of detail, we can represent a state by a set of fictitious “particles” representing the stabilizer endpoints, as depicted in Figure 6, where blue circles represent right endpoints and white circles represent left endpoints. It can be shown that, due to a gauge freedom in choosing the stabilizers labeling the state, the total number of endpoints (left plus right) on a site can be chosen to be exactly two. It is then sufficient to only keep track of the right endpoints (blue circles), while the left endpoints can be regarded simply as “holes.” Within this representation, a tensor product state corresponds to a uniform density of particles, since each site is the left and right endpoint of a local (on-site) stabilizer. Entanglement is then represented as a deviation from this uniform density. Indeed, as discussed in Reference nahum1, , the entanglement entropy associated with a partition at location is given by:

(65)

In other words, the entanglement entropy is given by the excess particle number on one side of the partition. As the system evolves under random unitary time evolution, the tendency is for stabilizers to grow, which amounts to particles ( right endpoints of stabilizers) to drift to the right. As shown in Reference nahum1, , unitary evolution causes the particles to undergo biased diffusion, such that the density evolves as:

(66)

where and are constants, and is a random variable representing noise. Eventually, pure unitary evolution would take the system to a maximally entangled state, in which all particles are as far to the right as possible, as seen in Figure 6.

However, this flow of particles to the right is interrupted in the presence of projective dynamics. The effect of a local measurement is to disentangle one spin from the rest of the system, such that the resulting state is stabilized by a local operator on that site. This indicates that the measured site now has one left and one right endpoint, such that . In other words, local measurement resets the local particle density to that of the unentangled system. We can easily account for this effect in the diffusion equation by adding a decay term on deviations from the mean density:

(67)

where and . This takes the form of a non-linear diffusion equation where the diffusing density can decay with constant probability. It is exponentially unlikely that significant density will diffuse very far to the right of the cut.

We now propose a hydrodynamic description for entanglement under unitary-projective dynamics in this universality class. We begin with the observation from Reference nahum1, that under random unitary time evolution, the entropy on a given site evolves according to

(68)

where is a noise term. We now wish to also account for the effects of projective measurements. When a site is measured, it becomes disentangled with the rest of the system, such that there is no difference in entanglement between the partitions to the immediate left and right of that site. In other words, measurement sets the value of to zero on that site. The projective portion of the evolution then acts as a decay term on the evolution of , which does not have a natural local form in terms of . We therefore take a derivative of Equation 68 and add an appropriate decay term to yield:

(69)

where we have defined . We see that obeys the same diffusion equation as the particle density in the case of Clifford evolution (see Equation 67), which also served as the derivative of entropy. This then provides yet another argument for saturation to area law entanglement under Clifford dynamics. We conjecture that this hydrodynamic equation should be valid not just for Clifford-projective dynamics, but for all models of unitary-projective dynamics in this universality class.

V Floquet Random Circuits

We now move from Clifford circuits to a fully random circuit model, similar to Ref. amos1, ; amos2, . Specifically, we consider two 1-dimensional -site Floquet (time-periodic) unitary circuits generated by Haar-distributed random unitaries, where the quantum states at each site span a -dimensional Hilbert space. We will be taking a large limit, in which the general argument advanced in section II does not bound the entropy (since the co-efficient of the area law bound derived in the section is singular in the limit ). Nonetheless, we find saturation to area law Renyi entropies for .

Our set-up for the unitary-projective time evolution is as follows. The unitary dynamics of the system is modelled by a Floquet circuit. A non-unitary measurement layer is applied after every layers of the unitary circuit. Each time we apply a measurement layer, we randomly draw fraction of sites to perform projective measurements. We make use of developments amos1 (); amos2 () allowing exact calculation of the ensemble average of exponential of the Renyi- entropies for with unitary-projective time evolution in the large- limit using diagrammatic techniques. Remarkably, the diagrammatic approach provides a mapping from ensemble averages of observables to emergent classical statistical mechanics problems. Our result states that in the limit at any finite (albeit arbitrarily large) , for a non-vanishing fraction and a finite period , there is an area law saturation for entanglement entropy. In the limit at large , we provide a heuristic argument for an area law saturation at least for sufficiently large .

v.1 Floquet Haar Random Unitary Circuit

First, we review the model and the results of half-system bipartite entanglement spreading without projective measurements. The model is defined by a Floquet operator , where and . Each is a unitary matrix acting on sites and . In Reference amos1, , is written as a -perturbative series in the large- limit, and is mapped to an emergent statistical mechanical problem, which, for , amounts to generating all minimal-length domain wall (DW) diagrams in Fig. 7. The solution gives

(70)

which suggests a linear growth of before the saturation time, and a volume-law saturation after the saturation time.

