Self-testing through EPR-steering

Given our formalism, the self-testing of quantum states is rendered extremely easy due to the purification principle: every density matrix ${\rho }_{A}$ on some system A can result as the marginal state of some bipartite pure state ${| \psi \rangle }_{{AB}}$ on the joint system AB such that ${\rho }_{A}={\rm{tr}}({| \psi \rangle }_{{AB}}{\langle \psi | }_{{AB}})$, and this pure state is uniquely defined up to an isometry on system B. Therefore, in our formalism, we can observe that given a reduced state ${\rho }_{C}={{\rm{tr}}}_{P}(| \psi \rangle \langle \psi | )$ we can describe the state $| \psi \rangle$ upto an isometry on provider's system. In particular, due to the Schmidt decomposition of the reduced state ${\rho }_{C}={\sum }_{i}{\lambda }_{i}| {\mu }_{i}\rangle \langle {\mu }_{i}|$ (such that ${\sum }_{i}{\lambda }_{i}=1$ and ${\lambda }_{i}\geqslant 0$ for all i) we have a purification of the form:

$$\begin{eqnarray*}&&| \psi \rangle =\displaystyle \sum _{i}\sqrt{{\lambda }_{i}}| {\mu }_{i}\rangle | {\nu }_{i}\rangle ,\end{eqnarray*}$$

where $\{| {\mu }_{i}\rangle \}{}_{i}$ ($\{| {\nu }_{i}\rangle \}{}_{i}$) is some set of orthogonal states in ${{ \mathcal H }}_{C}$ (${{ \mathcal H }}_{P}$). The local isometry ${\rm{\Phi }}\;:{{ \mathcal H }}_{P}\to {{ \mathcal H }}_{P}\otimes {{ \mathcal H }}_{P}^{\prime }$ then maps the set ($\{| {\nu }_{i}\rangle \}{}_{i}$) to another set of orthogonal states ($\{| {\nu }_{i}^{\prime }\rangle \}{}_{i}$).

As a consequence of our formalism, we can establish that $| \tilde{\psi }\rangle$ and $| \psi \rangle$ are equivalent solely by checking to see if the reduced state ${\tilde{\rho }}_{C}={{\rm{tr}}}_{P}(| \tilde{\psi }\rangle \langle \tilde{\psi }| )$ is equal to the reduced state ${\rho }_{C}={{\rm{tr}}}_{P}(| \psi \rangle \langle \psi | )$. Another obvious consequence for entanglement verification between the client and provider is that they share some entanglement if and only if ${\rho }_{C}$ is mixed. This is purely a consequence of the assumption that they share a pure state. Indeed, it is cryptographically well-motivated to say that the provider produces a pure state since this gives the provider maximal information about the devices that are used in a protocol.

Even though self-testing of states is rendered easy by our assumptions, the self-testing of measurements does not follow from only looking at the reduced state ${\tilde{\rho }}_{C}$. In other words, knowing the global pure $| \psi \rangle$ from the reduced state ${\tilde{\rho }}_{C}$, does not immediately imply that the provider is making the required measurements on a useful part of that pure state. It should be emphasised that in any one-sided device-independent quantum information protocol, measurements will be made on a state in any task to extract classical information from the systems, both trusted and untrusted. The self-testing of measurements made by an untrusted agent is, as explicitly stated in equation (1), crucial. We give a simple example to illustrate this point. This is an example of a physical system that a provider can prepare and a measurement they can perform.

Example 1. Establishing that the client and provider share a state that is equivalent to a reference state is not immediately useful. Consider the situation where the provider prepares the state $| \psi ^{\prime} \rangle =\tfrac{1}{\sqrt{2}}(| {0}_{C}\rangle | {0}_{{P}_{1}}\rangle | {0}_{{P}_{2}}\rangle +| {1}_{C}\rangle | {1}_{{P}_{1}}\rangle | {0}_{{P}_{2}}\rangle )$ where the subscripts ${P}_{1}$ and ${P}_{2}$ label two qubits that the provider retains and sends the qubit with the subscript $C$ to the client. The two qubits labelled by ${P}_{1}$ and ${P}_{2}$ can be jointly measured or individually measured. In this example the provider's measurement solely consists of measuring qubit ${P}_{2}$ and ignoring qubit ${P}_{1}$ such that measurement projectors are of the form ${{\mathbb{I}}}_{{P}_{1}}\otimes {({E}_{a| x})}_{{P}_{2}}$. Therefore, the reduced state of the client is ${\rho }_{C}=\tfrac{{\mathbb{I}}}{2}$ which indicates that the client and provider share a maximally entangled state. However, every element of the assemblage $\{{\sigma }_{a| x}\}{}_{a,x}$ is ${\sigma }_{a| x}=\tfrac{{\mathbb{I}}}{2}$, and thus unaffected by any measurement performed by the provider. Therefore we cannot say anything about the provider's measurements and, furthermore, the entanglement is not being utilised by the provider and will thus not be useful for any quantum information task.

