pith. sign in

arxiv: 2211.01064 · v2 · submitted 2022-11-02 · 🪐 quant-ph

Localizing genuine multiparty entanglement in noisy stabilizer states

Harikrishnan K. J. , Amit Kumar Pal This is my paper

Pith reviewed 2026-05-24 10:34 UTC · model grok-4.3

classification 🪐 quant-ph
keywords genuine multiparty entanglementstabilizer statesgraph stateslocalizable entanglementPauli noisetoric codebiseparable statescritical noise strength
0
0 comments X

The pith

Stabilizer states allow polynomial lower bounds on localized genuine multiparty entanglement, with a critical noise threshold beyond which post-measurement states are biseparable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper calculates lower bounds on genuine multiparty entanglement localized over chosen subsystems of multi-qubit stabilizer states, both without noise and under single-qubit Pauli noise. It adopts a graph-based technique that works on arbitrary graph states as representatives of stabilizer states and shows the operations scale polynomially with system size. Demonstrations cover linear, ladder, and square graphs, and the method is applied to a toric code on a square lattice. In the noisy case the calculations reveal a critical noise strength for a specific measurement setup, above which the post-measured states become biseparable. This provides a concrete way to bound entanglement in large noisy multiparty states that would otherwise be hard to characterize.

Core claim

For noiseless stabilizer states the lower bounds on localizable genuine multiparty entanglement are obtained via graph operations on representative graph states, with polynomial scaling in system size; when single-qubit Markovian or non-Markovian Pauli noise acts on all qubits, a specific lower bound corresponding to a chosen Pauli measurement setup vanishes above a critical noise strength, after which all post-measured states are biseparable; the same critical-noise feature appears for a toric code defined on a square lattice.

What carries the argument

Graph operations performed on graph states that represent stabilizer states via local unitary equivalence, together with Pauli measurements that localize the entanglement over a chosen multiparty subsystem.

If this is right

  • The polynomial scaling permits explicit calculation of the bounds for large linear, ladder, square, and toric-code graphs.
  • The local unitary link between stabilizer states and graph states extends the same bounds and critical-noise result to arbitrary large stabilizer states under the same noise.
  • The critical noise strength marks the point at which the specific lower bound reaches zero and all post-measured states are biseparable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The existence of the threshold suggests that protocols relying on localized genuine multiparty entanglement from stabilizer resources have a sharply defined noise tolerance.
  • Similar thresholds may appear for other noise channels or measurement choices once the same graph technique is applied.
  • The method supplies a practical diagnostic for when noise has destroyed the resource character of a stabilizer state for multiparty tasks.

Load-bearing premise

The chosen specific Pauli measurement setup yields a representative lower bound that continues to hold under noise, and arbitrary graph states cover all stabilizer states through local unitaries.

What would settle it

Direct computation or tomography of the post-measurement states at noise strengths above the reported critical value that finds genuine multiparty entanglement remaining in any of those states.

Figures

Figures reproduced from arXiv: 2211.01064 by Amit Kumar Pal, Harikrishnan K. J..

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Subsystems [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Transformation of a graph as per the discussion [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Connected subsystem of qubits situated at [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) A plaquette of four qubits as the chosen subsystem [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) A set of 2 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Connected subsystem of qubits of size [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The PMS used for the lower bound calculation in noisy (a) linear (boundary, bulk), (b) ladder (boundary, bulk) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Variations of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Variations of [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. A toric code of 18 qubits on a square lattice, [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Transformation of a stabiliser state [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Variations of [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. In (a)-(c), we depict variations of [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Variations of [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Attributes for the nodes of a graph according [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. (a) Local complementation operation on the [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Transformation of the graph [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Variation of the total number of graph opera [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Schematic representation of the class-structure [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. (a) Pauli measurement setup to create a di [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
read the original abstract

