Entanglement detection under coherent noise: Greenberger-Horne-Zeilinger-like states
You Zhou
quant-ph

APS/123-QED




             

zyqphy@gmail.com
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China


Entanglement is an essential resource in many quantum information tasks and entanglement witness is a widely used tool for its detection.
In experiments the prepared state generally deviates from the target state due to some noise. Normally the white noise model is applied to quantifying such derivation and in the same time reveals the robustness of the witness. However, there may exist
other kind of noise, in which the coherent noise can dramatically “rotate" the prepared state. In this way, the coherent noise is likely to lead to a failure of the detection, even though the underlying state is actually entangled. In this work, we propose an efficient entanglement detection protocol for N-partite Greenberger-Horne-Zeilinger (GHZ)-like states. The protocol can eliminate the effect of the coherent noise and in the same time feedback the corresponding noise parameters, which are beneficial to further improvements on the experiment system. In particular, we consider two experiment-relevant coherent noise models, one is from the unconscious phase accumulation on N qubits, the other is from the rotation on the control qubit. The protocol effectively realizes a family of entanglement witnesses by postprocessing the measurement results from N+2 local measurement settings, which only adds one more setting than the original witness specialized for the GHZ state.
Moreover, by considering the trade-off between the detection efficiency and the white-noise robustness, we further reduce the number of local measurements to 3 without altering the performance on the coherent noise.
Our protocol can enhance the entanglement detection under coherent noises and act as a benchmark for the state-of-the-art quantum devices.




















                             

                              
Entanglement detection under coherent noise: Greenberger-Horne-Zeilinger-like states

    You Zhou
    December 30, 2023
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§ INTRODUCTION

Entanglement, as a unique feature of quantum mechanics, plays an essential role in many quantum information processing tasks, such as quantum teleportation <cit.>, quantum cryptography <cit.>, non-locality test <cit.>, quantum computing <cit.>, quantum simulation <cit.> and quantum metrology <cit.>.
Consequently, it is quite significant to detect entanglement in experimental systems, which not only acts as benchmark and calibration of the underlying platform, but also certifies useful quantum resources for the further information processing. So far, tremendous efforts have been devoted to the realization of multipartite entanglement in various systems <cit.>. In particular, the genuine multipartite entanglement is witnessed in 14-ion-trap-qubit <cit.>, 10-superconducting-qubit <cit.>, and 12-photon-qubit systems <cit.>, with the target state being the Greenberger-Horne-Zeilinger (GHZ) state. 

The detection of genuine multipartite entanglement is generally a challenging task, since the dimension of the Hilbert space increases exponentially with respect to the system size. Compared with the unfeasible quantum state tomography <cit.>, the entanglement witness is an useful tool to realize it <cit.>. The witness is usually a Hermitian operator 𝒲, satisfying that Tr(𝒲σ_s)≥ 0 for all separable states σ_s ∈𝒮_sep, with 𝒮_sep the separable state set; Tr(𝒲|Ψ⟩⟨Ψ|)<0 for some entangled state |Ψ⟩, such as the GHZ state. Consequently, if W returns a negative value, one can confirm that the prepared state is entangled; a non-negative value tells nothing, denoted as a null result.

A straightforward way to construct a witness is based on the intuition that the prepared state ρ_pre is entangled if it is close to an entangled target state, say |Ψ⟩. To be specific,

    𝒲=α𝕀-|Ψ⟩⟨Ψ|,

where α is the maximal fidelity between |Ψ⟩ and all separable states σ_s, i.e., α=sup{⟨Ψ|σ_s|Ψ⟩|σ_s ∈ S_sep}. On account of the convexity of S_sep, α can be determined by the maximal Schmit coefficient of |Ψ⟩ optimized under all bipartitions <cit.>. For instance, α=1/2 for the GHZ state.
The expectation value of 𝒲 shows, Tr(𝒲ρ_pre)=α-Tr(|Ψ⟩⟨Ψ|ρ_pre), which is directly related to measuring fidelity.

Normally, the multipartite projector |Ψ⟩⟨Ψ| is decomposed with a few of local measurement settings (LMSs) <cit.>, for example the Pauli operator σ_x^⊗ N, which can be realized in experiments. Even for one LMS, it needs thousands of times of the measurement to obtain the estimation of the expectation value. Thus, the total number of LMSs characterize the efficiency of the witness. For the GHZ state, it needs N+1 LMSs <cit.>.








On the other hand, the robustness is another key feature of a witness, which benchmarks its detection ability. Generally, one applies white noise tolerance to characterize the robustness, i.e.,

    ρ=(1-p)|Ψ⟩⟨Ψ|+p𝕀/2^N,

which moves the target state towards the maximal mixed state. The maximal p_max such that Tr(𝒲ρ)≤0 describes the robustness of the witness. 

Since the witness 𝒲 shown in Eq. (<ref>) is designed specifically for the target state |Ψ⟩, it may return null results for some other entangled states. This phenomenon may become serious when the experiment system suffers from the coherent noise, i.e.,

    |Ψ_pre⟩=U_noise|Ψ⟩.

Since the unitary evolution can ”rotate" the state dramatically (not like the translation in the white noise case), the white noise tolerance corresponding to the result state can decrease, and is possibly outside the detection range of the witness in some case. See Fig. <ref> for an illustration.
Taking the GHZ state as an example, according to Eq. (<ref>), the fidelity-based witness shows,

    𝒲_GHZ=1/2𝕀-|GHZ⟩⟨GHZ|,

where |GHZ⟩=1/√(2)(|0⟩^⊗ N+ |1⟩^⊗ N). If the prepared state becomes |Ψ_pre⟩=1/√(2)(|0⟩^⊗ N-|1⟩^⊗ N) under some coherent error that affects the phase, the witness gives a null result Tr(𝒲_GHZ|Ψ_pre⟩⟨Ψ_pre|)=1/2>0. Note that here |Ψ_pre⟩ is entangled but one cannot confirm this by using the witness 𝒲_GHZ in Eq. (<ref>).

To the best of our knowledge, the entanglement detection under realistic coherent noises still lacks studying. The investigation along this direction can offer us two main advantages. On the one hand, it can supply useful tools to tackle with coherent noises and hence enhance our entanglement detection ability; on the other hand, it is also helpful to the benchmarking and even the calibration of experimental systems. This is beneficial for the ultimate goal—fault-tolerant quantum computation <cit.>, as the coherent noise leads to a much worse threshold than the stochastic ones <cit.>.