This example just highlights that in our scenario it only makes sense to establish equivalence between a physical experiment and reference experiment taking into account both the state and measurements. The example motivates the need to study the assemblage generated in our scenario and not just the reduced state. Also, as will be shown later, this allows us to construct explicit isometries demonstrating equivalence between a physical and reference experiment instead of just knowing that such an isometry exists. In colloquial terms, being able to explicitly construct an isometry allows one to be able to 'locate' their desired state within the physical state.

So far we have assumed perfect equivalence between the reference and physical experiment as described by equations (1). In section 2.2 we extend our discussion to the case where equivalence can be established approximately which is known as RST. Instead of using the reduced state of the client and assemblage, we may wish to study self-testing given the correlations resulting from measurements on the assemblage and we discuss this in section 2.3.

In this section we formally introduce robust AST and indicate its advantages and limitations. Before this we need to recall some mathematical notation in order to discuss 'robustness'. We need an appropriate distance measure between operators acting on a Hilbert space. To facilitate this we will use the Schatten 1-norm $| | A| {| }_{1}$ for $A\in { \mathcal L }({ \mathcal H })$ being a linear operator acting on ${ \mathcal H }$. This norm is directly related to $D(\rho ,\sigma )$, the trace distance between quantum states since $D(\rho ,\sigma )=\tfrac{1}{2}| | \rho -\sigma | {| }_{1}$ for $\rho ,\sigma \in { \mathcal D }({ \mathcal H })$. Equivalently, $D(\rho ,\sigma )=\tfrac{1}{2}{\sum }_{i}| {\lambda }_{i}|$ where ${\lambda }_{i}$ is the ith eigenvalue of the operator $(\rho -\sigma )$. Another property of the trace distance is that when $\rho =| a\rangle \langle a|$ and $\sigma =| b\rangle \langle b|$ are pure then $D(| a\rangle \langle a| ,| b\rangle \langle b| )=\sqrt{1-| \langle a| b\rangle {| }^{2}}$ [29].

The motivation for introducing a distance measure is clear when we consider imperfect experiments. That is, if our physical experiment deviates from the predictions of our reference experiment by a small amount can we be sure that our physical experiment is (up to a local isometry on ${{ \mathcal H }}_{P}$) close (in the trace distance) to our reference experiment? Now we can utilise the trace distance to describe closeness between the physical state $| \psi \rangle$ and reference state $| \tilde{\psi }\rangle$. To whit, if $D({\rho }_{C},{\tilde{\rho }}_{C})=\epsilon \gt 0$ where ${\tilde{\rho }}_{C}={{\rm{tr}}}_{P}(| \tilde{\psi }\rangle \langle \tilde{\psi }| )$ and allowing for isometries Φ on the provider's side, then the minimal distance between physical and reference states will be the minimal value of

$$\begin{eqnarray}&&D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )=\sqrt{1-| \langle { \mathcal A }| \langle \tilde{\psi }| {\rm{\Phi }}\rangle {| }^{2}}\end{eqnarray} \tag{ 2 }$$

for $| {\rm{\Phi }}\rangle ={\rm{\Phi }}(| \psi \rangle )$. Clearly, $D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )\geqslant D({\tilde{\rho }}_{C},{\rho }_{C})=\epsilon$ since the trace distance does not increase when tracing out the provider's sub-system.

This lower bound on the distance in equation (2) does not tell us that there is an isometry achieving this bound. We wish to be able to state that there exists an isometry for which the distance in equation (2) is small. Furthermore it would be preferable to be able to construct this isometry. This is, in essence, RST. We now formalise this intuition in the following definition:

