arXiv:1407.2442v2 [quant-ph] 15 Sep 2015 tt hsi fe eerdt sa eetv ie,de- size’, ‘effective an as to given referred a often in is extend this effects – quantum state which to scale largest quantum long-range or quantum correlations. many-body macroscopic has a that scale. necessarily atomic expects state the which one over words, yet extending other physics, effects In from quantum up by built only are include, explained systems are These matter condensed that of 8]. properties bulk [7, was example, phenomena Leggett for accumulated by out appreciated rule originally to ef- need quantum The that microscopic of scale fects. accumulation large an a simply at not behavior is nonclassical by display is given it to 1’ if able macroscopic definition is ‘working state quantum of a Fr¨owis D¨ur lines and [6]: the mo- along are tivated measures proposed typically, macroscopicity; tify macroscopic to also transition may the behavior. measure classical understand a better Such us help produced. macro- qualitatively are ‘quantum which states va- of in different experiments great measure compare this general to to a scopicity’ Due needs one interferometers [1–5]. riety, resonators molecular mechanical circuits, and superpositions superconducting macroscopic similar in or systems the photonic if or its from exists, system a limit isolating environment. size of noisy matter fundamental a a purely is if challenge mechanics decide macro- classical probe to and to create – quantum order to between in boundary attempt experiments, the theory. in to states quantum important quantum within scopic therefore possible states, is is dead and this It alive principle of in superposition a yet in existing of cat absurdity a the ex- highlights seen Schr¨odinger’s thought cat never well-known of macroscopic The periment are the objects superpositions. at quantum macroscopic down in particular, ex- break to In to is appears scale. problems it remaining why greatest plain its of one chanics, naporaemauesol hndsrb the describe then should measure appropriate An quan- to agreed generally measure single no is There in states’ ‘cat sought have experiments recent Some me- quantum of successes overwhelming the Despite unu arsoiiyvru itlaino macroscopi of distillation versus macroscopicity Quantum oosi oedti n s hte hyaerbs opertu to 03.65.Ud 03.67.Mn, robust 03.65.Ta, are numbers: PACS robustly they distilled whether not ask are states and superpositio cluster detail macroscopic one-dimensional into more distilled in s be tocols code can surface Kitaev but th and on quantum states cluster based Both macroscopicity quantum examples. of eral measure c existing classical and an to operations local by states Horne-Zeilinger esgetawyt uniyatp fmcocpcentangleme macroscopic of type a quantify to way a suggest We .INTRODUCTION I. 2 etefrQatmTcnlge,Ntoa nvriyo Si of University National Technologies, Quantum for Centre nvriyo xod ak od xod X P,UK 3PU, OX1 Oxford, Road, Parks Oxford, of University 1 tmcadLsrPyis lrno Laboratory, Clarendon Physics, Laser and Atomic ejmnYadin Benjamin Dtd coe ,2018) 7, October (Dated: 1 n ltoVedral Vlatko and i apnsot ttsia pnmdl.W conclude We models. V. Section spin answers in statistical finding onto imperfections, mappings distillation to via these sensitive information whether are Fisher ask we the protocols IV, by Section entangle- detected In not macroscopic measure. is have in this states but relate Kitaev ment, ground and these states how code cluster that investigate surface find and We via examples. III, specific Section entanglement in macroscopic states [13] (GHZ) state Greenberger-Horne-Zeilinger a of pure distillation propose of then macroscopicity, measure of measure information the on existing information. based an Fisher macroscopicity against quantum quantity of measure this compare widely-studied systems focus We characterizations continuous-variable (we for although these [10–12]. exist here), to macroscopicity qubits restricted of of currently systems is on work systems, our finite-dimensional about for so Statements easiest entanglement. quantifying are macroscopic of entanglement way distill- of a on kind based as a size superpositions, effective correla- macroscopic an quantum ing propose we about Thus statement tions. a as macroscopicity ρ where ihrifraincnb endby defined be can information Fisher N eeatpoet ftefml stesaigof scaling the is family the of property relevant obvious quantity some by size parameterized states of families consider hc au of value a which by here noted ecnie h class the consider We . h case The . ie state a Given nScinI,w rtitoueteqatmFisher quantum the introduce first we II, Section In viewing of consequences the explore we work, this In I IHRIFRAINMAUEOF MEASURE INFORMATION FISHER II. muiain eaayehwti relates this how analyze We ommunication. u ihrdmninlcutrsae are. states cluster higher-dimensional but ae r on ontb macroscopically be not to found are tates p a s elo tteedsilto pro- distillation these at look We ns. unu ihrifraini sev- in information Fisher quantum e and F( ,A ρ, btos n e euti that is result key One rbations. Greenberger- of distillation via nt N N S gpr,Snaoe117543 Singapore ngapore, N ψ ∗ ,2 1, a = ) ∗ MACROSCOPICITY eg h ubro uis.Te the Then qubits). of number the (e.g. = ⟩ N ρ ewl o moeaycto above cut-off any impose not will We . r h ievle n iesae of eigenstates and eigenvalues the are O ∗ n nobservable an and 2 ( onsa arsoi,btinstead but macroscopic, as counts Q a,b N ) ( p s‘aial arsoi’[9]. macroscopic’ ‘maximally is A superpositions c p a a − fosralswihcnbe can which observables of + p p b b ) 2 S⟨ ψ a S A A S ψ h quantum the , b ⟩S 2 , N ∗ with (1) 2