In this work, we study the entanglement detection under coherent noises and focus on the GHZ state, which is essential in many quantum information tasks, such as Bell-nonlocality <cit.>, multipartite quantum key distribution <cit.>, quantum secret sharing <cit.>, and quantum metrology <cit.>. We show an entanglement detection protocol that can effectively eliminate the influence of certain types of coherent noises for the GHZ state. Our protocol only adds 1 LMS than the original one, which needs N+1 LMSs. In particular, the protocol can effectively realize a family of entanglement witnesses with respective to the coherent noise, and one can select the optimal one by only postprocessing the measurement results. The protocol can also help us to estimate corresponding noise parameters and further give feedback to the experiment system. Moreover, we also consider the reduction of the number of LMSs, which makes the experimental realizations more efficient.

Our paper is organized as follows. In Sec. <ref>, two coherent noise models of the GHZ state are proposed, one is generated by the unconscious phase accumulation on N qubits, the other is due to the rotation on the control qubit. The overall noise model is the combination of the coherent part and the white noise part. In Sec. <ref>, we show the detection protocol with N+2 LMSs, used to witness the entanglement under coherent noises and further feed back the noisy parameters.
In Sec. <ref>, we further reduce the number of LMSs to 3, and propose more efficient witnesses.
Sec. <ref> is the conclusion and outlook.







§ THE NOISE MODEL

In this section, we show two realistic coherent noise models of the GHZ state, which will be analysed in the following Sec <ref>. One is caused by the unconscious phase accumulating on all qubits, the other is due to the single qubit rotation on the first control qubit.

Let us first review the white noise model.
Usually, one uses the white noise to analyse the noise tolerance of the entanglement witness, i.e., mixing the original state with the maximally mixed state,

    Φ_p(ρ)=(1-p)ρ+p𝕀/2^N.

For the GHZ state, the resulting state is

    ρ_w^p=(1-p)|GHZ⟩⟨GHZ|+p𝕀/2^N.

The corresponding noise tolerance is determined by Tr(𝒲_GHZρ_w^p)=0 where 𝒲_GHZ is defined in Eq. (<ref>), and it equals to,

    p_max=2^(N-1)/2^N-1  (≃ 0.5,N →∞).

The white noise is generated by the depolarizing channel, and it effectively displaces the original state |GHZ⟩ towards the maximally mixed state in the state space, as shown in Fig. <ref>.  However, generally speaking, the coherent noise could appear in the experiment due to some system errors, as we illustrated in the following sections.



 §.§ Model 1: Unconscious phase accumulation

In experimental realizations, the degree of freedom of N-qubit is generally encoded in N two-level subsystems, such as the ground state and the excited state of atoms. There might appear unconscious phase accumulation between |0⟩ and |1⟩ of qubits that dramatically transforms the state. To be specific, this kind of coherent error can be modeled as,

    U_z=⊗_j=1^N U_j^z,
       U_j^z=|0⟩_j⟨0|+e^iϕ_j|1⟩_j⟨1|,

where U_j^z denote the rotation around the Z-basis on the j-th qubit.
If we apply the above coherent noise on the GHZ state, it shows

    |Ψ_ϕ⟩   =U_z|GHZ⟩
       =⊗_j=1^N U_j^z|GHZ⟩
       =1/√(2)(|0⟩^⊗ N+e^i∑_j=1^Nϕ_j|1⟩^⊗ N)
       =1/√(2)(|0⟩^⊗ N+e^iϕ|1⟩^⊗ N),

where ϕ≐∑_j=1^Nϕ_j. Similar as the white noise case, the tolerance of  𝒲_GHZ in Eq. (<ref>) under this coherent noise is determined by

    Tr(𝒲_GHZ|Ψ_ϕ⟩⟨Ψ_ϕ|)=-cosϕ/2<0

which leads to ϕ∈ (-π/2, π/2). Thus if the absolute value of the phase |ϕ|≥π/2, the witness 𝒲_GHZ cannot properly detect the entanglement, while the prepared state |Ψ_ϕ⟩ is clearly an entangled one.

More generally, the realistic noise can be the combination of the white noise part Φ_p and the coherent part U_z, thus the output state shows,

    ρ_pre   =Φ_p ∘ U_z(|GHZ⟩⟨GHZ|)
       =(1-p)|Ψ_ϕ⟩⟨Ψ_ϕ|+p𝕀/2^N,

where U_z(·)=U_z· U_z^†, and note that Φ_p ∘ U_z= U_z ∘Φ_p.

In this joint noise model, the noise tolerance range is determined by Tr(𝒲_GHZρ_pre)<0 with ρ_pre in Eq. (<ref>). The result is given by the following formula including the coherent and white noise parameters ϕ and p,

    cosϕ >p/1-p,

as N →∞. The detailed derivation is left in Appendix <ref>.
Comparing to Eq. (<ref>), Eq. (<ref>) shows that the range of ϕ shrinks due to the introduction of white noise. On the other hand, Eq. (<ref>) can be rewritten as follows,

    p<1-1/cosϕ+1.

It indicates that the range of the white noise parameter p also decreases on account of the coherent noise, comparing to Eq. (<ref>).



 §.§ Model 2: Rotation on the first control qubit

The GHZ state is normarlly generated by the following circuit routine, as shown in Fig. <ref>.

  
  * Initialize all the N qubits to be |0⟩.
  
  * Apply a Hardmard gate H on the first (control) qubit, and transform it to |+⟩=1/2(|0⟩+|1⟩).
  
  * Apply Controled-NOT (CNOT) gate on qubit pairs (1,2), (2,3), (3,4),⋯(j,j+1)⋯⋯(N-1,N) in sequence, where j is the control qubit and j+1 is the target qubit.



It is clear to see that the CNOT gate sequence spreads the superposition information of the first qubit to all the qubits, and thus builds the quantum correlation on the whole system.
Hence, the quality of the rotation on the first qubit significantly affects the preparation of the final GHZ state.


Suppose besides the ideal H gate, there is also another uncontrolled unitary on the first qubit, i.e.,

    |ψ⟩_1=U_1H_1|0⟩_1=cosθ|0⟩_1 + e^iϕsinθ|1⟩_1,

with θ∈[0,π/2] and ϕ∈[-π,π).
Here the overall unitary U_1'=U_1H_1 in principle can be any single qubit unitary, thus |ψ⟩_1 describes any single qubit state after ignoring the irrelevant global phase. In addition, we also allow the
unconscious phase accumulation on the state at the final stage.