Definition 1. Given a reference experiment consisting of the state $| \tilde{\psi }\rangle \in {{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P}^{\prime }$ with reduced state ${\tilde{\rho }}_{C}$ and measurements $\{{\tilde{E}}_{a| x}\}{}_{a,x}$ such that the assemblage $\{{\tilde{\sigma }}_{a| x}\}{}_{a,x}$ has elements ${\tilde{\sigma }}_{a| x}={{\rm{tr}}}_{P}({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x}| \tilde{\psi }\rangle ),\forall a,x$. Also given a physical experiment with the state $| \psi \rangle \in {{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P}$, reduced state ${\rho }_{C}$ and measurements $\{{E}_{a| x}\}{}_{a,x}$ such that the assemblage $\{{\sigma }_{a| x}\}{}_{a,x}$ has elements ${\sigma }_{a| x}={{\rm{tr}}}_{P}({{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle ),\forall a,x$. If, for some real $\epsilon \gt 0,D({\tilde{\rho }}_{C},{\rho }_{C})\leqslant \epsilon$ and $| | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}\leqslant \epsilon ,\forall a,x$, then $f(\epsilon )$-robust AST ($f(\epsilon )$-AST) is possible if the assemblage $\{{\sigma }_{a| x}\}{}_{a,x}$ implies that there exists an isometry ${\rm{\Phi }}\;:{{ \mathcal H }}_{P}\to {{ \mathcal H }}_{P}\otimes {{ \mathcal H }}_{P}^{\prime }$ such that

$$\begin{eqnarray}&&D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )\leqslant f(\epsilon ),\\ &&\quad | | | {\rm{\Phi }},{E}_{a| x}\rangle \langle {\rm{\Phi }},{E}_{a| x}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes ({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x})| \tilde{\psi }\rangle \langle \tilde{\psi }| ({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x})| {| }_{1}\leqslant f(\epsilon )\\ &&\quad {\rm{for}}| {\rm{\Phi }}\rangle ={\rm{\Phi }}(| \psi \rangle ),| {\rm{\Phi }},{E}_{a| x}\rangle ={\rm{\Phi }}({{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle ),| { \mathcal A }\rangle \in {{ \mathcal H }}_{P}\;{\rm{and}}\;f\;:{\mathbb{R}}\to {\mathbb{R}}.\end{eqnarray} \tag{ 3 }$$

In this definition, in order to simplify matters, we have bounded both the distance between physical and reference states both with and without measurements by the function $f(\epsilon )$. It will often be the case that the trace distance between states (without measurements) will be smaller than the distance between measured states, but we are considering the worst case analysis. In further study, it could be of interest to give a finer distinction between these distance measures in the definition.

Note also that, in this definition, we only ask for the existence of an isometry. Later, in section 3, we will construct an isometry for RST which will be more useful for various protocols. Also, for this definition to be useful, a desirable function would be $f(\epsilon )\leqslant O\left({\epsilon }^{\tfrac{1}{p}}\right)$ where p is upper-bounded by a small positive integer. If $D({\tilde{\rho }}_{C},{\rho }_{C})=\epsilon$, as mentioned earlier this establishes a lower bound on the distance between physical and reference experiments, and so the ideal case would be $O(\epsilon )$-AST. We now give a simple example to show that, in general, this ideal case is not obtainable.

Example 2. The client has a three-dimensional Hilbert space ${{ \mathcal H }}_{C}$. The reference experiment consists of the state $| \tilde{\psi }\rangle =\tfrac{1}{\sqrt{2}}(| {0}_{C}{0}_{P}\rangle +| {1}_{C}{1}_{P}\rangle )$ with measurements $\{{\tilde{E}}_{0| 0}=| {0}_{P}\rangle \langle {0}_{P}| ,\;{\tilde{E}}_{1| 0}=| {1}_{P}\rangle \langle {1}_{P}| ,{\tilde{E}}_{0| 1}=| {+}_{P}\rangle \langle {+}_{P}| ,\;{\tilde{E}}_{1| 1}=| {-}_{P}\rangle \langle {-}_{P}| \}$ and $| {\pm }_{P}\rangle =\tfrac{1}{\sqrt{2}}(| {0}_{P}\rangle \pm | {1}_{P}\rangle )$ where ${{ \mathcal H }}_{P}^{\prime }$ is a two-dimensional Hilbert space. The assemblage for this reference experiment has the following elements:

$$\begin{eqnarray*}{\tilde{\sigma }}_{0| 0} & = & \displaystyle \frac{1}{2}| {0}_{C}\rangle \langle {0}_{C}| ,\qquad \quad {\tilde{\sigma }}_{1| 0}=\displaystyle \frac{1}{2}| {1}_{C}\rangle \langle {1}_{C}| ,\\ {\tilde{\sigma }}_{0| 1} & = & \displaystyle \frac{1}{2}| {+}_{C}\rangle \langle {+}_{C}| ,\qquad \quad {\tilde{\sigma }}_{1| 1}=\displaystyle \frac{1}{2}| {-}_{C}\rangle \langle {-}_{C}| .\end{eqnarray*}$$