Figure 7: Left: An example of minimal-length DW diagrams for at large at a time smaller than the saturation time , which can be generated as followsamos1 (): Draw a DW beginning at the position of bipartition from the top to the bottom; Turn the DW to the left or the right for a distance of lattice spacing when the DW encounters a 2-gate; Repeat until the DW reaches the bottom of the diagram. In general, there are of such diagrams, and each diagram is translated into an algebraic term as in the -perturbative expansion of , where is the number of horizontal wall segments along the DW. This implies that . Right: One of the two minimal-length DW diagrams at a time larger than . These two diagrams have DW directed solely to the left or the right, and each contributes a factor of . This implies that .

For the sake of simplicity, we will assume for the following derivations, but the proofs can be straightforwardly extended to general Renyi index . Now, we investigate the behaviour of entanglement entropy growth with unitary-projective time evolution at a time smaller than the saturation time. It is instructive to consider the effect of performing a single projection operator onto color state at site and time . In App. A, we prove that the relevant diagrams are the minimal-length DW diagrams whose DW passes through the space-time point and (Fig. 8 left), because this segment of DW does not give rise to a factor of (i.e. this DW segment is “free”), which makes this diagram more dominant in the -perturbative series. The algebraic factor associated to such diagrams is . Generally, for a finite period , fraction , and time smaller than (to be specified below), the leading diagrams of contain DW that passes through the location of a projection measurement every unitary layers, and the order is (Fig. 8 right), where is the floor division. An expression for the multiplicity of such leading order DW diagrams can be written in terms of a transfer matrix (acting on a Hilbert space labelled by the DW position) as described in App. B.

Figure 8: Left: An example of DW diagrams for at large and with a single measurement at and time (represented by the symbols in purple). These diagrams are a subset of the diagrams in Fig. 7 containing those with DW passing through point and . Each of such diagrams is algebraically translated to , where is the number of projection operators along the DW. Right: A simplified illustration of the set of minimal-length DW diagrams before saturation time with multiple layers of projective measurements. A leading order diagram (in red) belongs to the set of “path-integral” diagrams that scatter off the projective measurements: the DW begins from the top centre of the diagram, walks along one of the minimal-distance paths (in black), and pass through one of the measurements (purple dots) on every measurement layer until it reaches the bottom edge of the diagram.

At a time sufficiently large, the minimal-length DW diagrams are the ones with DW ending on the side of the diagrams. Without measurements, there are two leading diagrams where the DW-s are solely directed to the left or the right (Fig 7 right). With measurements, minimal-length DW diagrams are the ones with DW passing through a certain number of projection operators to reach the side of the diagrams. We prove in App. A the following equation.

(71)

where , ceil is the ceiling function, and where and are independent of and dependent of . On the LHS, we have implicitly averaged over the positions of projection operators in a given measurement layer.

The intuition behind the result is can be explained using Fig. 9. At large , the leading “staircase” diagrams are the ones with the DW reaching the side of the diagrams in the shortest distance, utilizing the “free” segments of walls provided by the projective measurements (purple segments in Fig. 9). The area-law saturation originates from the fact that the DW in these leading staircase diagrams pass through free DW segments. Take as an example, there exists a realization of projection measurements such that the DW can reach the side using 2 measurement layers each of which provides “free” DW segments (Fig. 9 right). So the orders of such diagrams are at least . In general, it takes number of “stairs” (and hence periods) to reach either side of the diagram. This explains Eq. V.1.