Consequently, the final prepared state shows,

    |Ψ_ϕ^θ⟩=cosθ|0⟩^⊗ N+e^iϕsinθ|1⟩^⊗ N,

where the accumulated phase at the final stage is also dropped into the parameter ϕ without confusion. Note that the noisy state |Ψ_ϕ^θ⟩ in Eq. (<ref>) is more general than |Ψ_ϕ⟩ in Eq. (<ref>), since |Ψ_ϕ^θ⟩ allows unbalanced state coefficients besides the relative phase.

The noise tolerance of 𝒲_GHZ in Eq. (<ref>) under this coherent noise is determined by

    Tr(𝒲_GHZ|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|)=1/2-1/2[1+sin(2θ)cos(ϕ)]<0

that is, sin(2θ)cos(ϕ)>0, which leads to ϕ∈ (-π/2, π/2).

As in Sec. <ref>, one can also consider the combination of the coherent noise and the white noise, and the final state shows,

    ρ_pre=(1-p)|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|+p𝕀/2^N,


Accordingly, the tolerance of 𝒲_GHZ in this scenario when N →∞ shows,

    sin(2θ)cos(ϕ)> p/1-p.

The detailed derivation is left in Appendix <ref>. Comparing to Eq. (<ref>), one can see that the noise tolerance range decreases further, after the introduction of the noise parameter θ. In addition, Eq. (<ref>) can be rewritten as,

    p<1-1/sin(2θ)cos(ϕ)+1,

and it is worse than Eq. (<ref>) and (<ref>). See Fig. <ref> for an illustration.




§ ENTANGLEMENT DETECTION PROTOCOL UNDER COHERENT NOISE

As shown in Sec. <ref>, the witness 𝒲_GHZ specialized for the GHZ state potentially returns a null result when the prepared state suffers from some coherent noise. Here we propose an entanglement detection protocol that can eliminate the effect of coherent noises shown in the above section. The protocol only involves N+2 LMSs, which only adds one LMS comparing to the previous witness 𝒲_GHZ specialized for the GHZ state <cit.>.


Since the resulting state of noise model 2 in Sec <ref> is more general than that of model 1 in Sec <ref>, for clearness, in the following we first apply the entanglement detection protocol on the model 1, and then generalize it to the model 2.




 §.§ Detection protocol under noise model 1

The protocol measures the fidelity between ρ_pre and |Ψ_ϕ⟩ in Eq. (<ref>) for any phase parameter ϕ with the same N+2 LMSs. As a result, one can effectively detect the  entanglement by choosing the optimal witness in the family,

    𝒲_Ψ_ϕ=1/2𝕀-|Ψ_ϕ⟩⟨Ψ_ϕ|,

by postprocessing the measurement results. See Fig. <ref> for an illustration. Hereafter qubit Pauli operators are denoted by {σ_x, σ_y, σ_z}, and we summarize the result into the following Theorem.

The family of witnesses 𝒲_Ψ_ϕ parameterized by ϕ in Eq. (<ref>) can be realized with totally N+2 LMSs, i.e., σ_z^⊗ N and

    ℳ_θ_k=(cosθ_kσ_x+ sinθ_kσ_y)^⊗ N,

where θ_k=kπ/N+1 and k=0,1,⋯,N.



The projector |Ψ_ϕ⟩⟨Ψ_ϕ| can be written as,

    |Ψ_ϕ⟩⟨Ψ_ϕ|=𝒵+𝒳,

where 𝒵 denotes the summation of diagonal terms, i.e.,

    𝒵=   1/2(|0⟩⟨0|^⊗ N+|1⟩⟨1|^⊗ N),

and 𝒳 is for off-diagonal terms

    𝒳=cosϕ𝒳_+ +sinϕ𝒳_-,

where

    𝒳_+=|0⟩⟨1|^⊗ N+|1⟩⟨0|^⊗ N/2,
       𝒳_-=|0⟩⟨1|^⊗ N-|1⟩⟨0|^⊗ N/2i.


The diagonal part 𝒵 can be measured with the LMS σ_z^⊗ N. The off-diagonal part







𝒳_+ and 𝒳_- involved in 𝒳 can be further decomposed with LMSs ℳ_θ_k given in Eq. (<ref>) as,






    𝒳_+=1/N+1∑_k=0^N (-1)^k cos(θ_k) ℳ_θ_k
       𝒳_-=-1/N+1∑_k=0^N (-1)^k sin(θ_k) ℳ_θ_k.

The proof of these decompositions is based on discrete Fourier transform, and we leave it in Appendix <ref>.


To eliminate the effect of the coherent noise due to the unconscious phase accumulation, one should maximize the fidelity between the prepared state ρ_pre and all possible |Ψ_ϕ⟩ based on measurement results, that is,

    max_ϕ⟨Ψ_ϕ|ρ_pre|Ψ_ϕ⟩   =⟨𝒵⟩+max_ϕ⟨𝒳⟩
       =⟨𝒵⟩ + max_ϕ{cosϕ⟨𝒳_+⟩ + sinϕ⟨𝒳_-⟩}
       =⟨𝒵⟩+√(⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2),

where ⟨·⟩ denotes the expectation value of the corresponding operator on ρ_pre, and in the final line we apply the Cauchy-Schwarz inequality. Note that ⟨𝒵⟩ and ⟨𝒳_±⟩ can be obtained from LMS σ_z and {ℳ_θ_k}_k=0^N, respectively. The optimal ϕ_opt to saturate the maximal value in Eq. (<ref>) is determined by

    tanϕ_opt   =⟨𝒳_-⟩/⟨𝒳_+⟩
       =-∑_k=0^N (-1)^k sin(θ_k) ⟨ℳ_θ_k⟩/∑_k=0^N (-1)^k cos(θ_k) ⟨ℳ_θ_k⟩.

where the second line is on account of Eq. (<ref>), and (cosϕ_opt, sinϕ_opt) is in the same quadrant with (⟨𝒳_+⟩, ⟨𝒳_-⟩).

For instance, for the noisy state shown in Eq. (<ref>), one can effectively choose the corresponding witness in Eq. (<ref>) to eliminate the effect of the coherent noise and detect the entanglement. Note that the parameter ϕ_opt is determined by the measurement results. It is clear that the noise tolerance now is the same as in the sole white noise case, p_max in Eq. (<ref>), no matter what value ϕ is.

Moreover, this protocol can further help to improve the experiment system. That is, one can apply an reverse unitary to amend the system according to the optimal ϕ_opt abstracted from the measurement results. In particular, one can add a corresponding Z-basis rotation on any qubit to eliminate the error.



 §.§ Detection protocol under noise model 2

In this section, we generalize the entanglement detection protocol proposed in Sec. <ref> and apply it to the noise model 2.

The main strategy is similar, and here we realize the following family of witnesses with the same N+2 LMSs.