The physical experiment consists of the state $| \psi \rangle =\sqrt{1-\epsilon }| \tilde{\psi }\rangle | {0}_{P^{\prime} }\rangle +\sqrt{\epsilon }| \xi \rangle | {1}_{P^{\prime} }\rangle$ where $| \xi \rangle =| {2}_{C}{0}_{P}\rangle$ and the subscript $P^{\prime}$ denotes a second qubit that the provider has in their possession. The measurements in the physical experiment are ${E}_{i| j}={\tilde{E}}_{i| j}\otimes | {0}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| +| {i}_{P}\rangle \langle {i}_{P}| \otimes | {1}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }|$ for $i\in \{0,1\}$. The state $| \psi \rangle$ has the reduced state ${\rho }_{C}=\tfrac{(1-\epsilon )}{2}(| {0}_{C}\rangle \langle {0}_{C}| +| {1}_{C}\rangle \langle {1}_{C}| )+\epsilon | {2}_{C}\rangle \langle {2}_{C}|$ thus implying that $D({\rho }_{C},{\tilde{\rho }}_{C})=\epsilon$. The assemblage for this physical experiment then has the elements:

$$\begin{eqnarray*}{\sigma }_{0| 0} & = & \displaystyle \frac{(1-\epsilon )}{2}| {0}_{C}\rangle \langle {0}_{C}| +\epsilon | {2}_{C}\rangle \langle {2}_{C}| ,\qquad \qquad {\sigma }_{1| 0}=\displaystyle \frac{(1-\epsilon )}{2}| {1}_{C}\rangle \langle {1}_{C}| ,\\ {\sigma }_{0| 1} & = & \displaystyle \frac{(1-\epsilon )}{2}| {+}_{C}\rangle \langle {+}_{C}| \;+\;\epsilon | {2}_{C}\rangle \langle {2}_{C}| ,\qquad \qquad {\sigma }_{1| 1}=\displaystyle \frac{(1-\epsilon )}{2}| {-}_{C}\rangle \langle {-}_{C}| .\end{eqnarray*}$$

From the above assemblages we observe that $| | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}\lt \tfrac{3}{2}\epsilon =\epsilon ^{\prime} ,\forall a,x$. Here we have just defined a new closeness parameter $\epsilon ^{\prime}$ for the convenience of our definitions. Given these physical and reference experiments, we now wish to calculate a lower bound on $D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )$ for all possible isometries Φ in the definition above; this will give a lower-bound on the function $f(\epsilon ^{\prime} )$ for $f(\epsilon ^{\prime} )$-AST. To do this, we introduce the notation $| \tilde{0}\rangle$ for the ancillae that the provider can introduce and ${U}_{P}$ as the unitary that they can perform jointly on the ancillae and their share of the physical state $| \psi \rangle$. Thisthen gives us:

$$\begin{eqnarray*}&&D({U}_{P}(| \psi \rangle \langle \psi | \otimes | \hat{0}\rangle \langle \hat{0}| ){U}_{P}^{\dagger },| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )=\sqrt{1-{F}^{2}},\end{eqnarray*}$$

where

$$\begin{eqnarray*}F & = & | \langle { \mathcal A }| \langle \tilde{\psi }| U| \psi \rangle | \hat{0}\rangle | \\ & = & \displaystyle \frac{\sqrt{1-\epsilon }}{2}| \langle { \mathcal A }| (\langle {0}_{C}{0}_{P}| ({{\mathbb{I}}}_{C}\otimes {U}_{P})| {0}_{C}{0}_{P}\rangle +\langle {1}_{C}{1}_{P}| ({{\mathbb{I}}}_{C}\otimes {U}_{P})| {1}_{C}{1}_{P}\rangle )| \hat{0}\rangle | ,\end{eqnarray*}$$

where ${{\mathbb{I}}}_{C}$ is the identity on the client's system. Thus maximising this quantity for all isometries, we obtain the maximal value ${F}^{* }=\sqrt{1-\epsilon }=\sqrt{1-\tfrac{2\epsilon ^{\prime} }{3}}$ and the lower bound $D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )\geqslant \sqrt{\tfrac{2\epsilon ^{\prime} }{3}}$.

This example excludes the possibility of having $O(\epsilon )$-AST given that the client's Hilbert space is three-dimensional. We will later return to this reference experiment in section 3.1 with the modification that the client's Hilbert space is two-dimensional.

