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arxiv: 2404.09014 · v2 · pith:TJ5YINI5new · submitted 2024-04-13 · 🪐 quant-ph

Lower bounds on bipartite entanglement in noisy graph states

Aqil Sajjad , Eneet Kaur , Kenneth Goodenough , Don Towsley , Saikat Guha This is my paper

Pith reviewed 2026-05-24 02:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords graph statescoherent informationdepolarizing noisebipartite entanglementdistillable entanglementquantum networksnoisy intermediate-scale quantum
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The pith

Certain graph states keep strictly positive coherent information under any non-maximal depolarizing noise in the pre-edge model.

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

The authors develop a calculable method for the coherent information across bipartitions of graph states when each qubit experiences independent depolarizing noise before the CZ gates that add edges. They show that particular families of these states keep the coherent information strictly above zero for every noise strength short of total depolarization. A sympathetic reader would care because coherent information lower-bounds the rate of entanglement distillation, so these states could remain useful resources on noisy hardware. The work also records how adding nodes or edges changes the value of this lower bound across chosen cuts.

Core claim

The paper establishes a family of graph states for which the coherent information across a chosen bipartition stays strictly positive under the stated noise model for any depolarizing probability p less than one. The method derives an explicit expression for this quantity in terms of the graph structure and the noise parameter, allowing direct verification that the lower bound on distillable entanglement never reaches zero before the noise becomes maximal.

What carries the argument

Coherent information across a bipartition of the noisy graph state, evaluated after independent depolarizing channels act on the qubits before the CZ operations that define the edges.

If this is right

  • Adding certain nodes or edges can preserve or raise the coherent-information lower bound across the chosen cut.
  • The explicit calculation method lets researchers enumerate larger families of noise-robust graph states systematically.
  • These states remain candidate resources for entanglement distillation even when every qubit experiences strong but incomplete depolarizing noise.
  • The bipartite distillable entanglement rate stays bounded away from zero for the identified families at all finite noise levels.

Where Pith is reading between the lines

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

  • The result suggests that the ordering of noise and entanglement-generating gates can be exploited to protect lower bounds on distillable entanglement.
  • Similar pre-gate noise models might be analyzed for other resource states such as cluster states or repeater graphs.
  • Hardware experiments that apply depolarizing noise before controlled-phase gates could directly test whether the positive lower bound survives real-device imperfections.

Load-bearing premise

Depolarizing noise acts independently on each qubit before the CZ gates create the graph edges.

What would settle it

Prepare one of the claimed graph states on hardware or in simulation, apply the pre-CZ depolarizing noise at a strength p = 0.9, and measure a quantity that upper-bounds the coherent information; if the result is consistent with zero while the paper's formula predicts a positive value, the claim is falsified for that state.

Figures

Figures reproduced from arXiv: 2404.09014 by Aqil Sajjad, Don Towsley, Eneet Kaur, Kenneth Goodenough, Saikat Guha.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: 4. If we fix nB = 2, and increase nA with an equi￾table distribution, we obtain purely positive co￾herent information for sufficiently large nA for any non-maximal noise parameter P. Thus the family of graphs with purely positive coherent information (for non-maximal noise) is larger than just star graphs or rank 2 type one graphs with nB = 2. More on this in section V A where we compare type 1 and type 2,… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

Graph states are a key resource for a number of applications in quantum information theory. Due to the inherent noise in noisy intermediate-scale quantum (NISQ) era devices, it is important to understand the effects noise has on the usefulness of graph states. We consider a noise model where the initial qubits undergo depolarizing noise before the application of the CZ operations that generate edges between qubits situated at the nodes of the resulting graph state. For this model we develop a method for calculating the coherent information -- a lower bound on the rate at which entanglement can be distilled, across a bipartition of the graph state. We also identify some patterns on how adding more nodes or edges affects the bipartite distillable entanglement. As an application, we find a family of graph states that maintain a strictly positive coherent information for any amount of (non-maximal) depolarizing noise.