The combinatoric factor arising from requiring a staircase configuration of projection measurement locations implies that the coefficient is suppressed in as , but independent in (to be discussed further in Sec. V.3). Taking on both side of Eq. V.1, and taking the limit for fixed but arbitrarily large , we have for large ,

(72)

which means that the Renyi- entropy for saturates according to the area law for finite and non-vanishing in the limit at any finite but arbitrarily large (in other words, for that does not grow in and for that is not suppressed in ). This is one of the first exact calculations that demonstrate the area law saturation of entanglement entropy in fully random quantum circuit under unitary-projective dynamics. Remarkably, an area law follows even though the general argument of Sec. II does not apply, because of the infinite local Hilbert space dimension.

Figure 9: Left: The simplified diagrammatic representation of a realization of projective measurements before ensemble-averaging, where the locations of projective measurements (purple) are scattered randomly along the measurement layers. Right: The minimal-length diagrams are staircase DW diagrams, which requires specific realizations of projection measurements. The DW connects the side of the diagram after number of periods. Therefore, the order is at least . The right diagram is drawn for .

Two questions naturally follow. Firstly, how does the result extend to finite ? While the problem is no longer analytically tractable at finite q, the general arguments of Sec.II will apply as long as is finite - and should guarantee an area law. Secondly, what happens when we take the limit of large , but after we take the thermodynamic limit ?

To address the second question, we provide a heuristic argument to show that there are exponential many staircases diagrams in Sec. V.3, and it is plausible for the area-law saturation to survive at least for large enough , even when we take the limit of large after we take the thermodynamic limit.

v.2 Floquet Random Phase Circuit

Figure 10: Left: The Floquet random phase circuit where the first layer contains Haar random unitary 1-gates, and the second layer contains diagonal 2-gates with random phases. Right: An example of the leading order minimal-length DW diagrams of in the regime when . Note that the diagonal 2-gates are not illustrated in this diagram. An analogous description concerning the enumeration and the order of DW diagrams given in Fig. 7 applies here.

The domain wall picture extends beyond the Floquet Haar random unitary circuit. Here we describe the analogous result (proved in App. A.2) in the Floquet random phase circuit first introduced in Ref. amos2, (Fig. 10). This model is similarly defined by a Floquet operator where is a tensor product of independent Haar random unitaries, and couples neighbouring sites using a diagonal 2-gates with entries

(73)

Each is a Gaussianly-distributed random phase with mean zero and variance . This circuit has two parameters: (i) , which allows us to obtain analytical results at , and (ii) , which allows us to tune how strongly nearest-neighbouring sites couple.

If the unit time is defined after the application of and , then in the strong-coupling limit , both Eq. 70 and V.1 apply, and hence we have again Eq. 72. This statement is proved in App. A.2.

v.3 Heuristics: Staircase Diagrams

Figure 11: Top: The illustration of -, - and - staircase diagrams for from the left to the right. The orders of the diagrams are where is the number of staircase. The positions of where the staircases begin give rise to the multiplicity of the -staircase diagram, which is proportional to . Bottom: Other variants of staircase diagrams which one may expect to contribute to the area law saturation of , but for each of these diagrams, one can identify another “non-Gaussian” diagramamos1 () that will lead to cancellations. These diagrams is not explicitly considered in Eq. 74.

We provide a heuristic argument for an exponential number of staircase diagrams, so that it is plausible for the area-law saturation of for to at least survive for large enough , when the limit is taken before the limit of large . For the sake of simplicity, we consider an alternative set-up where there is a probability for each site in a measurement layer to be projectively measured. Again, we explain the derivation explicitly for , but the argument holds for general . Lastly, the argument below is applicable to both the models in Sec. V.1 and  V.2.