Characterizing large noisy multiparty quantum states using genuine multiparty entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multiparty entanglement localized over a chosen multiparty subsystem of multi-qubit stabilizer states in the noiseless and noisy scenario. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states, and show that the graph operations required for the calculation has a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multiparty entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits, and demonstrate, for a specific lower bound of the localizable genuine multiparty entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the post measured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice, and computing a lower bound of localizable genuine multiparty entanglement over a non-trivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also, and discuss its interesting features.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper develops a graph-theoretic method to compute lower bounds on genuine multiparty entanglement (GME) localized to chosen subsystems of multi-qubit stabilizer states. In the noiseless case it shows that the required graph operations scale polynomially with system size and evaluates the bounds explicitly on linear, ladder and square graphs. Under single-qubit Pauli noise (Markovian or non-Markovian) it identifies, for a specific measurement choice, a critical noise strength at which the computed lower bound reaches zero and asserts that all post-measurement states are then biseparable; the same construction is applied to a toric-code loop via local-unitary equivalence to graph states.

Significance. If the central claims hold, the work supplies a concrete, polynomially scalable witness for localized GME in large noisy stabilizer states and supplies explicit noise thresholds for two families of states. The polynomial scaling and the toric-code demonstration are genuine strengths that could be useful for characterizing entanglement in fault-tolerant architectures.

major comments (1)
  1. [Abstract; noisy-graph-states section] Abstract and the section on noisy graph states: the claim that a vanishing lower bound on localizable GME implies that 'all of the post measured states are biseparable' is not supported by any additional argument. A lower bound of zero only shows that the chosen witness no longer detects GME; it does not establish biseparability. An independent upper bound, a direct factorization argument, or an alternative monotone is required to convert the numerical observation into the stated conclusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this important point of clarification. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract; noisy-graph-states section] Abstract and the section on noisy graph states: the claim that a vanishing lower bound on localizable GME implies that 'all of the post measured states are biseparable' is not supported by any additional argument. A lower bound of zero only shows that the chosen witness no longer detects GME; it does not establish biseparability. An independent upper bound, a direct factorization argument, or an alternative monotone is required to convert the numerical observation into the stated conclusion.

    Authors: We agree with the referee that a vanishing lower bound does not by itself establish biseparability of the post-measurement states. Our original wording in the abstract and the noisy-graph-states section overstated the conclusion. We will revise both locations to state that the critical noise strength is the point at which the specific lower-bound witness reaches zero (i.e., the witness no longer detects genuine multiparty entanglement). We will remove the claim that all post-measurement states are biseparable and, if length permits, add a short remark noting that a rigorous proof of biseparability would require an independent argument such as an upper bound or factorization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in graph operations and noise models

full rationale

The paper derives lower bounds on localizable genuine multiparty entanglement via explicit graph operations (vertex/edge deletions, local complementations) on graph states, which serve as LU-equivalent representatives of stabilizer states. These operations scale polynomially and are applied directly to linear, ladder, and square graphs without fitting parameters or renaming known results. The noisy-case extension applies single-qubit Pauli channels to all qubits and recomputes the same witness for specific measurement setups; the critical noise point is defined as the value where this particular lower bound reaches zero. No self-citation chain, ansatz smuggling, or uniqueness theorem imported from prior author work is invoked to force the result. The central claim therefore reduces to direct computation on the chosen representatives rather than to any input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5780 in / 1023 out tokens · 20653 ms · 2026-05-24T10:34:48.178815+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

140 extracted references · 140 canonical work pages · 3 internal anchors

  1. [1]

    Obtaining the reduced graph We now discuss the graph transformationG → G′ α for a specific choice of the PMS α, which is a key ingredient of the protocol for computing EP S over an arbitrary S in an arbitrary G, as discussed in Sec. III B 1. Starting from ρ corresponding to an ar- bitrary graph G, we achieve this in two steps, Step A and Step B. In Step A,...

    work page

  2. [2]

    The sizes of these sets are given by |Ddw| and |Lcw|, respectively

    obtain the set Ddw of all nodes with s = d, f = w (i.e., and nodes), and the set Lcw of all pairs (i, j) of nodes with s = c, f = w, (i.e., ( , ), ( , ), ( , ), and ( , ) links) such that the link (i, j) exists. The sizes of these sets are given by |Ddw| and |Lcw|, respectively. (a) while |Lcw| > 0 or |Ddw| > 0, do (I) operate B1 on all nodes i ∈ D dw (II...

