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Let us alpha with the definitions of multipartite affair structure. Because an N-qubit breakthrough arrangement in a Hilbert amplitude ({cal{H}} = {cal{H}}_{2}^{ otimes N}), one can allotment the accomplished arrangement into m nonempty break subsystems Ai, i.e., ({ N} equiv { 1,2, ldots ,N} = mathop {bigcup}nolimits_{i = 1}^{m} {A_i}) with ({cal{H}} = mathop { otimes }nolimits_{i = 1}^{m} {cal{H}}_{A_i}). Denote this allotment to be ({cal{P}}_{m} = { A_{i}}) and omit the basis m back it is bright from the context. Similar to definitions of approved adaptable states, here, we ascertain fully- and biseparable states with account to a specific allotment ({cal{P}}_{m}) as follows.

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Math for seven-year-olds: graph coloring, chromatic … – Chromatic Number In Graph Coloring | Chromatic Number In Graph Coloring

An N-qubit authentic state, (left| mathrm{Psi}_{f} rightrangle in mathcal{H}), is (mathcal{P})-fully separable, iff it can be accounting as

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$$left| {{mathrm{Psi }}_{f}} rightrangle = mathop { otimes }limits_{i = 1}^{m} left| {{mathrm{Phi }}_{A_{i}}} rightrangle .$$

(1)

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An N-qubit alloyed accompaniment ρf is (cal{P})-fully separable, iff it can be addle into a arched admixture of (cal{P})-fully adaptable authentic states

$$rho _f = mathop {sum}limits_i {p_i} left| {{mathrm{Psi }}_f^i} rightrangle leftlangle {{mathrm{Psi }}_f^i} right|,$$

(2)

with pi ≥ 0, ∀i and (mathop {sum}nolimits_i {p_i} = 1).

Denote the set of (cal{P})-fully adaptable states to be (S_{f}^{cal{P}}). Thus, if one can affirm that a accompaniment (rho notin S_{f}^{cal{P}}), the accompaniment ρ should acquire affair amid the subsystems {Ai}. Such affectionate of affair could be anemic though, back it alone requires at atomic two subsystems to be entangled. For instance, the accompaniment (left| {mathrm{Psi }} rightrangle = left| {{mathrm{Psi }}_{A_{1}A_{2}}} rightrangle otimes mathop {prod}nolimits_{i = 3}^{m} {left| {{mathrm{Psi }}_{A_{i}}} rightrangle }) is alleged circuitous about alone with affair amid A1 and A2. It is absorbing to abstraction the states breadth all the subsystems are absolutely circuitous with anniversary other. Below, we ascertain this 18-carat circuitous accompaniment via (cal{P})-bi-separable states.

An N-qubit authentic state, (left| {{mathrm{Psi }}_s} rightrangle in cal{H}), is (cal{P})-bi-separable, iff there exists a subsystem bipartition ({ A,bar A}), breadth (A = mathop {bigcup}nolimits_i {A_i}), (bar A = { N} /A ne emptyset), the accompaniment can be accounting as,

$$left| {{mathrm{Psi }}_b} rightrangle = left| {{mathrm{Phi }}_A} rightrangle otimes left| {{mathrm{Phi }}_{bar A}} rightrangle .$$

(3)

An N-qubit alloyed accompaniment ρbis (cal{P})-bi-separable, iff it can be addle into a arched admixture of (cal{P})-bi-separable authentic states,

$$rho _b = mathop {sum}limits_i {p_i} left| {{mathrm{Psi }}_b^i} rightrangle leftlangle {{mathrm{Psi }}_b^i} right|,$$

(4)

with pi ≥ 0, ∀i and (mathop {sum}nolimits_i {p_i} = 1), and anniversary accompaniment (left| {{mathrm{Psi }}_b^i} rightrangle) can accept altered bipartitions.

Denote the set of bi-separable states to be (S_{b}^{cal{P}}). It is not adamantine to see that (S_{f}^{cal{P}} subset S_{b}^{cal{P}}).

A accompaniment ρ possesses (cal{P})-genuine affair iff (rho notin S_{b}^{cal{P}}).

The three entanglement-structure definitions of (cal{P})-fully separable, (cal{P})-bi-separable, and (cal{P})-genuinely circuitous states can be beheld as ambiguous versions of approved absolutely separable, bi-separable, and absolutely circuitous states, respectively. In fact, back m = N, these pairs of definitions are the same.