N n n written as A = ∑i 1 Ai over local Ai, each acting nontriv- states GHZn ∶= 0 ⊗ + 1 ⊗ √2 as the target for dis- = ially on a single qubit i and with fixed norm YAiY = 1. tillation.S GHZ⟩ n (Sis⟩ oftenS described⟩ )~ as the typical qubit The effective size proposed by Fr¨owis and D¨ur [6] is model of aS macroscopic⟩ superposition, and is the max- imally macroscopic state of n qubits in the sense that F(ρ, A) N ∗ (ρ) ∶= max , (2) N ∗ = n. Furthermore, it is easy to motivate assigning an F A N F ∈A 4 effective size of n to such a state. and lies in the range 1,N . (A may be extended to ‘k- To define our measure, take a pure state ψ of N S ⟩ local’ A with A acting[ on] groups of k qubits, with k qubits, and consider acting on ψ with stochastic LOCC i S ⟩ bounded independent of N, in which case the denomina- (SLOCC), described by measurement operators Ma { } tor of equation (2) contains the number of groups instead corresponding to the outcomes √pa φa ∶= Ma ψ with † S ⟩ S ⟩ of N.) probabilities pa = ψ MaMa ψ . We denote this kind of ⟨ S S ⟩ Observables in A are supposed to model the kinds of transformation by ψ → φa ,pa . Now restrict these quantities that are easily measured at the macroscopic operations to the setS ⟩Dψ {Ssuch⟩ that} every outcome is of N Sa scale with coarse-grained, noisy classical detectors. For the form φa = GHZSa 0 −S S where Sa ⊆ 1, 2,...,N 1 S ⟩ S ⟩S ⟩ { } pure states, 4 F equals the variance, and N ∗ is seen to is some subset of N qubits of cardinality Sa . We asso- F quantify the largest quantum fluctuations of any macro- ciate with each φa a size na = Sa unless aS ‘trivial’S GHZ scopic observable – originally identified in [14, 15]. It has state of size 1 isS obtained,⟩ inS whichS case na = 0. Our been shown [6] that N ∗ is more inclusive than a variety of measure is measures [16–18] lookingF at ‘macroscopic superpositions’ of two states: maximal macroscopicity according to any N ∗ ψ max p n . (3) of these measures implies N ∗ = O N . D ∶= Q a a Ma Dψ F (S ⟩) { }∈ a In general, N ∗ describes the usefulness( ) of a state for F quantum metrology. Consider a family of states ρθ = This is supposed to describe the size of GHZ-type en- iθA iθA e− ρe encoding a parameter θ ∈ R. From n indepen- tanglement present in the state ψ . We have restricted dent copies of ρ, the quantum Cram´er-Rao bound sets a each final state to a single GHZ,S rather⟩ than a general lower limit on the uncertainty with which θ can be esti- product @i GHZni , since we are only interested in the mated: δθ ≥ 1 »nF ρ, A [19]. A macroscopic quantum largest GHZ;S the remaining⟩ parts could be converted de- state with N ∗~= O N( makes) δθ ∝ 1 N possible, a qual- terministically into product states. This prescription, F itative improvement( over) the classical~ δθ ∝ 1 √N. instead of summing the sizes, rules out the ‘accumu- The authors have recently provided a motivation~ for lated’ phenomena mentioned earlier. Thus a state like N this measure as a quantifier of macroscopic coherence in a 00 + 11 ⊗ has ND∗ = O 1 instead of O N . In gen- (S ⟩ S ⟩) ( ) ( ) precise sense [20]. (That work also notes some similarities eral, ND∗ @i ψi = maxi ND∗ ψi . ( S ⟩) (S ⟩) between this measure and other approaches motivated It is simple to show that ND∗ is an entanglement mono- from very different starting points [10, 21].) tone – it cannot increase on average under SLOCC. Sup- Furthermore, it has been demonstrated that large N pose ψ → χµ ,pµ by SLOCC. Then for each µ there ∗ S ⟩ {S ⟩ } is a witness of macroscopic entanglement in the followingF exists an optimal ensemble φµ,a ,pµ,a distilled from {S ⟩ } ways. For pure ‘k-producible’ states in which blocks of up χµ such that 1 S ⟩ to k sites may be entangled, F ψ ψ , A ≤ kN for 1- 4 N ∗ χ p n , (4) A (S ⟩⟨ S ) D µ = Q µ,a µ,a local (similar bounds exist for mixtures) [22, 23]. Also, (S ⟩) a N ∗ = O N for a pure state implies that macroscopically 2 manyF (i.e.( O) N ) pairs of sites have a nonvanishing O 1 where φµ,a contains a GHZ state of size nµ,a. By amount of localizable( ) entanglement [24, 25]. ( ) composingS the⟩ two SLOCC protocols, it follows that However, the converse is false: there are highly- ψ → φµ,a ,pµpµ,a is a valid distillation operation in S ⟩ {S ⟩ } entangled states with small N ∗ . To put this precisely, Dψ, so we will introduce a measure aimingF to quantify macro- N ∗ ψ ≥ p p n scopic entanglement. D Q µ µ,a µ,a (S ⟩) µ,a