    work page

  3. [3]

    Let us denote the set of these nodes i ∈ S′ as NS

    obtain the set EB of all boundary links (i, j) having node i ∈ S′, and node j ∈ S, such that i is either of , nodes. Let us denote the set of these nodes i ∈ S′ as NS. (a) for all (i, j) ∈ E B such that i ∈ N S, do (I) if j has s = c, apply B2 on (i, j) else first apply B1 to node j, and then to node i

    work page

  4. [4]

    extract the unitary operation V from the node attributes

    work page

  5. [5]

    output (1) The reduced graph G′ α, and (2) the uni- tary V in terms of the set of unitaries {Vi, i ∈ G′ α}

    replace all nodes to converting it to a normal graph, and extract the connectivity of G′ α. output (1) The reduced graph G′ α, and (2) the uni- tary V in terms of the set of unitaries {Vi, i ∈ G′ α}. Implementation of the algorithm presented in the pseudo code upto step 2 ensures the following for ρ′ α. (a) The neighborhood S′ 1 of S has at least one node...

    work page

  6. [6]

    deleting all the edges between the node i and its neighbours Ni, 24

    work page

  7. [7]

    replacing the node i with ( ) if the outcome is +1 ( −1), and

    work page

  8. [8]

    performing F sg Ni if the outcome is −1. If one now measures σ3 on a node i ∈ { , } belong- ing to S′ 2, due to the absence of connection between the node and the rest of the graph, the measure- ment leaves the graph unaltered, and the measure- ment outcome is +1 ( −1) for ( ) with certainty

    work page

  9. [9]

    p3S6vtT5I9zvgrgQiKUBevrqAhI=

    Scaling with system-size We now explore how the protocol discussed in Sec. III B 1 and Appendix A scales with the number of measured nodes |S′| = N − n = m in a graph G, which is given by the dependence of the total number of graph operations, C, performed during the proto- col on m. Since the graph G changes during each step of the protocol, it is in gen...

    work page

  10. [10]

    We can therefore regroup the outcome-index k as k ≡ lm with l ≡ l1l2

    by l (m), such that lj = kj (mj = kj) if j ∈ S′ 1 (S′ 2). We can therefore regroup the outcome-index k as k ≡ lm with l ≡ l1l2 . . . lL, and m ≡ mL+1mL+2 . . . mN, where we have as- sumed, without any loss in generality, that L is the size of S′ 1 such that S′ 2 has N − n − L qubits. In this notation, M α k ≡ M α lm, and using Eq. (23) in Eq. (21), we obt...

    work page

  11. [11]

    The graph transformation algorithm discussed in Appendix A ensures that the measure- ment on any of the qubits in S′ 2 can be either a σ3, or a σ1, i.e., αj = 1, 3 if j ∈ S′

    Note that Vα can change only the measurement basis, not the measure- ment outcome. The graph transformation algorithm discussed in Appendix A ensures that the measure- ment on any of the qubits in S′ 2 can be either a σ3, or a σ1, i.e., αj = 1, 3 if j ∈ S′

    work page

  12. [12]

    Now, application of M α lm on the graph state |G′ α⟩ results in the normal- 25 ized state of the form M α lm|G′ α⟩ ⟨G′ α| M α† lm = Wlm [ ( ⊗j∈S′ 1 |lj⟩ ⟨lj| ) ⊗ ( ⊗j∈S′ 2 |mj⟩ ⟨mj| ) ⊗ (|Gα S⟩ ⟨Gα S|) ] W† lm, (B2) where Gα S is the subgraph on S in G′′ α, which is ob- tained by performing the graph transformations cor- responding to the single-qubit Pau...

    work page

  13. [13]

    Let us now consider a bipartition of the system V as A ∪ B = V , and A ∩ B = ∅, where dA = dim(HA) (dB = HB), HA (HB) being the Hilbert space asso- ciated to A (B)