The origin of area law saturation can be related to DW diagrams that fulfil two criteria: (i) These diagrams have DW starting from the top centre of the diagram and ending on the side of the diagrams (otherwise the order of the diagrams decrease in time for ). (ii) These diagrams have DW passing through at least projective measurements (otherwise the diagram would have an order that scales in ). The diagrams that satisfy these criteria are the staircase diagrams (e.g. Fig. 9 right). We call a staircase diagram with number of staircases a -staircase diagrams. The -perturbative series of can be written in terms of the contribution of the -staircase diagrams (Fig. 11 top) as

(74)
(75)

where is the multiplicity of the -staircase diagram. As an example, the multiplicity of a 2-staircase diagram is of order because while the first staircase (counting from the top) always begin in the top center of the digram, the second staircase can begin anywhere between the center and the far right of the diagram (Fig. 11 top middle). The dots denote all other contributions to . For general , the above equation becomes

(76)

For this contribution to not be suppressed in , we must have the -dependent exponent to be greater than zero,

(77)

This contribution implies it is plausible that the area-law saturation of survive at least for large enough . However, this argument is not rigorous because we have not been able to systematically look at all sub-leading terms in the -perturbative series of . In particular, there can in principle be diagrams that are algebraically translated in negative terms which lead to cancellation with other positive terms (these are called “non-Gaussian” diagrams in Ref. amos1, ). To summarise, we have found an exponential numbers of staircase diagrams and argued that it is plausible for an area-law saturation of to survive at least for large enough , even when the limit is taken before the limit of large .

Vi Conclusions

In this work, we have investigated the entanglement dynamics of a system featuring a combination of unitary and projective time evolution, which have competing effects on quantum entanglement. We have argued that the effects of projection generically dominate over those of unitary evolution, with even a small amount of projection in the dynamics sufficient to keep the system in a state of low entanglement. Specifically, we have rigorously excluded the possibility of volume law Von Neumann entropy in the thermodynamic limit at any non-zero measurement rate, if the onsite Hilbert space dimension is finite. We have also provided arguments strongly suggesting that power laws intermediate between area law and volume law are also impossible. Our general arguments do however allow for logarithmic corrections to area laws, and for a phase transition between a high measurement phase characterized by an area law for Von Neumann entropy, and a low measurement phase characterized by an area law with a logarithmic correction. We have tested our understanding with explicit calculations of entanglement entropy in three progressively less fine tuned but analytically tractable models of unitary projective dynamics, beginning with a toy model for Bell pair dynamics. This toy model also allows us to determine the dynamics of entanglement, which appear to exhibit an overshoot phenomenon consistent with the numerical results reported in pcts (). Saturation to area law is also observed for Clifford evolution in one dimensional qubit systems, and Floquet random circuits in one dimension, in the limit of large onsite Hilbert space dimension. We find area laws for all Renyi entropies in the Clifford circuit case, and area laws for Renyi- entropies with for the Floquet random circuit, even though for the latter case, the system evades the general argument.

We thus conclude that any non-zero probability of measurement produces saturation to area law entanglement, potentially with logarithmic corrections. This implies - counter-intuitively - that ‘measurement’ of a quantum system can inhibit thermalization through local unitary time evolution, and help keep the system in a low entanglement state. This seems to be rather good news both for Fisher’s model of quantum cognition, and for efforts to store and manipulate quantum information more generally.

We now comment on two related parallel works — by Li, Chen, and Fisher LiChenFisher (), and by by Skinner, Ruhman, and Nahum SkinnerRuhmanNahum (); — which appeared on the arXiv simultaneously with our own. In Li, Chen, and Fisher, numerical data on unitary-projective evolution in systems of size up to was reported, and a phase transition was observed between a high measurement phase in which entanglement entropy saturated to an area law, and a low measurement phase in which it did not saturate, and continued to grow faster than an area law up to the largest system sizes accessible. This low measurement phase was conjectured to be volume law. However, the maximum entropies and system sizes accessed in the simulation were still below the values where our general argument from Sec.II predicts saturation of volume law scaling. Meanwhile, Skinner, Ruhman and Nahum reported numerics on system sizes up to , and observed an analogous transition. Again, the maximum entropies achieved were below the level where our general argument predicts saturation. Skinner, Ruhman and Nahum also advanced an analytic argument for the low measurement phase being volume law. However (i) this analytic argument applied not to Von Neumann entropy (the quantity of most physical interest), but rather to the Hartley entropy (Renyi-zero), which is not bounded by our general argument in Sec.II. In addition (ii) they did not directly calculate the Hartley entropy, but calculated instead a quantity called ‘minimal cut’ which provides an upper bound on the Hartley entropy. It is this upper bound that was analytically shown to have volume law scaling in the low measurement phase. Whether the bound is tight for unitary-projective evolution is unknown.