= Q pµND∗ χµ , (5) III. MEASURE OF MACROSCOPIC µ (S ⟩) ENTANGLEMENT which proves the monotonicity. The optimization involved in determining ND∗ will gen- For a pure quantum state, one can quantify the amount erally be intractable – the best we can do is find bounds. of bipartite entanglement by counting the number of A lower bound must come from an explicit construction maximally entangled states that can be distilled by lo- of distillation operations, and this will be difficult except cal operations and classical communication (LOCC) from for some particular cases. By extending a method in [28], many copies of the given state [26]. we present a simple upper bound for states that are sym- In the multipartite case there is no unique maximally metric under exchange of any two sites (see Appendix A). entangled state [27]. To give a reasonable notion of This comes from the geometric entanglement [29] of a sin- macroscopic entanglement, we suggest to use the GHZ gle site with the rest: EG ψ = 1−λmax ψ , where λmax (S ⟩) (S ⟩) 3

x is the largest eigenvalue of ρ1 ∶= Tr2,3,...,N ψ ψ . Using to local σ gates depending on the outcomes, where NB the monotonicity of EG under SLOCC, weS find⟩⟨ S is the number of B sites. Classical communication can remove these errors, making the final state deterministi- N ∗ ψ ≤ 2N 1 − λmax ψ . (6) D cally GHZNB . Therefore ND∗ CN ≥ NB = O N . It is (S ⟩) ( (S ⟩)) simpleS to see that⟩ this generalizes(S to⟩) cluster states( ) of any dimension. A. Generalized GHZ states

Generalized GHZ states [28] were our initial motivation C. Kitaev surface code states for considering GHZ distillation. These depend on N and a parameter ǫ ∈ R, and were suggested to be a reasonable Kitaev’s surface code model is the simplest lattice sys- description of the macroscopic current superpositions in ǫ tem displaying topological order, of great interest for con- superconducting qubits. They can be written as ψN ∝ N N S ⟩ densed matter physics and topological quantum compu- ǫ ⊗ + −ǫ ⊗ , where ±ǫ ∶= cos ǫ 2 0 ±sin ǫ 2 1 . For tation [32–34]. Its ground states are sensitive to global π N S ⟩ S ⟩ S ⟩ ( ~ ) S ⟩ ( ~⊗) S ⟩ ǫ = 2 we recover GHZN , while ǫ = 0 gives 0 . Thus topological properties of their lattice. Since they could we expect ǫ to varyS the macroscopicity⟩ smoothlyS ⟩ between exist on a macroscopic lattice, this makes them interest- the minimal and maximal values. Indeed, for ǫ 1 ing candidates for macroscopicity. 2 2 Nǫ , N ∗ Nǫ [6]. ≪ ≪ The distillationF ≈ protocol constructed in [28] produces 2 z ¡ 1

an average distilled size of approximately Nǫ 2 in the x z § 2 5 £ 4 above limit. Moreover, λmax cos ǫ 2 giving~ N ∗ s D z