    Bipartite entanglement Let us consider a system of qubits V ≡ {1, 2, · · · , N} forming a connected graph G. Let us now consider a bipartition of the system V as A ∪ B = V , and A ∩ B = ∅, where dA = dim(HA) (dB = HB), HA (HB) being the Hilbert space asso- ciated to A (B). For each pure state |G⟩ in HA ⊗HB, orthonormal basis {bi A} ∈ H A and {bj B} ∈ H B ...

    work page

  14. [14]

    A multi-qubit quantum state is gen- uinely multiparty entangled if it is not separable in any possible bipartition

    Multipartite entanglement We now discuss the multiparty entanglement in graph states. A multi-qubit quantum state is gen- uinely multiparty entangled if it is not separable in any possible bipartition. Connected pure graph states are known to be genuinely multiparty entan- gled [48], where the degree of entanglement can be quantified using the Schmidt meas...

    work page

  15. [15]

    Note further that an application of Eq

    A technical account of the prescription for calculating ˜ρS is given in the Appendix F for the interested readers. Note further that an application of Eq. (D4) on ˜ρS leads to another GD state on S. To determine this, we (a) first obtain the GD state resulting from the application of Eq. (D4) on |ψ⟩ ⟨ψ| ≡ | 0⟩ ⟨0| = |Gα S⟩ ⟨Gα S| (using Eq. (40)) as ∑ (S,s...

    work page

  16. [16]

    Since Uj = Vj = I corresponding to qubits j = 1, 2 in the case depicted in Fig

    When no outcomes are forbidden Let us consider σ3 measurements performed on qubits 1 and 2 constituting S′. Since Uj = Vj = I corresponding to qubits j = 1, 2 in the case depicted in Fig. 2(a), the reduced graph is equivalent to the original graph, and s′′ = s′ = s in Eqs. (41)-(47). The change l1 → l′ 1 and m2 → m′ 2 in Eq. (47) due to the values of s, a...

    work page

  17. [17]

    Since both qubits in S′′ 1 = {1, 2} are connected to the two qubits in S, the normalized state ˜ρS (see Eq. (D5)) corresponding to outcome k = (+1)(+1) ≡ 00, occurring with a prob- ability 1/4, can be written in a row vector notation as ∑ ψ λψ |ψ⟩ ⟨ψ| ≡ [ 1 − q + q2 2 , 0, 0, q − q2 2 ] (E1) where λψ is the value in the position ψ of the row. Similarly, e...

    work page

  18. [18]

    sHlBn3nzuTgUhUT0HUHU6PWCiko=

    When forbidden set of outcomes is present For α ∈ Γ, there exists a set of outcomes that are forbidden (see Sec. III B 2). In such cases, re- sults discussed in the previous section hold except for the calculation of ˜ ρS, and the explicit calcula- tion depends on the fact that the change lm → l′m′ may result in a transition between allowed and for- bidde...

    work page

  19. [19]

    the subsystem A, and ||A|| = Tr [√ A†A ] for the matrix A

    Negativity as a bipartite measure for a graph-diagonal state For a graph-diagonal state ρ representing a graph with two partitions A and B, the negativity between A and B is defined as [99] NAB = ||ρTA || − 1 2 , (G1) where TA denotes the transposition of ρ w.r.t. the subsystem A, and ||A|| = Tr [√ A†A ] for the matrix A

    work page

  20. [20]

    PBHRt8W9v2YgwMQb2Yjv6GOIEFA=

    Genuine multiparty entanglement in graph diagonal states Determination of the multiparty entanglement measures is difficult in the case of mixed states. How- ever, in the case of specific types of mixed states, such as the graph diagonal states of certain types or number of parties, criteria for the state to be gen- uine multiparty entangled, or biseparable,...

    work page

  21. [21]

    Genuine multiparty concurrence We also point out that the density matrix of a GHZD state has the form of an X state. For such states, one may compute the genuine multiparty con- currence [95](GMC) as a multiparty entanglement measure for X-states, which is defined as CGM = 2 maxj[0, λ] with λ = ⏐⏐ρ(j,2N−j−1) ⏐⏐ − 2N−1−1∑ i=1,i̸=j [ ρ(1+i,1+i)ρ(2N−i,2N−i) ]...

    work page

  22. [22]