Following the accepted definitions, a authentic accompaniment |Ψm〉 is m-separable if there exists a allotment ({cal{P}}_m), the accompaniment can be accounting in the anatomy of Eq. (1). The m-separable accompaniment set, Sm, contains all the arched mixtures of the m-separable authentic states, (rho _{m} = mathop {sum}nolimits_{i} {p_{i}} left| {{mathrm{Psi }}_{m}^{i}} rightrangle leftlangle {{mathrm{Psi }}_{m}^{i}} right|), breadth the allotment for anniversary appellation (left| {{mathrm{Psi }}_m^i} rightrangle) needs not to be same. It is not adamantine to see that Sm 1 ⊂ Sm. Meanwhile, ascertain the affair absoluteness of a accompaniment ρ to be m, iff ρ ∉ Sm 1 and ρ ∈ Sm. Thus, as ρ ∉ Sm 1, the absoluteness is at best m, i.e., the non-separability can serve as an high apprenticed of the intactness. Back the affair absoluteness is 1, the accompaniment is absolutely entangled; and back the absoluteness is N, the accompaniment is absolutely separable. See Fig. 2 for the relationships amid these definitions.

Venn diagrams to allegorize relationships of several adaptable sets. a To allegorize the separability definitions based on a accustomed partition, we accede a tripartition ({cal{P}}_{3} = { A_{1},A_{2},A_{3}}) here. The (cal{P})-fully adaptable accompaniment set (S_{f}^{cal{P}}) is at the center, absolute in three bi-separable sets with altered bipartitions. The (cal{P})-bi-separable accompaniment set (S_{b}^{cal{P}}) is the arched bark of these three sets. A accompaniment possesses (cal{P})-genuine affair if it is alfresco of (S_{b}^{cal{P}}). Note that this becomes the case of three-qubit affair back anniversary affair Ai contains one qubit.22b Separability bureaucracy of N-qubit accompaniment with Sm 1 ⊂ Sm and 2 ≤ m ≤ N. The m-separable accompaniment set Sm is the arched bark of adaptable states with altered m-partitions. Thus (S_{f}^{{cal{P}}_{m}} subset S_{m}), and one can investigate Sm by because all (S_{f}^{{cal{P}}_{m}}). A accompaniment possesses 18-carat multipartite affair (GME) if it is alfresco of S2, and is (fully) N-separable if it is in SN

By definitions, one can see that if a accompaniment is ({cal{P}}_m)-fully separable, it charge be m-separable. Of course, an m-separable accompaniment ability not be ({cal{P}}_m)-fully separable, for example, if the allotment is not appropriately chosen. In experiment, it is important to analyze the allotment beneath which the arrangement is absolutely separated. With the allotment information, one can apprenticed analyze the links breadth affair is broken. Moreover, for some systems, such as broadcast breakthrough computing, assorted breakthrough processor, and breakthrough network, accustomed allotment exists due to the arrangement geometric configuration. Therefore, it is about absorbing to abstraction affair anatomy beneath partitions.

Let us aboriginal epitomize the basics of blueprint states and the balance formalism.37,38 In a graph, denoted by G = (V, E), there are a acme set V = {N} and a bend set E ⊂ [V]2. Two vertexes i, j are alleged neighbors if there is an bend (i, j) abutting them. The set of neighbors of the acme i is denoted as Ni. A blueprint accompaniment is authentic on a blueprint G, breadth the vertexes represent the qubits initialized in the accompaniment of (left| rightrangle = (left| 0 rightrangle left| 1 rightrangle )/sqrt 2) and the edges represent a Controlled-Z (C-Z) operation, ({mathrm{CZ}}^{{ i,j} } = left| 0 rightrangle _ileftlangle 0 right| otimes {mathbb{I}}_j left| 1 rightrangle _ileftlangle 1 right| otimes Z_j), amid the two acquaintance qubits. Then the blueprint accompaniment can be accounting as,

$$left| G rightrangle = mathop {prod}limits_{(i,j) in E} {{mathrm{CZ}}^{{ i,j} }} left| rightrangle ^{ otimes N}.$$

(5)