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

0 major / 2 minor

Summary. The manuscript develops a calculational method for the coherent information (a lower bound on distillable entanglement) across bipartitions of graph states under a noise model in which independent depolarizing noise acts on each qubit prior to the CZ gates that define the graph edges. It identifies patterns in how the addition of nodes or edges affects this quantity and applies the method to exhibit a family of graph states for which the coherent information remains strictly positive for every depolarizing strength p < 1.

Significance. If the method and the explicit calculations for the identified family are correct, the result supplies concrete evidence that certain graph states remain useful entanglement resources under arbitrarily strong (but non-maximal) noise of this form. The development of an explicit calculational procedure tailored to the pre-CZ depolarizing channel is itself a useful technical contribution that may extend to related noise models.

minor comments (2)
  1. The abstract states the existence of the family but does not name the specific graphs or the bipartition; a one-sentence description or reference to the relevant figure or theorem in the main text would improve readability.
  2. Notation for the coherent information and the noise parameter p should be introduced consistently in the first section where the noise model is defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper develops a calculational method for coherent information under the stated pre-CZ independent depolarizing noise model and applies it to identify a graph-state family with strictly positive coherent information for p < 1. No quoted equations or steps reduce a claimed prediction or lower bound to a fitted parameter, self-definition, or load-bearing self-citation chain. The central existence result follows directly from the method without the patterns of circularity enumerated in the instructions. This is the expected honest non-finding for a paper whose method is purpose-built for the noise model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions from quantum information theory for graph states, depolarizing channels, and coherent information; no free parameters, new axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard quantum information assumptions including the definition of the depolarizing channel and coherent information as a lower bound on distillable entanglement.
    Invoked to define the noise model and the quantity being bounded.

pith-pipeline@v0.9.0 · 5680 in / 1172 out tokens · 20677 ms · 2026-05-24T02:20:03.308682+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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    Therefore, nB = 1 maximizes IA

    Given nA, IA is a decreasing function of nB. Therefore, nB = 1 maximizes IA. This corre- sponds to a star graph where Bob has only one qubit connected to all of Alice’snA qubits

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    We show in Fig

    Maximizing coherent information IA, requires nB = 1 and a sufficiently large nA to reach the asymptotic limit. We show in Fig. 3 that this gives purely positive coherent information for any non-maximal value of P

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    This gives H(ρA) = f1(1, P) = 1, with coherent information IA = 1 − nH2(P)

    nA = 1 is a trivial case for which Alice’s reduced density matrix is the maximally mixed state on her qubit. This gives H(ρA) = f1(1, P) = 1, with coherent information IA = 1 − nH2(P)

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    When nA > nB, IA > IB and vice versa. That is, for any fully connected graph, the side hav- ing the largest number of qubits has the highest subsystem entropy (since f1(nA, P) > f1(nB, P) when nA > nB), hence the largest coherent in- formation. See Fig. 3 for details. Most importantly , we present numerical evidence in Fig. 4 that it is possible to tolera...

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    We can take the first row in set 1 and add it to all the other rows of the same set to turn them into zero rows

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    Beyond this, the behavior as we change n1, n2 and nB is as follows:

    The coherent information IA in this case is thus simply 2 − nH2(P). Beyond this, the behavior as we change n1, n2 and nB is as follows:

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    This property directly follows from the fact that the coherent information for such a case can be ex- pressed as the sum of the coherent informations of two star graphs

    For nB = 2, we obtain a purely positive coher- ent information for any non-maximal value of P by taking n1 and n2 to be sufficiently large. This property directly follows from the fact that the coherent information for such a case can be ex- pressed as the sum of the coherent informations of two star graphs

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    We call such a distribution of nA, nB an equitable or optimal distribution

    Given nA and nB, n1 = ⌊nA/2⌋ and n2 = ⌈nA/2⌉ maximizes IA over all values of n1 and n2 such that n1 + n2 = nA. We call such a distribution of nA, nB an equitable or optimal distribution. The last property easily follows from the behavior of [f1(n1, P) − n1H2(P)] and [f1(n2, P) − n2H2(p)]. As we found during the rank 1 discussion, [f1(n1, P) − n1H2(P) incr...