    Quantum en- tanglement,

    Ryszard Horodecki, Pawe l Horodecki, Micha l Horodecki, and Karol Horodecki, “Quantum en- tanglement,” Rev. Mod. Phys. 81, 865–942 (2009)

    work page 2009

  23. [23]

    Entanglement de- tection,

    Otfried G¨ uhne and G´ eza T´ oth, “Entanglement de- tection,” Phys. Rep. 474, 1–75 (2009)

    work page 2009

  24. [24]

    A one- way quantum computer,

    Robert Raussendorf and Hans J. Briegel, “A one- way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001)

    work page 2001

  25. [25]

    Experimental one-way quantum computing,

    P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005)

    work page 2005

  26. [26]

    Measurement-based quantum computation,

    H. J. Briegel, D. E. Browne, W. D¨ ur, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nat. Phys. 5, 19 (2009)

    work page 2009

  27. [27]

    Quantum cryptography based on bell’s theorem,

    Artur K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991)

    work page 1991

  28. [28]

    Quantum entanglement for secret sharing and secret splitting,

    Anders Karlsson, Masato Koashi, and Nobuyuki Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162–168 (1999)

    work page 1999

  29. [29]

    Quantum cryptography with entangled photons,

    Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and Anton Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000)

    work page 2000

  30. [30]

    Entangled state quantum cryptography: Eavesdropping on the ek- ert protocol,

    D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat, “Entangled state quantum cryptography: Eavesdropping on the ek- ert protocol,” Phys. Rev. Lett. 84, 4733–4736 (2000)

    work page 2000

  31. [31]

    Quantum cryptography,

    Nicolas Gisin, Gr´ egoire Ribordy, Wolfgang Tittel, and Hugo Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002)

    work page 2002

  32. [32]

    Com- munication via one- and two-particle operators on einstein-podolsky-rosen states,

    Charles H. Bennett and Stephen J. Wiesner, “Com- munication via one- and two-particle operators on einstein-podolsky-rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992)

    work page 1992

  33. [33]

    Dense coding in experimen- tal quantum communication,

    Klaus Mattle, Harald Weinfurter, Paul G. Kwiat, and Anton Zeilinger, “Dense coding in experimen- tal quantum communication,” Phys. Rev. Lett. 76, 4656–4659 (1996)

    work page 1996

  34. [34]

    Quantum Advantage in Communication Networks

    Aditi Sen(De) and Ujjwal Sen, “Quantum advan- tage in communication networks,” Phys. News 40, 17–32 (2011), arXiv:1105.2412

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  35. [35]

    Distributed quantum dense coding,

    D. Bruß, G. M. D’Ariano, M. Lewenstein, C. Mac- chiavello, A. Sen(De), and U. Sen, “Distributed quantum dense coding,” Phys. Rev. Lett. 93, 210501 (2004)

    work page 2004

  36. [36]

    Dense coding with multipartite quantum states,

    Dagmer Bruß, Maciej Lewenstein, Aditi Sen(De), Ujjwal Sen, Giacomo Mauro D’Ariano, and Chiara Macchiavello, “Dense coding with multipartite quantum states,” International Journal of Quan- tum Information 04, 415–428 (2006)

    work page 2006

  37. [37]

    Multipartite dense coding versus quantum correlation: Noise inverts relative capa- bility of information transfer,

    Tamoghna Das, R. Prabhu, Aditi Sen(De), and Ujjwal Sen, “Multipartite dense coding versus quantum correlation: Noise inverts relative capa- bility of information transfer,” Phys. Rev. A 90, 022319 (2014)

    work page 2014

  38. [38]

    Distributed quantum dense coding with two receivers in noisy environments,

    Tamoghna Das, R. Prabhu, Aditi Sen(De), and Ujjwal Sen, “Distributed quantum dense coding with two receivers in noisy environments,” Phys. Rev. A 92, 052330 (2015)

    work page 2015

  39. [39]

    Entanglement in many-body sys- tems,

    Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral, “Entanglement in many-body sys- tems,” Rev. Mod. Phys. 80, 517–576 (2008)

    work page 2008

  40. [40]