As outlined earlier, EPR-steering can be studied from the point-of-view of the probabilities obtained from measurements performed on elements of an assemblage, i.e. known measurements made by the trusted party. This point-of-view is native to Bell non-locality and is suitable for making further parallels between non-locality and EPR-steering. In this regard one can construct EPR-steering inequalities (the EPR-steering analogues of Bell inequalities) which can be written as a linear combination of the measurement probabilities [30]. The two figures-of-merit, assemblages and measurement correlations, lead to a certain duality in the theory of EPR steering. The approach that one will use depends on the underlying scenario. In the case when correlations are obtained by performing a tomographically complete set of measurements (on the trusted system) the two approaches become completely equivalent. However, in some cases probabilities obtained by performing a tomographically incomplete set of measurements, or even just the amount of violation of some steering inequality can provide all necessary information. Another possibility is that a trusted party can perform only two measurements and nothing more, i.e. has no resources to perform complete tomography. In this section we consider the definition and utility of defining RST with respect to these probabilities for an appropriate notion of robustness. This approach to self-testing is not immediately equivalent to the notion of AST defined previously (even if tomographically complete measurements are made) for reasons that will be become clear.

Recall the probabilities $p(a,b| x,y)={\rm{tr}}({F}_{b| y}{\sigma }_{a| x})$ for ${F}_{b| y}$ being elements of general measurement associated with the outcome b for measurement choice y such that ${\sum }_{b}{F}_{b| y}={{\mathbb{I}}}_{C}$. Naturally, we can also obtain the probabilities $p(b| y)={\rm{tr}}({F}_{b| y}{\rho }_{C})$. In addition to the 'physical probabilities' $p(a,b| x,y)$, we have the 'reference probabilities' $\{\tilde{p}(a,b| x,y)\}$ which refer to the probabilities resulting from making the same measurements $\{{F}_{b| y}\}{}_{b,y}$ on a reference assemblage $\{{\tilde{\sigma }}_{a| x}\}$ as described above. Performing RST given these probabilities will be the focus of this section.

A useful definition of the Schatten 1-norm is $| | A| {| }_{1}={{\rm{sup}}}_{| | B| | \leqslant 1}| {\rm{tr}}({BA})|$ where $| | \cdot | |$ is the operator norm. Since ${F}_{b| y}$ is a positive operator with operator norm upper bounded by 1 and if $D({\rho }_{C},{\tilde{\rho }}_{C})\leqslant \epsilon$ and for all elements ${\sigma }_{a| x}$ of an assemblage $| | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}\leqslant \epsilon$ we can conclude that

$$\begin{eqnarray*}| \tilde{p}(a,b| x,y)-p(a,b| x,y)| & = & | {\rm{tr}}[{F}_{b| y}({\sigma }_{a| x}-{\tilde{\sigma }}_{a| x})]| \quad \leqslant \quad | | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}\leqslant \epsilon ,\\ | p(b| y)-\tilde{p}(b| y)| & = & | {\rm{tr}}[{F}_{b| y}({\rho }_{C}-{\tilde{\rho }}_{C})]| \leqslant 2D({\rho }_{C},{\tilde{\rho }}_{C})\leqslant 2\epsilon \end{eqnarray*}$$

for all $a,b,x,y$. This then establishes that knowledge of the assemblage and establishing its closeness to the assemblage associated with a reference experiment implies closeness in the probabilities obtained from both experiments. Clearly, the converse is not necessarily true and closeness in probabilities does not always imply closeness of reduced states and assemblages. Assemblages can be calculated from the statistics obtained from performing tomographically complete measurements, and then the distance (in Schatten 1-norm) between this assemblage and some ideal assemblage can be calculated. However, even for tomographically complete measurements $\{{F}_{b| y}\}{}_{b,y}$, we only have that $| {\rm{tr}}[{F}_{b| y}({\sigma }_{a| x}-{\tilde{\sigma }}_{a| x})]| \quad \leqslant \quad | | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}$ thus having $| {\rm{tr}}[{F}_{b| y}({\sigma }_{a| x}-{\tilde{\sigma }}_{a| x})]| \leqslant \epsilon$ does not imply $| | {\tilde{\sigma }}_{a| x}-{\sigma }_{a| x}| {| }_{1}\leqslant \epsilon$. This goes to show that the AST approach is distinct from solely looking at the difference between probabilities.