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    Secondly , these three vectors satisfy the property that their binary sum is the zero vector, i.e

    Type 2 subsystem: three types of rows In this type of subsystem, all row vectors of the bi- adjacency matrix GAB equal one of three distinct vec- tors V1, V2 and V3. Secondly , these three vectors satisfy the property that their binary sum is the zero vector, i.e. V1 + V2 + V3 = 0. Assuming that none of the qubits in the graph state are completely disconn...

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    Given n1, n2, and n3, IA is a decreasing function of nB

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    Given n1, n3, and nB, IA is an increasing function of n2

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    If nAmod3 = 2, n1 = (nA − 2)/3 + 1, n2 = (nA −2)/3 +1 and n3 = (nA −2)/3 gives the highest coherent information

    Keeping nB constant, and nA as the total num- ber of Alice’s qubits, ifnA is a multiple of 3, n1 = n2 = n3 = nA/3, gives the highest coherent infor- mation IA. If nAmod3 = 2, n1 = (nA − 2)/3 + 1, n2 = (nA −2)/3 +1 and n3 = (nA −2)/3 gives the highest coherent information. If nAmod3 = 1, n1 = ( nA − 1)/3 + 1, n2 = ( nA − 1)/3 and n3 = (nA − 1)/3 gives the ...

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    Thus the family of graphs with purely positive coherent information (for non-maximal noise) is larger than just star graphs or rank 2 type one graphs with nB = 2

    If we fix nB = 2, and increase nA with an equi- table distribution, we obtain purely positive co- herent information for sufficiently large nA for any non-maximal noise parameter P. Thus the family of graphs with purely positive coherent information (for non-maximal noise) is larger than just star graphs or rank 2 type one graphs with nB = 2. More on this...

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    Since either of the two subsystems can be of type one or two, there are four possible forms of rank 2 graphs

    Rank 2 graphs: considering both systems together Having described types 1 and 2 for the individual subsystems, we can now consider them together to cat- egorize graphs with rank 2 biadjacency matrices. Since either of the two subsystems can be of type one or two, there are four possible forms of rank 2 graphs. When both subsystems are type 1 Let us say Al...

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    The set of rows equal to V1 then constitute a rank 1 piece (as we can eliminate all but one of them through row additions), and the remaining rows constitute a rank 2 piece of type 2. That is, if we were to remove all the qubits in the graph corresponding to the rows equal to V1, then we would be left with a rank 2 type 2 subsystem for Alice. This way , t...

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    We can use Gaussian elimi- nation to eliminate all but one row within each set of identical rows

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    rank 1 graph, ifnA > nB, then IA > IB

    We have already shown in Section IV C that for any fully connected i.e. rank 1 graph, ifnA > nB, then IA > IB

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    Theory and Engineering of Large-Scale Dis- tributed Entanglement

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    The entries 16 of ⃗a give the initial qubit configuration associ- ated with integer d, with 0 (1) representing |+〉 (|−〉) for each qubit i

    Evaluate the binary representation of the integer d and store it in the binary vector ⃗a. The entries 16 of ⃗a give the initial qubit configuration associ- ated with integer d, with 0 (1) representing |+〉 (|−〉) for each qubit i

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    For this, first obtain the number of 1 entries in ai as ν− = νA−1X i=0 ai (A2) Then the probability of the configuration d is Pd = P νA−ν− (1 − P)ν− (A3)

    Calculate the probability Pd of the qubit config- uration associated with d. For this, first obtain the number of 1 entries in ai as ν− = νA−1X i=0 ai (A2) Then the probability of the configuration d is Pd = P νA−ν− (1 − P)ν− (A3)

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    Asign this value to the integer l

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    Return to the vectorw defined earlier. Add to the existing value of the l’th entry ofw the probabil- ity Pd of the qubit configuration d. That is, wl = wl + Pd (A5) This way , as we run the loop overd to go through all the qubit configurations, we iteratively add its probability contribution to wl, which stores the probabilities for the different bracket ...

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