    𝒲_Ψ_ϕ^θ=max{cos^2θ, sin^2θ}𝕀-|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|,

where max{cos^2θ, sin^2θ} is the maximal Schmidt coefficient of |Ψ_ϕ^θ⟩ defined in Eq. (<ref>). One can further choose the optimal witness in the family by post-processing the measurement results. We summarize this into the following Theorem.

The family of witnesses 𝒲_Ψ_ϕ^θ parameterized by ϕ and θ in Eq. (<ref>) can be realized with totally N+2 LMSs, i.e., σ_z^⊗ N and {ℳ_θ_k}_k=0^N defined in Eq. (<ref>).


As in Eq. (<ref>), the projector |Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ| can be decomposed as follows,

    |Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|=cos^2θ𝒵_0 + sin^2θ𝒵_1 +sin (2θ) 𝒳,

where 𝒵_0 and 𝒵_1 denote |0⟩⟨0|^⊗ N and |1⟩⟨1|^⊗ N, whose expectation values can be evaluated from the LMS σ_z^⊗ N; 𝒳 is given by Eq. (<ref>) and (<ref>), whose expectation value can be obtained from LMSs {ℳ_θ_k}_k=0^N, as shown in Eq. (<ref>).














Similar as Sec. <ref>, we should find the maximal fidelity between the prepared state and all possible |Ψ_ϕ^θ⟩ based on the measurement results,

    max_ϕ,θ⟨Ψ_ϕ^θ|ρ_pre|Ψ_ϕ^θ⟩=max_ϕ,θ{cos^2θ⟨𝒵_0⟩+ sin^2θ⟨𝒵_1⟩ + 
                 sin (2θ) ( cosϕ⟨𝒳_+⟩ + sinϕ⟨𝒳_-⟩)}
       =max_θ{cos^2θ⟨𝒵_0⟩+ sin^2θ⟨𝒵_1⟩ + sin (2θ) √(⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2)}
       =max_θ{⟨𝒵_0⟩+⟨𝒵_1⟩/2+ cos(2θ) ⟨𝒵_0⟩-⟨𝒵_1⟩/2 + 
                 sin (2θ) √(⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2)}
       =⟨𝒵_0⟩+⟨𝒵_1⟩/2+√(1/4(⟨𝒵_0⟩-⟨𝒵_1⟩)^2+⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2).

Here the maximization on the parameters ϕ and θ can be conducted independently. In the second line, we take the optimal ϕ_opt given by Eq. (<ref>). The last line is due to the Cauchy-Schwarz inequality, and the optimal θ_opt takes the value,

    tan (2θ_opt)=2√(⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2)/⟨𝒵_0⟩-⟨𝒵_1⟩,

with [cos(2θ_opt), sin(2θ_opt)] being in the same quadrant with [⟨𝒵_0⟩-⟨𝒵_1⟩, 2√(⟨𝒳_+⟩ ^2+ ⟨𝒳_-⟩^2)].
Then one can choose the optimal witness in the family of Eq. (<ref>) to detect the entanglement, based on the fidelity maximization in Eq. (<ref>) and the associated optimal parameters ϕ_opt and θ_opt.

For instance, for the noisy state in Eq. (<ref>), it is not hard to see that the noise tolerance shows

    p<min{cos^2θ_opt, sin^2θ_opt},

with N →∞, no matter what value ϕ is. The detailed derivation is left in Appendix <ref>. Note that the white noise tolerance is still a function of θ, even if one can obtain its value by postprocessing. The reason is because the parameter θ, not like ϕ, indeed affects the entanglement.

On the other hand, one can also choose the optimal witness in the family of Eq. (<ref>) in Sec. <ref> on the noise model 2 here. Since the optimization in Eq. (<ref>) can help to determine the corresponding noise parameter ϕ_opt, the optimal witness shows,

    𝒲_Ψ_ϕ_opt=1/2𝕀-|Ψ_ϕ_opt⟩⟨Ψ_ϕ_opt|.

As a result, for the noisy state in Eq. (<ref>), the corresponding white noise tolerance reads (see Appendix <ref> for the derivation),

    p<1-1/sin(2θ)+1.

which shows a clear advantage comparing to Eq. (<ref>) with the original witness 𝒲_GHZ. Note that the term cos(ϕ) is eliminated due to the postprocessing. Surprisingly, the white noise tolerance in Eq. (<ref>) is better than the one in Eq. (<ref>), as illustrated in Fig. <ref>. We give a detailed comparison in Appendix <ref>. The reason for this phenomenon may be as follows.  By using the family of witnesses in Eq. (<ref>), one maximizes the fidelity between |Ψ_ϕ^θ⟩ and the prepared state. However, the corresponding fidelity bound in the witness for the separable state, i.e., max{cos^2θ, sin^2θ}, becomes larger and harder to violate.






The entanglement detection protocol under the noise model 2 employs the same set of N+2 LMSs as that in Sec. <ref>, but abstracts both noise parameters ϕ_opt and θ_opt. This is because here we postprocess measurement results more delicately. Even though the white noise tolerance of the corresponding witness in Eq. (<ref>) is not better than the one in Eq. (<ref>), the experiment system can be further improved with the noise parameters ϕ_opt and θ_opt extracted from the measurement results. In particular, one can add an unitary U_1^† on the first qubit when preparing the GHZ state, which can be determined by ϕ_opt and θ_opt.



§ ENTANGLEMENT DETECTION WITH LESS LMS

In entanglement detection, the number of LMSs usually determines the efficiency of the witness, since even for one setting it could take thousands of measurements to obtain the accurate estimation of the expectation value. Thus, it is beneficial to reduce the number of LMSs and enhance the efficiency of the witness. In this section, by utilizing the stabilizer formulation, we show that one can detect entanglement of GHZ-like states with only 3 LMSs under realistic coherent noises.

Comparing to the witness 𝒲_GHZ using N+1 LMSs in Eq. (<ref>),
there is a more efficient witness using 2 LMSs <cit.>, i.e.,

    𝒲^2_GHZ=1/2𝕀-𝒵-1/4σ_x^⊗ N,

where 𝒵 is defined in Eq. (<ref>). However, there is a trade-off between the efficiency and the white noise tolerance <cit.>. For the target GHZ state, the white noise tolerance of 𝒲^2_GHZ is p=1/3 as N →∞ <cit.>, while the tolerance is p=1/2 of 𝒲_GHZ. Note that 𝒲^2_GHZ employs two settings, i.e., σ_z^⊗ N and σ_x^⊗ N.