We are satisfied as to the existence of a phase transition between a high measurement and a low measurement phase. Indeed, it was the numerics of References LiChenFisher, and SkinnerRuhmanNahum, that led us to reexamine our argument and discover a potential logarithmic correction to area laws. However, in light of our general argument Sec.II, we believe the low measurement phase cannot be volume law. Rather we conjecture that the transition that is being observed is between a high measurement phase characterized by area law Von Neumann entropy, and a low measurement phase characterized by a Von Neumann entropy that starts off scaling as volume at small system sizes, but then transitions through a range of intermediate power laws with increasing sytsem size, asymptoting at the largest system sizes (beyond reach of the numerics) to an area law with a logarithmic correction. Whether the Hartley entropy (Renyi-zero) obeys a volume law in the low measurement phase, saturating the upper bound derived in Reference SkinnerRuhmanNahum, , is a question on which we remain agnostic.

Acknowledgments

We acknowledge inspiration for this project from a talk given by Matthew Fisher at the PCTS conference on “Statistical Mechanics out of Equilibrium” in May 2018, which was funded partially by the Foundational Questions Institute (fqxi.org; grant no. FQXi-RFP-1617) through their fund at the Silicon Valley Community Foundation. We also acknowledge useful discussions with Yang-Zhi Chou, Mario Collura, Andrea De Luca, and Adam Nahum. This work was supported by NSF Grant 1734006 (GS,MP), by EPSRC Grant No. EP/N01930X/1 (AC), by a Simons Investigator Award to Leo Radzihovsky (MP), by the Foundational Questions Institute (fqxi.org; grant no. FQXi-RFP-1617) through their fund at the Silicon Valley Community Foundation (MP,RN), and by the Alfred P. Sloan foundation through a Sloan Research Fellowship (RN). Much of this work was carried out during the Boulder Summer School for Condensed Matter Physics, which is supported by NSF grant DMR-13001648.

Appendix A Evaluation of in Section V

a.1 Floquet Haar Random Unitary Circuit

Figure 12: (L) top: The block representation of a leading order DW diagram for for the Floquet model specified in Sec. V.1 in the large- limit in early time . The far left column labels the rows of blocks. The configuration of the top row of blocks is fixed by the boundary condition of the trace structure of the observable (see Ref. amos1, for details). We compute by evaluating the partition function of the following ensemble. Each realization in the ensemble has state variables living in each block and local Boltzmann weights between two vertically-neighbouring blocks. Only non-trivial Boltzmann weights (the ones smaller than unity) are drawn in red. The locations of projective measurements are drawn in purple. (L) bottom: A dictionary between the simplified diagrammatic representations in the main text and the ones in the appendices.
(C) top left: The diagrammatic representation of the local Boltzmann weight (in red) between two vertically-neighbouring blocks and . (C) top right: A block (not at the edges of a diagram) has four local Boltzmann weight with its neighbouring blocks. (C) bottom: This figure illustrates the derivation of Table 1. Ref. amos1, is required to understand the figures. (C) bottom left: The LHS is the top view of two vertically-neighbouring block with time axis vertical and space axis horizontal. The RHS is the side view of the block in terms of loops on site with time axis vertical and space axis pointing out of the page. The purple dots represent the projective operators. We have , because as long as the top and bottom blocks have local contractions, this region of the diagram has saturated its highest possible order. (C) bottom right: The loop on site with purple dots is of length at least 3 due to the non-local contraction. This implies that the existence of this loop reduces the overall order of the diagram by from the highest possible order, since the 3-loop could have been split into smaller loops. Consequently, we have .
(R): Examples of leading order configurations of different types of row of walls. (a) An even row of walls without projection operators along the rows. (b) An odd row of walls (which cannot have any projection operators). (c) Even rows of walls with projection operators. (c) bottom: If multiple projection operators are located next to each other in space, there can be a leading order configuration in which a DW between domains and horizontally extends over a number of sites.