2 ¢ ≈ ( ~ ) ≤ 2

Nǫ 2 – so the protocol is exactly optimal [30], and x x z

¦ ¤ p ~ ǫ 2 8 6 3 ND∗ ψN Nǫ 2. Here we have ND∗ N ∗ 2. F (S ⟩) ≈ ~ ≈ ~ x

¥ 7 B. Cluster states

FIG. 2. A star s and plaquette p in Kitaev’s model, and their It has already been noted for cluster states that N ∗ = z z z z x x x x O 1 , but it is nevertheless possible to deterministicallyF corresponding operators: zs = σ1 σ2 σ3 σ4 and xp = σ5 σ6 σ7 σ8 . distill( ) from them GHZ states of size O N [6, 12]. Recall that a d-dimensional cluster state is defined( ) for qubits as- The two-dimensional version is defined by a square lat- sociated with the vertices of (a subset of) a d-dimensional tice where each edge represents a qubit, with Hamiltonian N z square lattice. It can be constructed from 0 + 1 ⊗ by H = − ∑s zs − ∑p xp. Here, zs is the product of σ oper- z x applying a controlled-σ gate between each(S ⟩ neighboringS ⟩) ators over a ‘star’ and xp is the product of σ operators pair [31]. over a ‘plaquette’ – see fig. 2 – and the sums are over To see the scaling of N ∗ , note that the variance quan- all stars and plaquettes in the lattice. There may be a tifies the total amount ofF two-point correlations [24]. In number of degenerate ground states, depending on the cluster states, it can be shown that two regions are uncor- topology of the lattice; however, these have equivalent related unless they share a boundary. Since each region of entanglement properties [35] so we concentrate on the bounded O 1 size has a bounded number of neighbors, simplest ground state. it follows that( ) N ∗ = O 1 . It is possible to describe the structure of this state ex- F ( ) plicitly. We define a ‘configuration’ of the lattice to be a product state where the qubits lying on a certain set of curves (open or closed) are all in the state 1 while the rest are 0 . The ground state KN is an equalS ⟩ superpo- A sition ofS all⟩ configurations containingS ⟩ only closed curves B which are topologically trivial (contractible to a point). KN , like CN , has no correlations between non- neighboringS ⟩ regionsS ⟩ [36], so N ∗ = O 1 . As a further par- allel, one can deterministicallyF distill( ) macroscopic GHZ

FIG. 1. The graph used to define the cluster state SCN ⟩. Mea- states from KN by SLOCC: choose any topologically surements are performed on the A sites in order to distill the trivial non-self-intersectingS ⟩ loop B of NB = O N qubits B sites into a GHZ state. and perform local projective measurements( on) the re- mainder of the lattice in the z basis. Consider the case For simplicity, we focus on the family of N-qubit clus- where the outcome of each measurement is 0 – then B S ⟩ ter states CN defined by graphs of the type shown in ends up in GHZNB containing the only two consistent fig. 1 – theseS ⟩ are two-dimensional cluster states with a closed-curveS configurations.⟩ For different outcomes, the fraction of sites removed. A measurement of each site of final state differs by local σx operations, so we can obtain x set A in the σ eigenbasis projects B into GHZNB up GHZNB deterministically. Hence ND∗ KN = O N . S ⟩ S ⟩ (S ⟩) ( ) 4