In the following, we study the entanglement detection under coherent noises with less LMSs, by adding another LMS σ_y⊗σ_x ^⊗ N-1 to σ_z^⊗ N and σ_x^⊗ N. Actually, the σ_y operator can be on any qubit due to the symmetry of GHZ-like states. With loss of generality, we set it on the first qubit. Similar as Sec. <ref>, we first consider the entanglement detection under noise model 1, then noise model 2 that contains more possible noisy states.


 §.§ Efficient detection under noise model 1

We extend the witness 𝒲^2_GHZ in Eq. (<ref>) to a family of witnesses, parameterized by the noisy parameter ϕ. Similar as Sec. <ref>, one can effectively detect entanglement under the coherent noise by choosing the optimal witness in this family, by postprocessing the measurement results. We summarize the result in the following theorem.

The witness 𝒲_Ψ_ϕ^2 can detect entanglement near the state |Ψ_ϕ⟩,

    𝒲_Ψ_ϕ^2=1/2𝕀-𝒵-1/4(cosϕℳ_x + sinϕℳ'_x),

where 𝒵 is defined in Eq. (<ref>), ℳ_x=σ_x ^⊗ N, and ℳ_x'=σ_y⊗σ_x ^⊗ N-1.

It is clear that the family of witness {𝒲_Ψ_ϕ^2} parameterized by ϕ can be realized by 3 LMSs, σ_z ^⊗ N,σ_x ^⊗ N,σ_y⊗σ_x ^⊗ N-1.

As given in Eq. (<ref>), the possible state under the coherent noise shows |Ψ_ϕ⟩=1/√(2)(|0⟩^⊗ N+e^iϕ|1⟩^⊗ N), and it can be transformed from the standard GHZ state by applying a single qubit unitary,

    |Ψ_ϕ⟩=U_1^z⊗𝕀^⊗ N-1|GHZ⟩,

where U_1^z=|0⟩_1⟨0|+e^iϕ|1⟩_1⟨1|=e^iϕ/2e^-iσ_zϕ/2. For the density matrix, one has the relation Ψ_ϕ=U_1^z GHZ U_1^z†. Since the witness 𝒲^2_GHZ can detect entanglement near the GHZ state, we have the following witness that can detect entanglement near |Ψ_ϕ⟩ based on Observation <ref>,

    𝒲_2^ϕ   =U_1^z 𝒲^2_GHZ U_1^z†
       =1/2𝕀-𝒵-1/4(e^-iσ_zϕ/2σ_xe^iσ_zϕ/2)⊗σ_x^⊗ N-1,
       =1/2𝕀-𝒵-1/4(cosϕσ_x + sinϕσ_y)⊗σ_x^⊗ N-1.

In this way, we obtain a family of entanglement witnesses parameterized by the phase ϕ, and they all need 2 LMSs, i.e. σ_z^⊗ N and (cosϕσ_x - sinϕσ_y)⊗σ_x^⊗ N-1. In fact, one can realize these witnesses with only 3 LMSs, ℳ_z=σ_z^⊗ N, ℳ_x=σ_x^⊗ Nand ℳ_x'=σ_y⊗σ_x^⊗ N-1, since the result of (cosϕσ_x - sinϕσ_y)⊗σ_x^⊗ N-1 can be obtained by the linear combination of ℳ_x and ℳ_x'.



Suppose an entanglement witness 𝒲 can detect a entangled state ρ, i.e., Tr(𝒲ρ)<0 and Tr(𝒲σ_s)≥0, ∀σ_s ∈ S_sep, the state ρ'=U_locρ U_loc^† after the transformation with local unitary operation U_loc can be detected by the corresponding witness

    𝒲'=U_loc𝒲U_loc^†

where U_loc=U_1⊗ U_2⋯ U_N and U_i is the unitary on the i-th qubit.


Note that ρ' is still an entangled state, as local unitary operations does not alter entanglement property. Tr(𝒲'ρ')=Tr(U_loc𝒲U_loc^†U_locρ U_loc^†)=Tr(𝒲ρ)<0. In addition, Tr(𝒲'σ_sep)=Tr(U_loc𝒲U_loc^†σ_sep)=Tr(𝒲U_loc^†σ_sepU_loc)≥0,  since U_loc^†σ_sepU_loc is still a separable state.

To eliminate the effect of the coherent noise, similar as Eq. (<ref>), one should find the minimal expectation value of all the witnesses in the family, i.e., min_ϕTr(𝒲_Ψ_ϕ^2ρ_pre). Equivalently,

    max_ϕ{cosϕ⟨ℳ_x⟩+sinϕ⟨ℳ_x' ⟩}.

The optimal ϕ is determined by

    tanϕ_opt=⟨ℳ_x'⟩/⟨ℳ_x⟩.

and (cosϕ_opt, sinϕ_opt) is in the same quadrant with (⟨ℳ_x⟩, ⟨ℳ'_x⟩).

For instance, for the noisy state shown in Eq. (<ref>), one can effectively choose the corresponding witness in Eq. (<ref>) to detect entanglement. Now the noise tolerance is the same as in the sole white noise case, i.e., p=1/3 as N →∞, no matter what value ϕ is. In addition, this protocol can further help to improve the experiment system by applying an correcting unitary according to the optimal ϕ_opt.


 §.§ Efficient detection under noise model 2

We further extend the witness 𝒲^2_GHZ in Eq. (<ref>) to a family of witnesses, and have the following theorem.

The witness 𝒲_Ψ_ϕ^θ^2 can detect entanglement near the state |Ψ_ϕ^θ⟩,

    𝒲_Ψ_ϕ^θ^2=   2max{cos^2θ, sin^2θ}+1/4𝕀-𝒵
       -1/4cos (2θ) ℳ_z
    -1/4sin (2θ)(cosϕℳ_x + sinϕℳ'_x),

where 𝒵 is defined in Eq. (<ref>), ℳ_z=σ_z⊗𝕀^⊗  N-1, ℳ_x=σ_x ^⊗ N, and ℳ_x'=σ_y⊗σ_x ^⊗ N-1.

It is clear that in general |Ψ_ϕ^θ⟩ can not be transformed from the standard GHZ state by local unitary operations.

Thus, we cannot prove Theorem <ref> with the approach used in Theorem <ref>, and the following proof is based on the generalized stabilizer formula.