Denote the Pauli operators on qubit i to be Xi, Yi, Zi. An N-partite blueprint accompaniment can additionally be abnormally bent by N absolute stabilizers,

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Graph Coloring in Graph Theory | Chromatic Number of … – Chromatic Number In Graph Coloring | Chromatic Number In Graph Coloring

$$S_i = X_imathop { otimes }limits_{j in N_i} Z_j,$$

(6)

which drive with anniversary added and Si|G〉 = |G〉, ∀i. That is, the blueprint accompaniment is the altered eigenstate with eigenvalue of 1 for all the N stabilizers. Here, Si contains character operators for all the qubits that do not arise in Eq. (6). As a result, a blueprint accompaniment can be accounting as a artefact of balance projectors,

$$left| G rightrangle leftlangle G right| = mathop {prod}limits_{i = 1}^N {frac{{S_i {mathbb {I}}}}{2}} .$$

(7)

The allegiance amid ρ and a blueprint accompaniment |G〉 can be acquired from barometer all accessible articles of stabilizers. However, as there are exponential agreement in Eq. (7), this action is about inefficient for ample systems. Hereafter, we accede the affiliated graph, back its agnate blueprint accompaniment is absolutely entangled.

Now, we adduce a analytic adjustment to ascertain affair structures based on blueprint states. First, we accord allegiance apprenticed amid adaptable states and blueprint states as the afterward proposition.

Given a blueprint accompaniment |G〉 and a allotment ({cal{P}} = { A_{i}}), the allegiance amid |G〉 and any (cal{P})-fully adaptable accompaniment is high belted by

$${mathrm{Tr}}left( {left| G rightrangle leftlangle G right|rho _f} right) le min _{{ A,bar A} }2^{ – S(rho _A)};$$

(8)

and the allegiance amid |G〉 and any (cal{P})-bi-separable accompaniment is high belted by

$${mathrm{Tr}}(left| G rightrangle leftlangle G right|rho _b) le max _{{ A,bar A} }2^{ – S(rho _A)},$$

(9)

where ({ A,bar A}) is a bipartition of {Ai}, and S(ρA) = −Tr[ρA log2ρA] is the von Neumann anarchy of the bargain body cast (rho _A = {mathrm{Tr}}_{bar A}(left| G rightrangle leftlangle G right|)).

The apprenticed in Eq. (9) is tight, i.e., there consistently exists a (cal{P})-bi-separable accompaniment to bathe it. The apprenticed in Eq. (8) may not be apprenticed for some allotment (cal{P} = { A_{it{i}}}) and some blueprint accompaniment |G〉. In addition, we acknowledgment that to extend Assumption 1 from the blueprint accompaniment to a accepted accompaniment |Ψ〉, one should acting the anarchy in the apprenticed of Eqs. (8) and (9) with the min-entropy S∞(ρA) = −logλ1 with λ1 the better eigenvalue of (rho _A = {mathrm{Tr}}_{bar A}(left| Psi rightrangle leftlangle Psi right|)).

Next, we adduce an able adjustment to lower-bound the allegiance amid an alien able accompaniment and the ambition blueprint state. A blueprint is k-colorable if one can bisect the acme set into k break subsets ({bigcup} {V_l} = V) such that any two vertexes in the aforementioned subset are not connected. The aboriginal cardinal k is alleged the bright cardinal of the graph. (Note that the colorability is a acreage of the blueprint (not the state), one may abate the cardinal of altitude settings by belted Clifford operations.38) We ascertain the balance projector of anniversary subset Vl as

$$P_l = mathop {prod}limits_{i in V_l} {frac{S_i {mathbb{I}}}{2}} ,$$

(10)

where Si is the balance of |G〉 in subset Vl. The apprehension amount of anniversary Pl can be acquired by one belted altitude ambience (mathop { otimes }nolimits_{i in V_l} X_imathop { otimes }nolimits_{j in V/V_l} Z_j). Then, we can adduce a allegiance appraisal arrangement with k belted altitude settings, as the afterward proposition.

For a blueprint accompaniment (left| G rightrangle leftlangle G right|) and the projectors Pldefined in Eq. ( 10 ), the afterward asperity holds,

$$left| G rightrangle leftlangle G right| ge mathop {sum}limits_{l = 1}^k {P_l} – (k – 1){mathbb{I}},$$

(11)

where A ≥ B indicates that (A − B) is absolute semidefinite.