    Genuine quantum correlations in quantum many-body sys- tems: a review of recent progress,

    Gabriele De Chiara and Anna Sanpera, “Genuine quantum correlations in quantum many-body sys- tems: a review of recent progress,” Reports on Progress in Physics 81, 074002 (2018)

    work page 2018

  41. [41]

    Quantum entan- glement in photosynthetic light-harvesting com- plexes,

    Mohan Sarovar, Akihito Ishizaki, Graham R. Flem- ing, and K. Birgitta Whaley, “Quantum entan- glement in photosynthetic light-harvesting com- plexes,” Nat. Phys. 6, 462 (2010)

    work page 2010

  42. [42]

    Multi- partite quantum entanglement evolution in photo- synthetic complexes,

    Jing Zhu, Sabre Kais, Al ˜A¡n Aspuru-Guzik, Sam Rodriques, Ben Brock, and Peter J. Love, “Multi- partite quantum entanglement evolution in photo- synthetic complexes,” J. Chem. Phys. 137, 074112 (2012)

    work page 2012

  43. [43]

    Quantum biology,

    Neill Lambert, Yueh-Nan Chen, Yuan Chung Cheng, Che-Ming Li, Guang Yin Chen, and Franco Nori, “Quantum biology,” Nat. Phys. 9, 10–18 (2013)

    work page 2013

  44. [44]

    Multiboundary wormholes and holographic en- tanglement,

    Vijay Balasubramanian, Patrick Hayden, Alexan- der Maloney, Donald Marolf, and Simon F Ross, “Multiboundary wormholes and holographic en- tanglement,” Classical and Quantum Gravity 31, 185015 (2014)

    work page 2014

  45. [45]

    The AdS/CFT correspon- 33 dence,

    Veronika E Hubeny, “The AdS/CFT correspon- 33 dence,” Classical Quant. Grav. 32, 124010 (2015)

    work page 2015

  46. [46]

    Holographic quan- tum error-correcting codes: toy models for the bulk/boundary correspondence,

    Fernando Pastawski, Beni Yoshida, Daniel Har- low, and John Preskill, “Holographic quan- tum error-correcting codes: toy models for the bulk/boundary correspondence,” J. High. Energy Phys. 2015, 149 (2015)

    work page 2015

  47. [47]

    Bulk locality and quantum error correction in ads/cft,

    Ahmed Almheiri, Xi Dong, and Daniel Harlow, “Bulk locality and quantum error correction in ads/cft,” J. High Energy Phys. 4, 163 (2015)

    work page 2015

  48. [48]

    Holography and criticality in matchgate tensor networks,

    Alexander Jahn, Marek Gluza, Fernando Pastawski, and Jens Eisert, “Holography and criticality in matchgate tensor networks,” Science advances 5, eaaw0092 (2019)

    work page 2019

  49. [49]

    Multipartite entanglement and topology in holography,

    Jonathan Harper, “Multipartite entanglement and topology in holography,” Journal of High Energy Physics 2021, 116 (2021)

    work page 2021

  50. [50]

    Creation of a six- atom ‘schr¨ odinger cat’state,

    D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Re- ichle, and D. J. Wineland, “Creation of a six- atom ‘schr¨ odinger cat’state,” Nature438, 639–642 (2005)

    work page 2005

  51. [51]

    14-qubit en- tanglement: Creation and coherence,

    Thomas Monz, Philipp Schindler, Julio T. Bar- reiro, Michael Chwalla, Daniel Nigg, William A. Coish, Maximilian Harlander, Wolfgang H¨ ansel, Markus Hennrich, and Rainer Blatt, “14-qubit en- tanglement: Creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011)

    work page 2011

  52. [52]

    Controlled collisions for multi-particle entangle- ment of optically trapped atoms,

    Olaf Mandel, Markus Greiner, Artur Widera, Tim Rom, Theodor W. Hansch, and Immanuel Bloch, “Controlled collisions for multi-particle entangle- ment of optically trapped atoms,” Nature 425, 937–940 (2003)

    work page 2003

  53. [53]