Inspired by the literature in standard self-testing (see, e.g. [5, 6]), it should still be possible to attain RST based on probabilities for measurements on assemblages and with this in mind, we give the following definition:

Definition 2. Given a reference experiment consisting of the state $| \tilde{\psi }\rangle \in {{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P}^{\prime }$ with reduced state ${\tilde{\rho }}_{C}$ and measurements $\{{\tilde{E}}_{a| x}\}{}_{a,x}$ such that the assemblage $\{{\tilde{\sigma }}_{a| x}\}{}_{a,x}$ has elements ${\tilde{\sigma }}_{a| x}={{\rm{tr}}}_{P}({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x}| \tilde{\psi }\rangle ),\forall a,x$. Also given a physical experiment with the state $| \psi \rangle \in {{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P}$, reduced state ${\rho }_{C}$ and measurements $\{{E}_{a| x}\}{}_{a,x}$ such that the assemblage $\{{\sigma }_{a| x}\}{}_{a,x}$ has elements ${\sigma }_{a| x}={{\rm{tr}}}_{P}({{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle ),\forall a,x$. Additionally given a set $\{{F}_{b| y}\}{}_{b,y}$ of general measurements that act on ${{ \mathcal H }}_{C}$ such that $p(a,b| x,y)={\rm{tr}}({F}_{b| y}{\sigma }_{a| x})$ and $\tilde{p}(a,b| x,y)={\rm{tr}}({F}_{b| y}{\tilde{\sigma }}_{a| x})\forall a,x$. If, for some real $\epsilon \gt 0$,

$$\begin{eqnarray*}| \tilde{p}(a,b| x,y)-p(a,b| x,y)| & \leqslant & \epsilon ,\\ | \tilde{p}(b| y)-p(b| y)| & \leqslant & \epsilon ,\\ | \tilde{p}(a| x)-p(a| x)| & \leqslant & \epsilon ,\end{eqnarray*}$$

$\forall a,x,b,y$, then $f(\epsilon )$-robust CST ($f(\epsilon )$-CST) is possible if the probabilities imply that there exists an isometry ${\rm{\Phi }}\;:{{ \mathcal H }}_{P}\to {{ \mathcal H }}_{P}\otimes {{ \mathcal H }}_{P}^{\prime }$ such that

$$\begin{eqnarray*}&&D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle | \tilde{\psi }\rangle \langle { \mathcal A }| \langle \tilde{\psi }| )\leqslant f(\epsilon ),\\ &&\qquad | | | {\rm{\Phi }},{E}_{a| x}\rangle \langle {\rm{\Phi }},{E}_{a| x}| -| { \mathcal A }\rangle ({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x})| \tilde{\psi }\rangle \langle { \mathcal A }| \langle \tilde{\psi }| ({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{a| x})| {| }_{1}\leqslant f(\epsilon )\\ &&\qquad \mathrm{for}\;| {\rm{\Phi }}\rangle ={\rm{\Phi }}(| \psi \rangle ),| {\rm{\Phi }},{E}_{a| x}\rangle ={\rm{\Phi }}({{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle ),| { \mathcal A }\rangle \in {{ \mathcal H }}_{P}\;{\rm{and}}\;{\rm{}}\;f\;:{\mathbb{R}}\to {\mathbb{R}}.\end{eqnarray*}$$

Instead of directly bounding the distance between reference and physical probabilities, we can indirectly bound this distance by utilising an EPR-steering inequality. In the literature on standard self-testing, probability distributions that near-maximally violate a Bell inequality robustly self-test the state and measurements that produce the maximal violation [5, 6]. As a first requirement, there needs to be a unique probability distribution that achieves this maximal violation, and we now have many examples of Bell inequalities where this happens. The same applies to EPR-steering inequalities: there needs to be a unique assemblage that produces the maximal violation of an EPR-steering inequality. Furthermore this unique assemblage needs to imply a unique reference experiment (up to a local isometry). For EPR-steering inequalities of the form ${\sum }_{a| x}{\alpha }_{a,x}{\rm{tr}}({F}_{a| b}{\sigma }_{a| x})\geqslant 0$ for real numbers ${\alpha }_{a,x}$, any assemblage that violates this inequality is necessarily steerable. If all quantum assemblages satisfy ${\sum }_{a| x}{\alpha }_{a,x}{\rm{tr}}({F}_{a| b}{\sigma }_{a| x})\geqslant -\beta$ for some positive real number β then $-\beta$ is the maximal violation of the EPR-steering inequality. If we consider probabilities of the form $p(a,b| x,y)={\rm{tr}}({F}_{b| y}{\sigma }_{a| x})$ that satisfy ${\sum }_{a| x}{\alpha }_{a,x}{\rm{tr}}({F}_{a| b}{\sigma }_{a| x})\leqslant -(\beta -\epsilon )$ then they are at most -far from the reference experiment that produces the maximal violation of $-\beta$. We will make use of this approach to CST in section 3.2.