In this section, we prove Eq. V.1 for for the model described in Sec. V.1. The case of is discussed in App. A.3. We take for granted what is proven in Sec. IV C, App. B 2 and E in Ref. amos1, . We begin by reviewing the emergent statistical mechanical problem described in Sec. IV.C. in Ref. amos1, without projective measurements.

can be expressed as a –perturbative series in the large- limit, which can in turn be mapped to a partition function of the following ensemble at zero temperature. (This mapping is exact only in the large- limit.) The ensemble consists of configurations (diagrammatically represented in Fig. 12 L) whose state variables live in blocks and take values from the set (Fig. 14 in Ref. amos1, ). Between every pair of vertically-neighbouring blocks and , there is a local Boltzmann weight (explicitly derived and written in Table 1 in Ref. amos1, ) which is diagrammatically represented as the horizontal boundary (with a width of lattice spacing) between the two blocks (Fig. 12 C top). Without projective measurements, the weight is unity if and only if , so it is useful to distinguish the boundaries between domains of blocks of different values, which we call domain walls (DW). The associated global Boltzmann weight of diagram (which we also refer to as the order of ) is the product of all local Boltzmann weights of walls between neighbouring blocks and ,

(78)

In the limit , the partition function is dominated by diagrams with the largest Boltzmann weight or the highest order. It is provenamos1 () that the leading order diagrams are minimal-length DW diagrams with DW separating domains of -blocks and -blocks (Fig. 7).

The presence of projection operators effectively provides locations where DW can form without lowering the order of a diagram. In other words, DW-s that pass through projection operators are “free”. To be precise, we state, in Table 1, the local Boltzmann weight function for two vertically-neighbouring blocks that sandwich a projection measurement in-between. This function is derived using the same method introduced in Ref. amos1, and two examples are provided in Fig. 12 C bottom. Importantly, differs in the following way: aside from the diagonal entries of the table, there is a single entry, namely , in the table of that gives a Boltzmann weight of unity. This implies that the projection operators effectively provide locations at which DW can form without reducing the overall order of the diagram. We will use this observation to show that saturates to an area law in late time in the large- limit.

We proceed in the proof with three steps: (i) We analyse the diagrams row-by-row, and show an upper bound in the order for each row of walls (which is defined by two neighbouring rows of blocks). (ii) We identify the leading order diagrams by invoking the “sink-source” argumentsamos1 (), and by showing that these diagrams saturate the bounds found in (i). (iii) We show that all diagrams with the highest order are algebraically translated into positive factors (so there can be no cancellation between these contributions).

We label each row of blocks as in the far left of Fig. 12 L, and a row of walls by the label of the row of blocks above. For step (i), consider 3 types of rows of walls: (a) Even rows of walls without projection operators along the rows (e.g. row 2 in Fig. 12 L); (b) Odd rows of walls which cannot have any projection operators (row 3 in Fig. 12 L); (c) Even rows of walls with projection operators (row 4 in Fig. 12 L).

The following upper bounds in the order of rows of type (a) and (b) are proved in Ref. amos1, using Table 1 in the reference. For case (a), if there are two types of blocks, and , on the top row of blocks, the upper bound of the order of the row of walls is , given rise by a single factor of (while all the other local Boltzmann weights are , see Fig. 12 R (a)). Note that if there is only a single type of blocks, say , along the top row of blocks, then the upper bound of unity is always saturated by choosing the bottom row of blocks identical to the top one, i.e. also . For case (b), regardless of the number of block types in the top row of blocks, one can always find a configuration of row of walls with order unity, by choosing the bottom row of blocks identical to the top row of blocks (Fig. 12 R (b)). For case (c), the upper bound of order is unity even if there are two types of blocks, say and , on the top row of blocks (c.f. case (a)), because the DW between domains of block and can occur at the position of the projective measurement. Furthermore, depending on the realization of positions of projection operators, a leading row of walls can be an extended segments of horizontal DW (Fig. 12 R (c) bottom). This concludes step (i).

1 1
1
1
1
1
1
Table 1: Upper bounds for the local Boltzmann weight associated with the boundaries between two vertically-neighbouring blocks that sandwich a projective measurement in-between. The matrix is symmetric and so only the upper triangle is written explicitly. The upper bounds are saturated by all Boltzmann weights that appear in the leading order diagrams of