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tags: robust self-testing, bell inequality, self-testing, quantum computing, quantum information, quantum internet, 2016
---

# [Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities](https://arxiv.org/abs/1604.08176)
Jędrzej Kaniewski

## Abstract
Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance observing the maximal violation of the CHSH inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (with one exception which we discuss in detail). In this work we present a new technique for proving analytic self-testing bounds of practically relevant robustness. We obtain improved bounds for the case of self-testing the singlet using the CHSH inequality (in particular we show that non-trivial fidelity with the singlet can be achieved as long as the violation exceeds β∗=(16+142‾√)/17≈2.11). In case of self-testing the tripartite GHZ state using the Mermin inequality we derive a bound which not only improves on previously known results but turns out to be tight. We discuss other scenarios to which our technique can be immediately applied.

## Backgrounds

### self-testing

> One of the most striking consequences of Bell’s theorem is the fact that the non-classical nature of two (or more) devices can be verified by a classical user.

Bellの定理の最も重要な事実の一つ: 2つ以上のパーティの非局在性を古典的に検証することができる点

> we find that certain extremal quantum correlations (almost) uniquely identify the state and measurements under consideration, a phenomenon known as self-testing

量子相関が極大になる値を観測することが、その量子状態の同定につながる = **self-testing**

two inherent limitations of self-testing:
- the inability to see local unitaries
- the inability to detect auxiliary systems (on which the measurements act trivially)

### 既存のself-testingの例
- singlet
- graph state
- hight-dimensional maximally entangled states
- non maximally entangled states of two qubits

これらはrobustではない点で使えない(誤差の値が$\varepsilon < 10^{-4}$とかじゃないと意味をなさない)

### 実用的なロバストなself-testingは
- single analytic result
- swap trick

に限られる。

> **swap trick**
> The “swap trick” relies on explicitly constructing a circuit extracting the desired state into an extra register and using the hierarchy for quantum correlations [NPA07, DLTW08] to place a lower bound on the resulting fidelity.

量子相関(の階層?)をFidelityに変換

### swap trickの問題点:
> the computational cost grows rapidly with dimensionality, so far it has only been applied to quantum systems of small dimensions (two qubits/qutrits [YVB+14, BNS+15] or three/four qubits [WCY+14, PVN14]).

計算コストが高いので、小さいサイズにしか適用できていない

### 観測のself-testing

先行研究(とても少ない)
- CHSH inequality is violated maximally only if the observables anticommute (which corresponds to maximal incompatibility)
- how to certify more than two binary observables
- which Bell inequalities are well-suited for self-testing
- non-maximally anticommuting observables can be self-tested using the tilted CHSH inequality
- all measurements lying in a single plane of the Bloch sphere can be self-tested through chained inequalities

> To the best of our knowledge nothing is known about certifying measurements with more than two outcomes.

2出力より多い観測について検証する方法は見つかっていない

### self-testingの他の話題

- self-testing in parallel: producing a complete list of self-tests (within a particular Bell scenario)
- analysing semi-device independent scenarios

## Contributions

> In this Letter we present a new technique for proving analytic self-testing statements for quantum states.

> (i) understanding how to construct extraction channels from measurement operators
> (ii) analysing the resulting operator

量子相関をFidelityに変換できるのは大きなBreakthrough.

## Methods

### Settings
AとBが量子状態を共有している。
そこで、untrusted measurement devicesを考える。
> もし trusted measurement devices を持っていたら、state tomographyをすれば良い

できることは、
- choosing the measurement setting
- observing the outcome

手に入れるのは、観測した際の条件付き確率
> get the conditional probability distribution: the probability of observing outputs a, b for inputs x, y)

仮定:
- A natural solution is to require that they are capable of locally (without communication) extracting the desired state

定義:
For an arbitrary bipartite input state $\rho_{AB}$ we define the extractability of $\Psi_{A'B'}$ from $\rho_{AB}$ as

$$
\Xi\left(\rho_{A B} \rightarrow \Psi_{A^{\prime} B^{\prime}}\right):=\max _{\Lambda_{A}, \Lambda_{B}} F\left(\left(\Lambda_{A} \otimes \Lambda_{B}\right)\left(\rho_{A B}\right), \Psi_{A^{\prime} B^{\prime}}\right)
$$
over all CPTP channels $\Lambda_{A}, \Lambda_{B}$.

### extractability-violation trade-off

- $\Psi_{A^{\prime} B^{\prime}}$: a target state (which achieves the maximal quantum violation)
- $\mathcal{B}$: a Bell inequality
- $\beta_{C}$: the maximal values of the inequality $\mathcal{B}$ achieved within the classical theory
- $\beta_{Q}$: the maximal values of the inequality $\mathcal{B}$ achieved within the quantum theory

The extractability-violation trade-off is
$$
\mathcal{Q}_{\Psi, \mathcal{B}}:\left[\beta_{C}, \beta_{Q}\right] \rightarrow[0,1]
$$
such that
$$
\mathcal{Q}_{\Psi, \mathcal{B}}(\beta):=\inf _{\rho_{A B} \in \mathcal{S}_{\mathcal{B}}(\beta)} \Xi\left(\rho_{A B} \rightarrow \Psi_{A^{\prime} B^{\prime}}\right)
$$

where $\mathcal{S}_{\mathcal{B}}(\beta)$ is the set of bipartite states (of arbitrary dimension) that achieve the value of (at least) $\beta$ on the inequality $\mathcal{B}$.

計算を進めると、$\mathcal{Q}_{\Psi, \mathcal{B}}(\beta)$の上限と下限は以下のようになる。
$$
\lambda_{max}\leq \mathcal{Q}_{\Psi, \mathcal{B}}(\beta) \leq \lambda_{\max }+\left(1-\lambda_{\max }\right) \cdot \frac{\beta-\beta_{C}}{\beta_{Q}-\beta_{C}}
$$

> Our goal is to self-test the target state $\Psi_{A^{\prime} B^{\prime}}$ using a particular Bell inequality $\mathcal{B}$ and here we show how to obtain linear self-testing statements of the form
> $$\mathcal{Q}_{\Psi, \mathcal{B}}(\beta) \geq s \beta+\mu$$

**目的は、特定のBell不等式$\mathcal{B}$を用いた、状態$\Psi_{A^{\prime} B^{\prime}}$のself-testingをすること。そのために、$$\mathcal{Q}_{\Psi, \mathcal{B}}(\beta) \geq s \beta+\mu$$の形の不等式を得たい。**

証明の道筋: Fidelityについて、$K \geq s W+\mu\mathbb1$を証明する。

### 結果
- **CHSH不等式に対して:**

- **Mermin不等式に対して:**

## Comments

> Our technique hinges on the idea that measurement operators can be used to construct local extraction maps. This gives rise to a family of operators and placing a lower bound on the spectrum of these operators immediately yields a self-testing statement.

## Open Problems

> An immediate follow-up problem is to certify the n-partite GHZ state using one of the Werner-WolfŻukowski-Brukner inequalities
> one could consider generalisations of the CHSH inequality with more than two settings per party