D. Dicke states In the unperturbed case, projective measurements on A give full GHZ states on B with equal probability for each Dicke states [37] have recently been studied for their in- outcome. From this symmetry it is clear that we need NA teresting multipartite entanglement properties [6, 22, 38– examine only a single branch, say the outcome E⊗ . NA a b x b z 40]. The N-qubit versions N, k are defined by sym- The final state is ψ ∝ ∑b E⊗ A GHZ B . † N k k S ⟩ The most generalS ⟩ form (up( to normalization)S ( ) ⟩) S ( for)E⟩ E metrizing 0 ⊗( − ) 1 ⊗ . For any observable of the form S ⟩ x S ⟩ y in the x-basis is A = ∑i cos θ σi + sin θ σi we obtain the maximal vari- ( ) ( ) ance giving N ∗ = 1 + 2f 1 − f N where the quantity † 1 δ∗ C R F ( ) E E = ,δ ∈ ,ǫ ∈ . (7) f = k N controls the macroscopicity in the same way ‹δ ǫ  ~ ǫ as ǫ for ψN . z For the observable ZB = ∑i B σi , F depends on ǫ but not Our upperS ⟩ bound gives N N, k ≤ 2 min k,N − k . ∈ D∗ δ: We are not aware of any SLOCC(S protocols⟩) to distill{ GHZ} states from N, k , and so cannot provide a lower bound. 1 1 a b 2 F ψ ψ ,ZB = Q ǫS ( )S NB − 2 b , (8) S ⟩ We can only state that ND∗ ≤ 2 N ∗ − 1 , using min f, 1 − 4 (S ⟩⟨ S ) Z b ( S S) ( F ) { f ≤ 2f 1 − f . Thus N ∗ O N ⇒ ND∗ O N . How- a b } ( ) F < ( ) < ( ) where Z = ∑b ǫS ( )S and a ∶= ∑ ai. We interpret this ever we do not know if N ∗ = O N ⇒ ND∗ = O N . i F ( ) ( ) by mapping the problem ontoS S a two-dimensional square- lattice ferromagnetic classical Ising model. Each b maps onto a spin configuration with individual magnetic mo- IV. ROBUSTNESS OF DISTILLATION ments of ±1 and a total magnetization M = NB −2 b , and a b is the corresponding bond configuration. ThereS S is In the above cases of cluster states and Kitaev surface an( energy) cost of 2J for each ai = 1, so we associate ǫ 2βJ code states, the distillation accomplishes something of with e− and Z with the partition function. Therefore 1 2 practical value: it extracts states which are useful for 4 F ψ ψ ,ZB = M . quantum metrology from initial states which are not, This(S ⟩⟨ modelS ) has⟨ a⟩ low-temperature ferromagnetic without increasing entanglement. Equivalently, it am- phase; from a bound by Griffiths [41] we obtain 1 2 4 plifies the macroscopicity of the states according to the 4 F ψ ψ ,ZB ≥ NB 1 − O ǫ – so the distillation is measure N ∗ . If this is to be regarded as something with robust.(S ⟩⟨ S ) ( ( )) practical significance,F then the final states must be robust This fails for one-dimensional cluster states, because with respect to imperfections in the protocol. Indeed, one the Ising model has no T 0 phase transition in one di- might doubt the physical meaning of the distillation if it mension [42]. So distillation> from one-dimensional cluster is not robust in this sense. states is not robust. To be specific, we neglect environmental noise since the GHZ states produced are maximally sensitive to de- coherence [28], so any experiment taking advantage of B. Kitaev surface code states them must have tolerably low noise. Instead, we suppose that the measurement device operating on the individual KN works similarly, and the closed-loop restriction qubits may couple to them imprecisely. Therefore we say letsS us⟩ interpret the configurations as domain walls of an that: Ising model with a spin on each plaquette. The robust- The distillation of GHZ states is robust if the average ness of distillation again relies on T 0 ferromagnetic Fisher information of the final state retains the same scal- order. However, the relevant statistical> model is more ing with N for any small perturbation to the measurement complex and depends on δ. As discussed in Appendix operators. C, the disorder associated with δ maps onto probabilisti- The definition of the perturbation will be clear from cally ‘turning off’ the bonds, giving a dilute Ising model. the following examples. This has a ferromagnetic phase [43], letting us conclude 2 2 ZB = O NB for sufficiently small δ and ǫ. ⟨ An⟩ important( ) caveat is that this only works when the loop B divides the lattice into two-dimensional regions – A. Cluster states for the same reason as the failure of one-dimensional clus- ter states. Therefore it seems we can only conclude ro- For cluster states, we perturb the projective measure- bustness when NB = O √N . The distilled states, while ments to generalized measurement operators E, E¯ sat- robust, have F = O N( and) do not offer a metrological isfying E†E + E¯†E¯ = I. Via the action of{ controlled-} advantage over the initial( ) state. z σ gates, the initial state can be written as CN ∝ a b x b z S ⟩ ∑b A GHZ B where A and B are expressed in theS ( x)−⟩Sand z(−bases) ⟩ respectively, a ∈ 0, 1 NA , b ∈ V. CONCLUSIONS NB { } 0, 1 and GHZ b ∶= b + b √2 with bi ∶= 1 − bi. {Each} value ofSa is determined( )⟩ (S ⟩ byS b⟩)~, and without loss of In summary, we have proposed a notion of macroscopic generality we fix b1 = 0 – see Appendix B for details. entanglement which measures the size of GHZ states that 5 can be extracted by SLOCC, and compared this with an equal weighting. As a result, the probability pa for each existing measure of macroscopicity. We find that clus- GHZSa is a function of na only. We denote by qn the ter states and Kitaev model ground states are macro- Stotal probability⟩ of obtaining any GHZ state of size n – N scopically entangled by our definition but are not macro- it is clear that qn = n pa for any a such that Sa = n. scopically quantum by the Fisher information. How- ‰ Ž N SaS S For a distilled state φa = GHZS 0 ⊗ −S S the geo- ever, we lack statements examining the converse: whether a metric entanglement ESG of⟩ aS site i with⟩S ⟩ the rest of the N ∗ = O N ⇒ ND∗ = O N . Extensions to our measure 1 F system is 2 if i ∈ Sa and 0 otherwise. The number of beyond qubits( ) are also important( ) – for finite-dimensional N 1 k 1 k 1 n subsets of size n containing i is − . systems, one could distill GHZ = ∑ − i ⊗ . We n 1 n √k i 0 The average E for the final ensemble‰ − Ž is therefore S ⟩ = S ⟩ 2 G suggest ascribing the same size to this as to GHZn , as m m N N 1 2 ⊗ 2 S ⟩ − GHZn looks like GHZn at the m-qubit level. n 1 1 Q paEG φa = Q qn ‰ − Ž × S The⟩ distillation canS also be⟩ interpreted as increasing N 2 a (S ⟩) n 2 n the advantage of these states in high-precision parameter = ‰ Ž 1 N n estimation. A criterion requiring the distillation to be = Q qn of practical utility, by being robust against errors, tells 2 n 2 N = us that two-dimensional, but not one-dimensional, clus- 1 ter states are useful. This dependence on dimensionality = ND∗ ψ . (A1) 2N (S ⟩) also appears in the statement that two-dimensional clus- ter states are universal for measurement-based quantum By the monotonicity of EG under SLOCC, we have computation, while the one-dimensional versions are not ND∗ ψ ≤ 2NEG ψ . (A2) [44]. Kitaev model ground states are robust, but this can (S ⟩) (S ⟩) be proved only up to a non-macroscopic O √N distilled ( ) size. It is noteworthy that our analysis for these states Appendix B: Distillation from cluster states SCN ⟩ is very close to the proof that surface codes can be er- ror corrected [45]. Thus we speculate that our results x x To construct CN , we define 0 , 1 ∶= + , − might be related to the ability of these states to encode S ⟩ N {S ⟩ S ⟩} {S ⟩ S ⟩} quantum information and do so robustly. and write the state + ⊗ before applying the UCZ gates S ⟩ As suggested by others [12], it would be helpful to de- as velop a resource theory for macroscopicity, requiring an x z z 0x bz A 0i ⊗ A 0j + 1j = Q A ⊗ B . (B1) understanding of the operations unable to increase the i A S ⟩ j B(S ⟩ S ⟩) b S ⟩ S ⟩ effective size. The SLOCC distillation operations used ∈ ∈ here are not included in the set of ‘free’ operations for The action of UCZ on two qubits i, j is determined by macroscopic coherence defined in [20], under which the x z x z Fisher information cannot increase. The crucial point UCZ ai bj = ai ⊕ bj bj , (B2) S ⟩S ⟩ S( ) ⟩S ⟩ is that the final σx gates, conditioned on measurement where ⊕ denotes addition modulo 2. Therefore we have outcomes, are not free in that framework and instead en- that tail manipulation of coherence between eigenstates of ZB. This is where the present notion of macroscopic entangle- a b x bz CN ∝ Q A B , (B3) ment departs from macroscopic coherence. S ⟩ b S ( ) ⟩S ⟩