    Exploring quantum matter with ultracold atoms in optical lattices,

    Immanuel Bloch, “Exploring quantum matter with ultracold atoms in optical lattices,” J. Phys. B: At. Mol. Opt. Phys. 38, S629–S643 (2005)

    work page 2005

  54. [54]

    Quantum information processing in opti- cal lattices and magnetic microtraps,

    P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Man- del, A. Widera, T. Rom, I. Bloch, and TheodorW. H¨ ansch, “Quantum information processing in opti- cal lattices and magnetic microtraps,” Fortschritte der Physik 54, 702–718 (2006)

    work page 2006

  55. [55]

    Spatial entanglement of bosons in optical lattices,

    M. Cramer, A. Bernard, N. Fabbri, L. Fallani, C. Fort, S. Rosi, F. Caruso, M. Inguscio, and M. B. Plenio, “Spatial entanglement of bosons in optical lattices,” Nature Communications 4, 2161 (2013)

    work page 2013

  56. [56]

    Experimental realization of dicke states of up to six qubits for multiparty quantum networking,

    R. Prevedel, G. Cronenberg, M. S. Tame, M. Pa- ternostro, P. Walther, M. S. Kim, and A. Zeilinger, “Experimental realization of dicke states of up to six qubits for multiparty quantum networking,” Phys. Rev. Lett. 103, 020503 (2009)

    work page 2009

  57. [57]

    Experimental demonstration of a hyper- entangled ten-qubit schr¨ odinger cat state,

    Wei-Bo Gao, Chao-Yang Lu, Xing-Can Yao, Ping Xu, Otfried G¨ uhne, Alexander Goebel, Yu-Ao Chen, Cheng-Zhi Peng, Zeng-Bing Chen, and Jian- Wei Pan, “Experimental demonstration of a hyper- entangled ten-qubit schr¨ odinger cat state,” Nature Physics 6, 331–335 (2010)

    work page 2010

  58. [58]

    Observation of eight-photon entanglement,

    Xing-Can Yao, Tian-Xiong Wang, Ping Xu, He Lu, Ge-Sheng Pan, Xiao-Hui Bao, Cheng-Zhi Peng, Chao-Yang Lu, Yu-Ao Chen, and Jian-Wei Pan, “Observation of eight-photon entanglement,” Na- ture Photonics 6, 225–228 (2012)

    work page 2012

  59. [59]

    18-qubit entanglement with six photons’ three degrees of freedom,

    Xi-Lin Wang, Yi-Han Luo, He-Liang Huang, Ming- Cheng Chen, Zu-En Su, Chang Liu, Chao Chen, Wei Li, Yu-Qiang Fang, Xiao Jiang, Jun Zhang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian- Wei Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018)

    work page 2018

  60. [60]

    Bench- marking quantum control methods on a 12-qubit system,

    C. Negrevergne, T. S. Mahesh, C. A. Ryan, M. Ditty, F. Cyr-Racine, W. Power, N. Boulant, T. Havel, D. G. Cory, and R. Laflamme, “Bench- marking quantum control methods on a 12-qubit system,” Phys. Rev. Lett. 96, 170501 (2006)

    work page 2006

  61. [61]

    Superconducting quantum circuits at the surface code threshold for fault tolerance,

    R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,...

    work page 2014

  62. [62]

    M. A. Nielsen and I. L. Chuang, Quantum Compu- tation and Quantum Information (Cambridge Uni- versity Press, 2010)

    work page 2010

  63. [63]

    Entangle- ment of multiparty-stabilizer, symmetric, and anti- symmetric states,

    Masahito Hayashi, Damian Markham, Mio Murao, Masaki Owari, and Shashank Virmani, “Entangle- ment of multiparty-stabilizer, symmetric, and anti- symmetric states,” Phys. Rev. A77, 012104 (2008)

    work page 2008

  64. [64]

    Entanglement in graph states and its applications,

    M Hein, W D¨ ur, Jens Eisert, Robert Raussendorf, M Van den Nest, and H J. Briegel, “Entanglement in graph states and its applications,” arXiv:quant- ph/0602096 (2006)

    work page arXiv 2006

  65. [65]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’s theorem, quantum theory and conceptions of the universe (Kluwer, Netherlands, 1989)

    work page 1989

  66. [66]