We now briefly return to the issue of complex conjugation. As mentioned above and discussed in appendix A, the AST approach is advantageous to the standard self-testing approach in that we can rule out the state and measurements in the reference experiment both being the complex conjugate of our ideal reference experiment. One issue with CST is that since we are reconsidering probabilities for a fixed set of measurements made by the client, if the measurements are invariant under complex conjugation then the provider can prepare a state and make measurements that are both the complex conjugate of the ideal case without altering the statistics. This can be remedied by the client choosing measurements that have complex entries as long as it does not drastically affect the ability to achieve $f(\epsilon )$-CST.

We first set-out the reference experiment that we will be studying for the rest of this section. It consists of the experiment described in section 2.2 but now with the client's Hilbert space being two-dimensional. Recall that the state is $| \tilde{\psi }\rangle =\tfrac{1}{\sqrt{2}}(| 00\rangle +| 11\rangle )$ and the measurements are $\{{\tilde{E}}_{0| 0}=| 0\rangle \langle 0| ,\;{\tilde{E}}_{1| 0}=| 1\rangle \langle 1| ,{\tilde{E}}_{0| 1}=| +\rangle \langle +| ,\;{\tilde{E}}_{1| 1}=| -\rangle \langle -| \}$ and $| \pm \rangle =\tfrac{1}{\sqrt{2}}(| 0\rangle \pm | 1\rangle )$ where we have dropped the subscripts for reasons of clarity. The assemblage for this reference experiment has the following elements:

$$\begin{eqnarray*}{\tilde{\sigma }}_{0| 0} & = & \displaystyle \frac{1}{2}| 0\rangle \langle 0| ,\qquad \qquad {\tilde{\sigma }}_{1| 0}=\displaystyle \frac{1}{2}| 1\rangle \langle 1| ,\\ {\tilde{\sigma }}_{0| 1} & = & \displaystyle \frac{1}{2}| +\rangle \langle +| ,\qquad \qquad {\tilde{\sigma }}_{1| 1}=\displaystyle \frac{1}{2}| -\rangle \langle -| .\end{eqnarray*}$$

We will henceforth call this reference experiment the EPR experiment. We can now state a result about AST for this experiment.

Theorem 1. For the EPR experiment, $f(\epsilon )$-robust AST is possible for $f(\epsilon )=24\sqrt{\epsilon }+\epsilon$.

Before proving this theorem we will present two useful observations that will be used in the proof. The first observation is a lemma about the norm that we are using while the second is specific to the self-testing of the EPR experiment. We require the notation $| | | v\rangle | | \quad =\quad \sqrt{\langle v| v\rangle }$.

Lemma 1. For any two vectors $| u\rangle ,| v\rangle$ where $| | | u\rangle | | \quad \leqslant \quad 1$ and $| | | v\rangle | | \quad \leqslant \quad 1$, if $| | | u\rangle -| v\rangle | | \quad \leqslant \quad \eta \leqslant 1$, then for another vector $| t\rangle$ such that $| | | t\rangle | | \quad \leqslant \quad \beta ,| | (| u\rangle -| v\rangle )\langle t| | {| }_{1}\leqslant \beta \eta$ and $| | | t\rangle (\langle u| -\langle v| )| {| }_{1}\leqslant \beta \eta$

Proof. This fact essentially follows from the definition of $| | \cdot | |$. That is, $| | | u\rangle -| v\rangle | |$ = $\sqrt{\langle u| u\rangle +\langle v| v\rangle -\langle u| v\rangle -\langle v| u\rangle }$ and since the rank of $B=(| u\rangle -| v\rangle )\langle t|$ is 1 then the $| | B| {| }_{1}=\sqrt{{\rm{tr}}({{BB}}^{\dagger })}=| | | t\rangle | | \sqrt{\langle u| u\rangle +\langle v| v\rangle -\langle u| v\rangle -\langle v| u\rangle }$ which concludes our proof (along with the fact that $| | B| {| }_{1}=| | {B}^{\dagger }| {| }_{1}$).