where each a b is determined by the following rule: ai = 0 when the neighboring( ) bi are equal, and ai = 1 when they ACKNOWLEDGMENTS 1 2 are different. One can see that a b( ) = a b( ) if and only if b 2 = b 1 or b 1 . This leads( to) ( ) The authors acknowledge funding from the National ( ) ( ) ( ) Research Foundation (Singapore), the Ministry of Ed- a b x b z CN ∝ Q A GHZ B , (B4) ucation (Singapore), the EPSRC (UK), the Templeton S ⟩ b S ( ) ⟩S ( ) ⟩ Foundation, the Leverhulme Trust, the Oxford Martin School and Wolfson College, University of Oxford, and where we fix b1 = 0 without loss of generality – any single thank Wonmin Son for helpful discussions. site in B could be fixed. It is clear from this expression that a measurement of every A site in the x-basis will NB 1 project B into a GHZ state; there are 2 − different ∗ outcomes. Appendix A: Upper bound on ND for symmetric states With imperfect measurements, as discussed in the main text, we only need to consider a single branch with the outcome E for each measurement, so we use the final First note that for symmetric states, we may always state construct a symmetric optimal distillation protocol: if 1 an asymmetric optimal protocol is found, one just needs NA a b x b z ψ = Q E⊗ A GHZ B . (B5) to probabilistically perform all its permutations with S ⟩ √Z b ( S ( ) ⟩) S ( ) ⟩ 6