    Multi- particle generalization of entanglement swapping,

    S. Bose, V. Vedral, and P. L. Knight, “Multi- particle generalization of entanglement swapping,” Phys. Rev. A 57, 822–829 (1998)

    work page 1998

  67. [67]

    Quantum secret sharing,

    Mark Hillery, Vladim´ ır Buˇ zek, and Andr´ e Berthi- aume, “Quantum secret sharing,” Phys. Rev. A59, 1829–1834 (1999)

    work page 1999

  68. [68]

    Quantum-enhanced measurements: Beating the standard quantum limit,

    Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004)

    work page 2004

  69. [69]

    Multiparty entanglement in graph states,

    M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004)

    work page 2004

  70. [70]

    Multipartite state generation in quantum net- works with optimal scaling,

    J. Walln¨ ofer, A. Pirker, M. Zwerger, and W. D¨ ur, “Multipartite state generation in quantum net- works with optimal scaling,” Scientific Reports 9, 314 (2019)

    work page 2019

  71. [71]

    Gottesman, Stabilizer Codes and Quantum Er- ror Correction (Ph.D Thesis, CalTech, Pasadena, 1997)

    D. Gottesman, Stabilizer Codes and Quantum Er- ror Correction (Ph.D Thesis, CalTech, Pasadena, 1997)

    work page 1997

  72. [72]

    An introduction to quantum error correction and fault-tolerant quantum com- putation,

    Daniel Gottesman, “An introduction to quantum error correction and fault-tolerant quantum com- putation,” in Quantum information science and its contributions to mathematics, Proceedings of Sym- posia in Applied Mathematics , Vol. 68 (2010) pp. 13–58

    work page 2010

  73. [73]

    Mul- tipartite secure state distribution,

    W. D¨ ur, J. Calsamiglia, and H.-J. Briegel, “Mul- tipartite secure state distribution,” Phys. Rev. A 34 71, 042336 (2005)

    work page 2005

  74. [74]

    Multi-partite quantum cryptographic protocols with noisy ghz states,

    Kai Chen and Hoi-Kwong Lo, “Multi-partite quantum cryptographic protocols with noisy ghz states,” Quantum Info. Comput.7, 689–715 (2007)

    work page 2007

  75. [75]

    Esti- mating purity and entropy in stabilizer state ex- periments,

    Harald Wunderlich and Martin B. Plenio, “Esti- mating purity and entropy in stabilizer state ex- periments,” International Journal of Quantum In- formation 08, 325–335 (2010)

    work page 2010

  76. [76]

    Quantum computations on a topologi- cally encoded qubit,

    D. Nigg, M. M¨ uller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, “Quantum computations on a topologi- cally encoded qubit,” Science 345, 302–305 (2014)

    work page 2014

  77. [77]

    Experimental demonstration of a graph state quantum error-correction code,

    B. A. Bell, D. A. Herrera-Mart´ ı, M. S. Tame, D. Markham, W. J. Wadsworth, and J. G. Rar- ity, “Experimental demonstration of a graph state quantum error-correction code,” Nature Communi- cations 5, 3658 (2014)

    work page 2014

  78. [78]

    State preservation by repetitive error detection in a su- perconducting quantum circuit,

    J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mu- tus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wen- ner, A. N. Cleland, and John M. Martinis, “State preservation by repetitive error detection in a su- perconducting q...

    work page 2015

  79. [79]

    Practical and efficient experimental characteriza- tion of multiqubit stabilizer states,

    Chiara Greganti, Marie-Christine Roehsner, Ste- fanie Barz, Mordecai Waegell, and Philip Walther, “Practical and efficient experimental characteriza- tion of multiqubit stabilizer states,” Phys. Rev. A 91, 022325 (2015)

    work page 2015

  80. [80]

    Fault-tolerant quantum error detection,

    Norbert M. Linke, Mauricio Gutierrez, Kevin A. Landsman, Caroline Figgatt, Shantanu Debnath, Kenneth R. Brown, and Christopher Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, 10 (2017)

    work page 2017

Showing first 80 references.