The GHZ state is a stabilizer state that is uniquely determined by the following N independent stabilizer operators,

    S_1=σ_x ^⊗ N,S_2=σ_z^1σ_z^2, S_3=σ_z^2σ_z^3, ⋯, S_N=σ_z^N-1σ_z^N

and the witness 𝒲^2_GHZ in Eq. (<ref>) can be equivalently written as,

    2𝒲^2_GHZ=3/2𝕀-(P_1+P_2)

with P_1 and P_2 two projectors determined by the stabilizers,

    P_1   =𝕀+S_1/2=𝕀+σ_x ^⊗ N/2,
    
    P_2   =∏_i=2^N𝕀+S_i/2≡2𝒵.



Due to the fact (𝕀-P_1)(𝕀-P_2)≥ 0, one has

    |GHZ⟩⟨GHZ|=P_1P_2≥ (P_1+P_2-𝕀),

As a result, the witness 𝒲_GHZ^2 is valid since 2𝒲^2_GHZ≥𝒲_GHZ given in Eq. (<ref>) <cit.>.


In the following, we construct the witness 𝒲_Ψ_ϕ^θ^2 in Eq (<ref>) by finding generalized stabilizers of |Ψ_ϕ^θ⟩. Here “generalized” means that the stabilizer may be not in the Pauli tensor form.

It is not hard to see that the last N-1 stabilizers S_2,S_3,⋯ S_N in Eq (<ref>) also stabilize |Ψ_ϕ^θ⟩. Thus, we only need to find the first updated one S_1'. The construction is based on the following fact: if S stabilizes |Ψ⟩, i.e., S|Ψ⟩=|Ψ⟩, USU^† stabilizes U|Ψ⟩. Note that |Ψ_ϕ^θ⟩ can be prepared from the noisy circuit described in Fig. <ref>.

Initially, the stabilizers of the product state |0⟩^⊗ N is σ_z^1, σ_z^2,⋯, σ_z^N; after the single qubit unitary on the first qubit, σ_z^1 becomes

    U_1'σ_z^1U_1'^†=cos(2θ)σ_z^1 + sin(2θ)(cosϕσ_x^1+sinϕσ_y^1);

finally, after the successive application of CNOT gates, the stabilizer shows,

    S_1'=U_CNOTU_1'σ_z^1U_1'^†U_CNOT^†

where U_CNOT=[C_N-1X_N]⋯ [C_2X_3] [C_1X_2] is the CNOT gate sequence, with C_iX_j denoting the CNOT gate on the qubit j controlled by the qubit i. One can find that

    S_1'=   cos(2θ)σ_z^1⊗𝕀^⊗  N-1+
       sin(2θ)(cosϕσ_x^⊗ N+sinϕσ_y^1⊗σ_x^⊗ N-1)

on account the following relations,

    C_iX_j σ_x^i C_iX_j=σ_x^iσ_x^j,
       C_iX_j σ_y^i C_iX_j=σ_y^iσ_x^j,
       C_iX_j σ_z^i C_iX_j=σ_z^i.


Similar as Eq. (<ref>), we can define two projectors P_1' and P_2' associated with the stabilizers of the state |Ψ_ϕ^θ⟩, that is, P_1'=1/2(𝕀+S_1') and P_2'=P_2. Then based on the witness 𝒲_Ψ_ϕ^θ in Eq. (<ref>), we have the new witness with less LMSs,

    𝒲_Ψ_ϕ^θ^2   =1/2[(max{cos^2θ, sin^2θ}+1) 𝕀-(P_1'+P_2')]
       =2max{cos^2θ, sin^2θ}+1/4𝕀-𝒵-1/4S_1',

with S_1' given in Eq. (<ref>). Similar as Eq. (<ref>), one can verify 2𝒲_Ψ_ϕ^θ^2≥𝒲_Ψ_ϕ^θ and the witness 𝒲_Ψ_ϕ^θ^2 is valid.


Similar as Sec. <ref>, one should find the minimal expectation value of all the witnesses in the family of Eq. (<ref>), i.e., min_ϕ,θTr(𝒲_Ψ_ϕ^θ^2ρ_pre), parameterized by ϕ and θ. Equivalently,

    max_ϕ,θ   {cos (2θ) ⟨ℳ_z⟩+sin (2θ)(cosϕ⟨ℳ_x⟩ + sinϕ⟨ℳ'_x⟩)}
       =√(⟨ℳ_z⟩^2+⟨ℳ_x⟩^2+⟨ℳ'_x⟩^2)

The optimal ϕ_opt to saturate the maximal value is given in Eq. (<ref>), and the optimal θ_opt satisfies

    tan (2θ_opt)=√(⟨ℳ_x⟩^2+⟨ℳ'_x⟩^2)/⟨ℳ_z⟩,

and [cos(2θ_opt), sin(2θ_opt)] is in the same quadrant with (⟨ℳ_z⟩, √(⟨ℳ_x⟩^2+⟨ℳ'_x⟩^2)).

Consequently, one can choose the optimal witness in the family of Eq. (<ref>) to detect the entanglement, based on the associated optimal parameters ϕ_opt and θ_opt. For instance, for the noisy state shown in Eq. (<ref>), the corresponding white noise tolerance shows,

    p<2/3min{cos^2θ_opt, sin^2θ_opt},

for N →∞ no matter what value ϕ is, and the proof is left in Appendix <ref>.

On the other hand, similar as the discussion at the end of Sec. <ref>, one can also apply the detection protocol given in Sec. <ref> on the noise model 2 here. That is, using the optimal witness in the family of Eq. (<ref>). The optimization in Eq. (<ref>) can determine the corresponding noise parameter ϕ_opt and thus the optimal witness. For the noisy state in Eq. (<ref>), the corresponding white noise tolerance reads,

    p<1-2/sin(2θ)+2.

as N→∞. The proof is also left in Appendix <ref>. Similar as the comparison at the end of Sec. <ref>, the white noise tolerance in Eq. (<ref>) is better than the one in Eq. (<ref>).

The efficient detection protocol under the noise model 2 employs the same set of 3 LMSs as the one in Sec. <ref>, but abstracts both noise parameters ϕ_opt and θ_opt. This is because here we post-process measurement results more delicately. Even though the white noise tolerance of the corresponding witness in Eq. (<ref>) is not better than the one in Eq. (<ref>), the experiment system can be further improved with the noise parameters ϕ_opt and θ_opt extracted from the measurement results.


§ CONCLUSION AND OUTLOOK

In this paper, by focusing on GHZ-like states, we propose an entanglement detection protocol to enhance the detection ability under some practical coherent noises, which only adds one LMS comparing to the original witness method. Our protocol can feedback the noisy parameters by postprocessing and further help to improve the experimental system.
The main idea behind the protocol is that we construct a set of measurements which can tomography all possible states affected by the coherent noise, and thus realize a family of entanglement witnesses. In addition, we further reduce the number of LMSs to 3, which makes the experimental realization more efficient.