To calculate the Fisher information in the variable ZB = z 2 2 (o) ∑i B σi , we just need the variance ψ ZB ψ − ψ ZB ψ . σj Now∈ it is easy to see from symmetry⟨ S thatS ⟩ψ ⟨Z S ψ S =⟩0, B b while ⟨ S S ⟩ j (i) z z σj z bz b bi bz b Q σi ± = Q −1 ∓ i (S ⟩ S ⟩) i ( ) (S ⟩ S ⟩) z z = NB − 2 b b ∓ b (B6) ( S S)(S ⟩ S ⟩) gives Z2 GHZ b z = N − 2 b 2 GHZ b z . There- B B B B (o) bi (i) fore S ( ) ⟩ ( S S) S ( ) ⟩ σi σi 1 2 a b † NA a b ψ ZB ψ = Q E E ⊗ ′ ′ ⟨ S S ⟩ Z b,b′ ⟨ ( )S( ) S ( )⟩ b z 2 b z × GHZ B ZB GHZ ′ B ⟨ ( ) S S ( ) ⟩ FIG. 3. The highlighted loop B is the set of qubits to be 1 † NA = Q a b E E ⊗ a′ b′ distilled into a GHZ state. Each segment bi has neighboring Z b,b′ ⟨ ( )S( ) S ( )⟩ (i) (o) Ising spins σi and σi . 2 × δb,b′ NB − 2 b ( S S) 1 a b b 2 = Q ǫS ( )S NB − 2 , (B7) To define the required mapping, we recall that the Z b ( S S) pre-measurement state is a sum of (topologically trivial) given that 0 E†E 0 = 1, 1 E†E 1 = ǫ. Similarly, by closed-loop configurations in the z-basis: ⟨ S S ⟩ ⟨ S S ⟩a b setting ψ ψ = 1 we obtain Z = ∑b ǫS ( )S. ⟨ S ⟩ 1 KN ∝ Q cA cB , (C2) As described in the main text, 4 F ψ ψ ,ZB can 2 S ⟩ c ∂c 0 S ⟩S ⟩ be interpreted as M for a classical Ising(S ⟩⟨ modelS ) with ∶ = 2βJ ǫ = e− . Griffiths⟨ [41]⟩ establishes a bound of the form where ∂c is the boundary of c and cA and cB are the M ≥ NB 1 − f T where f is independent of NB parts of c existing on A and B respectively. (For the re- ⟨S S⟩ ( ( )) 2 2 and limT →0 f T = 0. Using 0 ≤ NB − M = NB − mainder of this section, a quantity such as c or c will ( 2 ) ⟨(2 S S) ⟩ 2 A 2NB M + M , it follows that M ≥ 2NB M −NB ≥ 2 8βJ be understood to be a vector, while ci is a single compo- N 1⟨S− 2S⟩f ⟨T .⟩ To leading order,⟨ f⟩ T ≈ e⟨S− S⟩ which B nent.) Since B is a closed loop, every distinct cA in this gives( the bound( )) ( ) sum corresponds to exactly two values of cB, which are the opposites of each other. Therefore, by choosing to fix 1 2 4 a single site of B, say b1 = 0 (which we will do implicitly F ψ ψ ,ZB ≥ NB 1 − O ǫ . (B8) 4 (S ⟩⟨ S ) ( ( )) from now on), we can write the state as

KN ∝ Q cA GHZ cB . (C3) Appendix C: Distillation from Kitaev model ground S ⟩ c ∂c 0 S ⟩S ( )⟩ states SKN ⟩ ∶ =