Note that Hypothesis 2 with k = 2 case has additionally been advised in literature.34 Combining Propositions 1 and 2, we adduce entanglement-structure assemblage with k belted altitude settings, as presented in the afterward theorem.

Given a allotment ({cal{P}} = { A_{i}}), the abettor (W_{f}^{cal{P}}) can attestant non-(cal{P})-fully separability (entanglement),

File:Petersen graph 3-coloring.svg - Wikimedia Commons - Chromatic Number In Graph Coloring

File:Petersen graph 3-coloring.svg – Wikimedia Commons – Chromatic Number In Graph Coloring | Chromatic Number In Graph Coloring

$$W_{f}^{cal{P}} = left( {k – 1 min _{{ A,bar{A}} }2^{ – S(rho _A)}} right){mathbb{I}} – mathop {sum}limits_{l = 1}^{k} {P_{l}} ,$$

(12)

with (langle W_{f}^{cal{P}}rangle ge 0) for all (cal{P})-fully adaptable states; and the abettor (W_{b}^{cal{P}}) can attestant (cal{P})-genuine entanglement,

$$W_{b}^{cal{P}} = left( {k – 1 max _{{ A,bar A} }2^{ – S(rho _{A})}} right){Bbb I} – mathop {sum}limits_{l = 1}^{k} {P_{l}} ,$$

(13)

with (langle W_{b}^{cal{P}}rangle ge 0) for all (cal{P})-bi-separable states, breadth ({ A,bar A}) is a bipartition of {Ai}, (rho _{A} = {mathrm{Tr}}_{bar A}(left| G rightrangle leftlangle G right|)), and the projectors Plis authentic in Eq. (10).

The proposed entanglement-structure assemblage accept several favorable features. First, accustomed an basal blueprint state, the accomplishing of the assemblage is the aforementioned for altered partitions. This affection allows us to abstraction altered affair structures in one experiment. Note that the attestant operators in Eqs. (12) and (13) can be disconnected into two parts: The altitude after-effects of Pl acquired from the agreement await on the able alien accompaniment and are absolute of the partition; The bounds, (1 min {mkern 1mu} (max )_{{ A,bar A} }2^{ – S(rho _A)}), on the added hand, await on the allotment and are absolute of the experiment. Hence, in the abstracts postprocessing of the altitude after-effects of Pl, we can abstraction assorted affair structures for altered partitions by artful the agnate apprenticed analytically or numerically.

Second, besides investigating the affair anatomy amid all the subsystems, one can additionally administer the aforementioned beginning ambience to abstraction that of a subset of the subsystems, by assuming altered abstracts postprocessing. For example, accept the blueprint G is abstracted into three parts, say A1, A2, and A3, and alone the affair amid subsystems A1 and A2 is of interest. One can assemble new attestant operators with projectors (P_{l}^{prime}), by replacing all the Pauli operators on the qubits in A3 in Eq. (10) to identities. Such altitude after-effects can be acquired by processing the altitude after-effects of the aboriginal Pl. Then the affair amid A1 and A2 can be detected via Assumption 1 with projectors (P_{l}^{prime}) and the agnate apprenticed of the blueprint accompaniment (left| {G_{A_{1}A_{2}}} rightrangle). Details are discussed in Supplementary Note 1.

Third, back anniversary subsystem Ai contains alone one qubit, that is, m = N, the assemblage in Assumption 1 become the accepted ones. It turns out that Eq. (13) is the aforementioned for all the blueprint states beneath the N-partition ({cal{P}}_{N}), as apparent in the afterward corollary. Note that, a appropriate case of the corollary, the 18-carat affair attestant for the GHZ and 1-D arrangement states, has been advised in literature.34

The abettor (W_{b}^{{cal{P}}_{N}}) can attestant 18-carat multipartite entanglement,

$$W_{b}^{{cal{P}}_{N}} = left( {k – frac{1}{2}} right){mathbb {I}} – mathop {sum}limits_{l = 1}^{k} {P_{l}} ,$$

(14)

with (langle W_{b}^{{cal{P}}_{N}}rangle ge 0) for all bi-separable states, breadth Plis authentic in Eq. (10) for any blueprint state.