There are a few prospective problems that can be explored in the future.

First, it is shown in the paper that even if one can obtain more parameters about the prepared state by delicate postprocessing, it may be not beneficial to the entanglement detection as shown by the noise tolerance comparison in Fig. <ref>. Thus it is significant to investigate further whether it is a general phenomenon. Second, it is interesting to extend the current protocol to more general states, such as permutation-invariant states <cit.> and stabilizer states <cit.>, where quantum error correction or mitigation methods can be applied to eliminate or reduce the effect of coherent noises. Third, it is also intriguing to study the entanglement detection under other types of coherent noises, which appear in certain experimental systems. In addition, the detection of more detailed multipartite entanglement structures <cit.> under coherent noises is significant to investigate.

We thank Chenghao Guo, Xiongfeng Ma and Qi Zhao for the insightful discussions. This work was supported by the National Natural Science Foundation of China Grants No. 11875173 and No. 11674193, and the National Key R&D Program of China Grant No. 2017YFA0303900.




§ DERIVATION OF MISCELLANEOUS NOISE TOLERANCES IN EQ. (<REF>), (<REF>) , (<REF>) AND (<REF>)

First, let us focus on Eq. (<ref>) and (<ref>), which are noise tolerances of 𝒲_GHZ in Eq. (<ref>) for the noise model 1 and 2 respectively. Since the noisy state in Eq. (<ref>) is more general than that in Eq. (<ref>), we only show the derivation of Eq. (<ref>) here.


    Tr(𝒲_GHZρ_pre)   =Tr{[1/2𝕀-|GHZ⟩⟨GHZ|]
               [(1-p)|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|+p𝕀/2^N]}
       =1/2-(1-p)|⟨GHZ|Ψ_ϕ^θ⟩|^2-p/2^N
       =1/2-1-p/2[1+sin(2θ)cos(ϕ)]-p/2^N<0.

As N →∞, one has

    sin(2θ)cos(ϕ)> p/1-p.


Then, let us consider Eq. (<ref>), which is the tolerance of the optimal witness in Eq. (<ref>) for the noisy state in Eq. (<ref>). Since the fidelity optimization in Eq. (<ref>) helps us to determine the parameters ϕ and θ of the prepared state in Eq. (<ref>), one can choose the witness with the same parameters in Eq. (<ref>) and the noise tolerance shows,

    Tr(𝒲_Ψ_ϕ^θρ_pre)   =Tr{[max{cos^2θ, sin^2θ}𝕀-|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|]
                [(1-p)|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|+p𝕀/2^N]}
       =max{cos^2θ, sin^2θ}-(1-p)-p/2^N<0.

As N →∞, one has

    p<1-max{cos^2θ, sin^2θ}=min{cos^2θ, sin^2θ}.


Finally, let us derive Eq. (<ref>), which is the tolerance of the optimal witness in Eq. (<ref>) for the noisy state in Eq. (<ref>). Since the fidelity optimization in Eq. (<ref>) helps us to determine the parameter ϕ of the prepared state in Eq. (<ref>), one can choose the witness with the same parameter ϕ in Eq. (<ref>) and the noise tolerance shows,

    Tr(𝒲_Ψ_ϕρ_pre)   =Tr{[1/2𝕀-|Ψ_ϕ⟩⟨Ψ_ϕ|]
                [(1-p)|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|+p𝕀/2^N]}
       =1/2-(1-p)|⟨Ψ_ϕ|Ψ_ϕ^θ⟩|^2-p/2^N
       =1/2-1-p/2[1+sin(2θ)]-p/2^N<0.

As N →∞, one has

    p<1-1/1+sin(2θ).



§ PROOF OF DECOMPOSITIONS IN EQ. (<REF>)









First, note that the matrix form of cosθ_k σ_x+ sinθ_k σ_y shows,

    cosθ_k σ_x+ sinθ_k σ_y   =(
     [       0 e^-iθ_k;  e^iθ_k       0;         ])
       =e^-iθ_k|0⟩⟨1|+e^iθ_k|1⟩⟨0|,


Let l(b)=∑_i=0^N b_i denote the weight of the binary string b∈{0,1}^N, and b̅ is the bitwise inverse of b with b̅_i=(b_i+1)  2. We can further rewrite the product operator ℳ_θ_k in Eq. (<ref>) in the computational basis as follows.

    ℳ_θ_k=   ∑_b e^i[l(b)-l(b̅)]θ_k|b⟩⟨b̅|
    
    =   ∑_b e^i[2l(b)-N]θ_k|b⟩⟨b̅|
    
    =   (
     [                   e^-iNθ_k;                 ⋯         ;  e^iNθ_k                  ;          ]).

Here in the second line we use the fact l(b)+l(b̅)=N and l(b)∈{0,1,⋯,N}.  Note that ℳ_θ_k only possesses terms on off-diagonal positions. For the clearness of the latter decomposition, we add a corresponding phase on ℳ_θ_k as,

    ℳ'_θ_k   ≡ e^iNθ_kℳ_θ_k=∑_b e^i2 θ_k  l(b) |b⟩⟨b̅|
       =∑_b e^i2π k/N+1 l(b)|b⟩⟨b̅|.


From Eq. (<ref>) and (<ref>) in Main Text, 𝒳 contains two terms 𝒳_+and 𝒳_-, and we rewrite them as,

    𝒳_+   =1/2𝒳_+',
    𝒳_-   =1/2i𝒳_-',

where 𝒳_+' and 𝒳_-' having matrix forms in the computational basis as,

    𝒳_+'=   (
     [         1;       0  ;     ⋯    ;   0      ; 1      ;   ]),
    𝒳_-'=   (
     [              1;           0   ;        ⋯      ;     0         ; -1         ;    ]).

Note that they show specific forms on the off-diagonal positions.


In the following, we derive the decomposition of 𝒳_+' and 𝒳_-' in terms of ℳ_θ_k' using discrete Fourier transform.

Note that ℳ_θ_k' shows the same coefficient on the terms |b⟩⟨b̅|, if they share the same l(b). Thus we only need to care about the weight of the binary b and denote t=l(b), which is the analog of the “time” domain, with t=[0,1,⋯,N]. It is clear that
the function of ℳ_θ_k' on this domain is the Fourier basis function e^i2π k/N+1 t, with the parameter k being the analog of the “frequency" domain. The corresponding functions of 𝒳_+' and 𝒳_-' on the time domain are f_+(t)=[1,0⋯,1] and f_-(t)=[1,0⋯,-1], respectively. By applying discrete Fourier transform, one has the coefficients showing

    F_+(k)=1/N+1∑_t=0^N e^-i2π k/N+1tf_+(t)=1+e^-i2π k N/N+1/N+1,
    
    F_-(k)=1/N+1∑_t=0^N e^-i2π k/N+1tf_-(t)=1-e^-i2π k N/N+1/N+1.