Furthermore, for each cB we can generate all the cor- 2 As above, we need to calculate ZB and verify that it responding cA by choosing a particular representative 2 ⟨ ⟩ remains O NB for sufficiently small perturbations. Our cA cB and adding to this all the possible closed loops approach will( again) involve a map onto a statistical model zA(lying) strictly within A. (Note that adding zA to cA where a spin σi is placed at the center of each plaquette. creates a configuration cA ⊕ zA in which cA is deformed B is chosen to be a rectangular loop of size NB = O √N through the region boundedS by⟩ zA.) This provides an which cuts the remainder A of the lattice into two( inde-) additional representation pendent two-dimensional regions. If we label each edge in B by bi = 0, 1 with neighboring ‘inside’ and ‘outside’ Ising i o i o KN ∝ Q ⎛ Q cA cB ⊕ zA ⎞ GHZ cB . (C4) spins σi( ), σi( ) = ±1 (see fig. 3), then σi( )σi( ) = 1−2bi and S ⟩ cB zA ∂zA 0 S ( ) ⟩ S ( )⟩ 2 ∶ = ZB will be given by ⎝ ⎠ ⟨ ⟩ 2 i o 2 Hence we can write the post-measurement state NB − 2 b = σ( )σ( ) Q i i NA ⟨( S S) ⟩ ⟨(i B ) ⟩ E⊗ KN as ∈ i i o o S ⟩ N σ( )σ( ) σ( )σ( ) , = B + 2 Q i j i j (C1) 1 i j ⟨ ⟩ ⟨ ⟩ NA < ψ = Q E⊗ cA GHZ cB (C5) S ⟩ √Z c ∂c 0 S ⟩S ( )⟩ where the expectation value is taken with respect to the ∶ = statistical model to be defined below. Hence it will fol- 1 NA 2 2 = Q ⎛ Q E⊗ cA cB ⊕ zA ⎞ low that ZB = O NB if our model results in an or- √Z cB zA ∂zA 0 S ( ) ⟩ dered phase⟨ where⟩ the( two-point) correlation functions are ⎝ ∶ = ⎠ bounded strictly above zero independent of N. × GHZ cB (C6) S ( )⟩ 7 such that Z gives the normalization ψ ψ = 1. We can ability of a configuration f,b is use these two forms simultaneously to⟨ determineS ⟩ Z: ( ) fe βH f,b p P f,b ∝ e− ( ) M ( ) e ‹1 − p f Z = c′ c′ ⊕ zA E cA e Q Q Q A B βJ fe 2be 1 fe p ′ ∝ e− ( + ( − )) c ∂c 0 cB zA ∂zA 0 ⟨ ( ) S S ⟩ M ∶ = ∶ = e ‹1 − p × GHZ cB′ GHZ cB fe p βJ 2βJ be 1 fe ⟨ ( )S ( )⟩ = M e− e− ( − ). (C12) = Q Q cA ⊕ zA E cA , (C7) e ‹1 − p  ( ) c ∂c 0 zA ∂zA 0 ⟨ S S ⟩ ∶ = ∶ = Therefore we can make the identification cA ⊕ zA E cA → P f,b as long as we map † NA ⟨ S S ⟩ ( ) where E ∶= E E ⊗ . In the second line, we have used p βJ the fact that( c′ =)cB to choose the representative c′ c′ δ → e− , B A B 1 − p to equal cA. Similarly we find ( ) ‹  2βJ ǫ → e− ,

zi → fe, 1 2 2 ci → be. (C13) ZB = Q Q cA ⊕ zA E cA NB − cB . ⟨ ⟩ Z c ∂c 0 zA ∂zA 0 ⟨ S S ⟩ ( S S) ∶ = ∶ = (C8) From the general form of E†E described in the main text, we have [46]

zi ci 1 zi cA ⊕ zA E cA = M δ ǫ ( − ). (C9) ⟨ S S ⟩ i A ∈

We shall see that this calculation can be mapped onto FIG. 4. An example of a loop to be distilled into a size O(N) a statistical spin model with the following Hamiltonian: GHZ state.

The last of these is compatible with the restriction ∂c = 0 H = Q Jij σiσj , (C10) because each bond configuration b for the spins must i,j> < be formed of closed loops. However, our condition that ∂zA = 0 means that we must similarly restrict the pat- terns of f in the spin model; it is nevertheless clear that where the sum is over neighboring pairs of spins, and this only reduces the disorder and so must preserve the Jij = 0 with probability p and Jij = −J J 0 with ferromagnetic phase. There are also no bonds lying on probability 1 − p. Such a model describes an( Ising> ) ferro- the curve B, explaining our earlier claim that B cuts the magnet with random disorder caused by removing some remainder A into two non-interacting sections. of the bonds; we take this disorder to be ‘annealed’ (as Therefore, to sum up, each term cA ⊕ zA E cA can be opposed to ‘quenched’), meaning that the Jij are consid- interpreted as the probability of a⟨ configurationS S ⟩ in this ered dynamical variables. It is known that this model is disordered spin model, so that Z becomes the partition in a ferromagnetic phase for sufficiently small T and p 2 function and ZB is related to the two-point correlators [43]. as described⟨ earlier.⟩ The existence of a ferromagnetic phase then lets us conclude that Z2 = O N 2 for suffi- Let us first rewrite the Hamiltonian in terms of the B B ciently small δ and ǫ. ⟨ ⟩ ( ) variables fij = 0, 1 with probabilities 1−p ,p respectively, 1 ( ) This argument assumed B to be a rectangular loop of and bij = 2 1 − σiσj : ( ) size O √N , such that the distilled GHZ states have a Fisher( information) of only O N – the same as the initial state KN . If we want an improvement( ) for metrology, we H f,b = −J Q 1 − fe 1 − 2be , (C11) wouldS need⟩ to choose a loop of size O N . As depicted ( ) e ( )( ) in fig. 4, it seems to us that any such loop( ) will divide the lattice into one-dimensional rather than two-dimensional regions, in which case the above argument does not ap- where we replace ij with e labeling the edges of the ply. By analogy with the cluster state example, we might lattice dual to the( spins.) Up to normalization, the prob- conjecture that no O N loop is robust to perturbations. ( ) 8

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