Fourth, the attestant in Eq. (12) can be artlessly continued to analyze non-m-separability, by investigating all accessible partitions ({cal{P}}_{m}) with anchored m. In fact, according to the analogue of m-separable states and Eq. (8), the allegiance amid any m-separable accompaniment ρm and the blueprint accompaniment |G〉 can be high belted by ({mathrm{max}}_{{cal{P}}_{m}}{mathrm{min}}_{{ A,bar{A}} }2^{ – S(rho _{A})}), breadth the access is over all accessible partitions with m subsystems. As a result, we accept the afterward assumption on the non-m-separability.

The abettor Wmcan attestant non-m-separability,

$$W_{m} = left( {k – 1 max _{{cal{P}}_{m}}min _{{ A,bar A} }2^{ – S(rho _{A})}} right){mathbb{I}} – mathop {sum}limits_{l = 1}^{k} {P_{l}} ,$$

(15)

with 〈Wm〉 ≥ 0 for all m-separable states, breadth the access takes over all accessible partitions ({cal{P}}_{m}) with m subsystems, the abuse takes over all bipartition of ({cal{P}}_{m}), (rho _A = {mathrm{Tr}}_{bar A}(left| G rightrangle leftlangle G right|)), and the projector Plis authentic in Eq. (10).

The robustness assay of the assemblage proposed in Theorems 1 and 2 beneath the white babble is presented in Methods. It shows that our entanglement-structure assemblage are absolutely able-bodied to noise. Moreover, the access problems in Theorems 1 and 2 are about hard, back there are exponentially abounding altered accessible partitions. Surprisingly, for several broadly acclimated types of blueprint states, such as 1-D, 2-D arrangement states, and the GHZ state, we acquisition the analytic solutions to the access problem, as apparent in the afterward section.

In this section, we administer the accepted affair apprehension adjustment proposed aloft to several archetypal blueprint states, 1-D, 2-D arrangement states, and the GHZ state. Note that for these states the agnate graphs are all 2-colorable. Thus, we can apprehend the assemblage with alone two belted altitude settings. For clearness, the vertexes in the subsets V1 and V2 are associated with red and dejected colors respectively, as apparent in Fig. 3. We address the balance projectors authentic in Eq. (10) for the two subsets as,

$$begin{array}{l}P_1 = mathop {prod}limits_{{mathrm{red}},i} {frac{{S_i {mathbb{I}}}}{2}} ,\ P_2 = mathop {prod}limits_{{mathrm{blue}},i} {frac{{S_i {mathbb{I}}}}{2}} .end{array}$$

(16)

The added accepted case with k-chromatic blueprint states is presented in Supplementary Note 1.

Graphs of the a 1-D arrangement accompaniment |C1〉, b 2-D arrangement accompaniment |C2〉, and c GHZ accompaniment |GHZ〉. Note that the blueprint accompaniment anatomy of the GHZ accompaniment is agnate to its approved form, ((left| 0 rightrangle ^{ otimes N} left| 1 rightrangle ^{ otimes N})/sqrt 2), up to belted unitary operations

We alpha with a 1-D arrangement accompaniment |C1〉 with balance projectors in the anatomy of Eq. (16). Accede an archetype of tripartition ({cal{P}}_{3} = { A_{1},A_{2},A_{3}}), as apparent in Fig. 3a, there are three means to bisect the three subsystems into two sets, i.e., ({ A,bar A}) = {A1, A2A3}, {A2, A1A3}, {A3, A1A2}. It is not adamantine to see that the agnate affair entropies are (S(rho _{A_{1}}) = S(rho _{A_{3}}) = 1) and (S(rho _{A_{2}}) = 2). Note that in the calculation, anniversary burst bend will accord 1 to the entropy, which is a apparent of the breadth law of affair entropy.44 According to Assumption 1, the operators to attestant ({cal{P}}_{3})-entanglement anatomy are accustomed by,

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$$begin{array}{l}W_{f,C_1}^{{cal{P}}_3} = frac{5}{4}{mathbb{I}} – (P_1 P_2),\ W_{b,C_1}^{{cal{P}}_3} = frac{3}{2}{mathbb{I}} – (P_1 P_2),end{array}$$

(17)

where the two projectors P1 and P2 are authentic in Eq. (16) with the blueprint of Fig. 3a.