Combing these coefficients with the operators, we have,

    𝒳_+   =1/2∑_k=0^N  F_+(k) ℳ_θ_k',
       = 1/2∑_k=0^N 1+e^-i2π k N/N+1/N+1 e^iNθ_kℳ_θ_k,
       =1/N+1∑_k=0^Ncos(π k N/N+1) ℳ_θ_k,
       =1/N+1∑_k=0^N (-1)^k cos(θ_k) ℳ_θ_k,

where the last line is on account of π kN/N+1=kπ-θ_k. Similarly,

    𝒳_-   =1/2i∑_k=0^N  F_-(k) ℳ_θ_k',
       = 1/2i∑_k=0^N 1-e^-i2π k N/N+1/N+1 e^iNθ_kℳ_θ_k,
       =1/N+1∑_k=0^Nsin(π k N/N+1) ℳ_θ_k,
       =-1/N+1∑_k=0^N (-1)^k sin(θ_k) ℳ_θ_k.



§ COMPARISON BETWEEN THE WHITE NOISE TOLERANCES IN EQ. (<REF>) AND (<REF>)

Here, we compare the white noise tolerance in Eq. (<ref>) using a family of witnesses 𝒲_Ψ_ϕ^θ with that using
a family of witnesses 𝒲_Ψ_ϕ in Eq. (<ref>), for the noisy state in Eq. (<ref>).  In the following, we show the difference of them denoted by the function,

    g(θ)=[1-1/sin(2θ)+1]-min{cos^2θ, sin^2θ}≥ 0,

where θ∈[0,π/2] and g(θ)=0 as θ=0,π/4, π/2. Note that g(θ) is symmetric with respective to θ=π/4. Thus we only need to consider the regime θ∈[0,π/4],

    g(θ)=   [1-1/sin(2θ)+1]-sin^2θ
       =cos^2θ-1/sin(2θ)+1
       =cos^2θ-1/(cosθ+sinθ)^2≥ 0

equivalently, cosθ(cosθ+sinθ)-1≥ 0, that is, cosθsinθ≥sin^2θ→1≥tanθ or sinθ=0. This true since θ∈[0,π/4].



§ DERIVATION OF NOISE TOLERANCES IN EQ. (<REF>), (<REF>) AND THE COMPARISON

First, let us derive the noise tolerances in Eq. (<ref>). Hereafter, we use ϕ and θ to denote ϕ_opt and θ_opt without confusion, and also denote f(θ)=max{cos^2θ, sin^2θ} in 𝒲_Ψ_ϕ^θ^2 for simplicity.

Note that in Eq. (<ref>), 𝒲_Ψ_ϕ^θ^2 is written in the following form,

    2𝒲_Ψ_ϕ^θ^2   =(f(θ)+1) 𝕀-(P_1'+P_2'),

where P_1' and P_2' are two projectors determined by the stabilizers of the state |Ψ_ϕ^θ⟩.

The noise tolerance is determined by Tr(𝒲_Ψ_ϕ^θ^2ρ_pre)<0. Inserting the witness of Eq. (<ref>) and the noisy state ρ_pre of Eq. (<ref>), one has

    Tr{[(f(θ)+1) 𝕀-(P_1'+P_2')]ρ_pre}
       =(f(θ)+1)-Tr{(P_1'+P_2')[(1-p)|Ψ_ϕ^θ⟩⟨Ψ_ϕ^θ|+p𝕀/2^N]}
       =(f(θ)+1)-2(1-p)-p/2^NTr( P_1'+ P_2'  )
       =(f(θ)+1)-2(1-p)-p(2^N-1+2)/2^N<0,

where in the third line we use the fact that P_1' and P_2' stabilize |Ψ_ϕ^θ⟩, and in the final line Tr(P_1')=2^N-1, Tr(P_2')=2. From Eq. (<ref>), it is not hard to see that

    p=2^N-1/3*2^N-2-1(1-f(θ))≃2/3min{cos^2θ, sin^2θ},

as N →∞.

Second, let us derive Eq. (<ref>), which is the tolerance of the optimal witness in Eq. (<ref>) for the noisy state in Eq. (<ref>). Since the optimization in Eq. (<ref>) helps to determine the parameter ϕ of the prepared state in Eq. (<ref>), one can choose the witness with the same parameter ϕ in Eq. (<ref>). Consequently, the white noise tolerance is the same with the noisy state,

    ρ_pre=(1-p)|Ψ^θ⟩⟨Ψ^θ|+p𝕀/2^N,

where |Ψ^θ⟩=cosθ|0⟩^⊗ N+sinθ|1⟩^⊗ N, under the detection of the witness 𝒲^2_GHZ in Eq. (<ref>).

    Tr(𝒲^2_GHZρ_pre)
       =Tr{[3/2𝕀-(P_1+P_2)][(1-p)|Ψ^θ⟩⟨Ψ^θ|+p𝕀/2^N]}
       =3/2-(1-p)[3/2+1/2sin(2θ)]-p/2^NTr( P_1+ P_2 )
       =3/2-(1-p)[3/2+1/2sin(2θ)]-p(2^N-1+2)/2^N<0.

Here in the second line we applies the formula of 𝒲^2_GHZ in Eq. (<ref>). The second line is due to the fact that P_2 stabilizes |Ψ^θ⟩ and Tr(P_1|Ψ^θ⟩⟨Ψ^θ|)=1/2+1/2sin(2θ), and the final line Tr(P_1)=2^N-1, Tr(P_2)=2. From Eq. (<ref>), it is not hard to see that

    p<sin(2θ)/sin(2θ)+2-2^2-N≃1-2/sin(2θ)+2,

as N →∞.

Finally, let us compare the two noise tolerances in Eq. (<ref>) and (<ref>). Similar as Appendix <ref>, we define the function l(θ) as the subtraction and consider the regime θ∈(0,π/4) due to the symmetry,

    l(θ)=   [1-2/sin(2θ)+2]-2/3sin^2θ
       =2/3(1/2+cos^2θ-3/sin(2θ)+2) ≥ 0.

That is, (1/2+cos^2θ)(sin(2θ)+2)≥ 3. After simplification, it is equivalent to 2+cos(2θ)≥ 2tan(θ), which is right for θ∈(0,π/4).




















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