Next, we booty an archetype of 2-D arrangement accompaniment |C2〉 authentic in a 5 × 5 filigree and accede a tripartition, as apparent in Fig. 3b. Similar to the 1-D arrangement accompaniment case with the breadth law, the agnate affair entropies are (S(rho _{A_{1}}) = S(rho _{A_{3}}) = 5) and (S(rho _{A_{2}}) = 4). According to Assumption 1, the operators to attestant ({cal{P}}_{3})-entanglement anatomy are accustomed by,

$$begin{array}{l}W_{f,C_2}^{{cal{P}}_3} = frac{{33}}{{32}}{mathbb{I}} – (P_1 P_2),\ W_{b,C_2}^{{cal{P}}_3} = frac{{17}}{{16}}{mathbb{I}} – (P_1 P_2),end{array}$$

(18)

where the two projectors P1 and P2 are authentic in Eq. (16) with the blueprint of Fig. 3b. Similar assay works for added partitions and added blueprint states.

Now, we accede the case breadth anniversary subsystem Ai contains absolutely one qubit, ({cal{P}}_{N}). Then, assemblage in Eq. (12) become the accepted ones, as apparent in the afterward Corollary.

The abettor (W_{f,C}^{{cal{P}}_{N}}) can attestant non-fully separability (entanglement),

$$W_{f,C}^{cal{P}_N} = (1 2^{ – leftlfloor {frac{N}{2}} rightrfloor }){Bbb I} – (P_1 P_2),$$

(19)

with (langle W_{f,C}^{{cal{P}}_{N}}rangle ge 0) for all absolutely adaptable states, breadth the two projectors P1and P2are authentic in Eq. (16) with the stabilizers of any 1-D or 2-D arrangement state.

Here, we alone appearance the cases of 1-D and 2-D arrangement states. We assumption that the attestant holds for any (such as 3-D) arrangement states. For a accepted blueprint state, on the added hand, the aftereffect does not hold. In fact, we accept a adverse archetype of the GHZ accompaniment apparent in Fig. 3c. It is not adamantine to see that for any GHZ state, the affair anarchy is accustomed by,

$$S(rho _A^{GHZ}) = 1,;;;forall { A, bar A} .$$

(20)

Then, Eqs. (12) and (13) crop the aforementioned witnesses. That is, the attestant complete by Assumption 1 for the GHZ accompaniment can alone acquaint 18-carat affair or not.

Following Assumption 2, one can fix the cardinal of the subsystems m and investigate all accessible partitions to ascertain the non-m-separability. The access botheration can be apparent analytically for the 1-D and 2-D arrangement states, as apparent in Aftereffect 3 and 4, respectively.

The abettor (W_{m,C_{mathrm{1}}}) can attestant non-m-separability,

$$W_{m,C_1} = (1 2^{ – leftlfloor {frac{m}{2}} rightrfloor }){mathbb{I}} – (P_1 P_2),$$

(21)

with (langle W_{m,C_1}rangle ge 0) for all m-separable states, breadth the two projectors P1and P2are authentic in Eq. (16) with the stabilizers of a 1-D arrangement state.

In particular, back m = 2 and m = N, (W_{m,C_{mathrm{1}}}) becomes the affair assemblage in Eqs. (14) and (19), respectively.

The abettor (W_{m,C_{mathrm{2}}}) can attestant non-m-separability for N ≥ m(m − 1)/2,

$$W_{m,C_2} = left( {1 2^{ – leftlceil {frac{{ – 1 sqrt {1 8(m – 1)} }}{2}} rightrceil }} right){mathbb{I}} – (P_1 P_2),$$

(22)

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with (langle W_{m,C_2}rangle ge 0) for all m-separable states, breadth the two projectors P1and P2are authentic in Eq. (16) with the stabilizers of a 2-D arrangement state.

We acknowledgment that the assemblage complete in Corollaries 1, 2, and 3 are tight. Booty the attestant (W_{m,C_{mathrm{1}}}) in Aftereffect 3 as an example. There exists an m-separable accompaniment ρm that saturates ({mathrm{Tr}}(rho _mW_{m,C_1}) = 0). In addition, as m ≤ 5, the attestant (W_{m,C_{mathrm{2}}}) in Aftereffect 4 is additionally tight. Detailed discussions are presented in Supplementary Methods 1–4.

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