The next observation follows from the conditions outlined in the definition of $f(\epsilon )$-AST and is as follows:

Lemma 2. If $| | {\sigma }_{a| x}-{\tilde{\sigma }}_{a| x}| {| }_{1}\leqslant \epsilon$ and $D({\rho }_{C},{\tilde{\rho }}_{C})\leqslant \epsilon$ then

$$\begin{eqnarray*}&&| | {{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle -{\tilde{E}}_{a| x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | | \quad \leqslant \quad 2\sqrt{\epsilon }.\end{eqnarray*}$$

Proof. The proof follows from a series of basic observations:

$$\begin{eqnarray*}| | {{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle -{\tilde{E}}_{a| x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | | & = & \sqrt{\langle \psi | {{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle +\langle \psi | {\tilde{E}}_{a| x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle -2\langle \psi | {\tilde{E}}_{a| x}\otimes {E}_{a| x}| \psi \rangle }\\ & \leqslant & \sqrt{1+2\epsilon -2\langle \psi | {\tilde{E}}_{a| x}\otimes {E}_{a| x}| \psi \rangle }\\ & \leqslant & \sqrt{1+2\epsilon -2\left(\displaystyle \frac{1}{2}-\epsilon \right)}\\ & = & 2\sqrt{\epsilon }.\end{eqnarray*}$$

The first inequality results from the fact that $\langle \psi | {{\mathbb{I}}}_{C}\otimes {E}_{a| x}| \psi \rangle ={{\rm{tr}}}_{P}({\sigma }_{a| x})$ and $\langle \psi | {\tilde{E}}_{a| x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle ={{\rm{tr}}}_{P}({\tilde{E}}_{a| x}{\rho }_{C})$ and that $| {\rm{tr}}({\sigma }_{a| x}-{\tilde{\sigma }}_{a| x})| \leqslant \epsilon$ and $| {\rm{tr}}({\tilde{E}}_{a| x}{\rho }_{C}-{\tilde{E}}_{a| x}{\tilde{\rho }}_{C})| \leqslant \epsilon$. The second inequality follows from the observation that $| {\rm{tr}}({\tilde{E}}_{a| x}{\sigma }_{a| x}-{\tilde{E}}_{a| x}{\tilde{\sigma }}_{a| x})| \leqslant \epsilon$.

We are now in a position to prove theorem 1.

Proof. Recall that we are promised that

$$\begin{eqnarray*}D({\rho }_{C},{\tilde{\rho }}_{C}) & \leqslant & \epsilon ,\\ | | {\sigma }_{a| x}-{\tilde{\sigma }}_{a| x}| {| }_{1} & \leqslant & \epsilon ,\end{eqnarray*}$$

for all $a,x$ where ${\tilde{\rho }}_{C}={{\rm{tr}}}_{P}(| \tilde{\psi }\rangle \langle \tilde{\psi }| )$. The aim is now to find an explicit isometry Φ that gives a non-trivial upper bound for the following expression:

$$\begin{eqnarray}&&| | | {\rm{\Phi }},{Q}_{a| x}\rangle \langle {\rm{\Phi }},{Q}_{a| x}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes ({{\mathbb{I}}}_{C}\otimes {\tilde{Q}}_{a| x})| \tilde{\psi }\rangle \langle \tilde{\psi }| ({{\mathbb{I}}}_{C}\otimes {\tilde{Q}}_{a| x})| {| }_{1},\end{eqnarray} \tag{ 4 }$$

for ${Q}_{a| x}\in \{{\mathbb{I}},{E}_{a| x}\},{\tilde{Q}}_{a| x}\in \{{\mathbb{I}},{\tilde{E}}_{a| x}\}$ and $| {\rm{\Phi }},{Q}_{a| x}\rangle$ as defined before. We first focus on the cases where ${Q}_{a| x}={{\mathbb{I}}}_{P}$ and ${\tilde{Q}}_{a| x}={\mathbb{I}}={{\mathbb{I}}}_{C}$ and use this to argue the more general result.

The isometry that we use is the so-called SWAP isometry that has been used multiple times in the self-testing literature. In this isometry (see figure 2) an ancilla qubit is introduced in the state $| {+}_{P^{\prime} }\rangle \in {{ \mathcal H }}_{P^{\prime} }$ where $P^{\prime}$ denotes the ancilla register on the provider's side in addition to the provider's Hilbert space ${{ \mathcal H }}_{P}$. After introducing the ancilla a unitary operator is applied to both the provider's part of the physical state and the ancilla, i.e. $| \psi \rangle | {+}_{P^{\prime} }\rangle \to ({{\mathbb{I}}}_{C}\otimes {VHU})| \psi \rangle | {+}_{P^{\prime} }\rangle$ where $U=| {0}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| \otimes {{\mathbb{I}}}_{P}+| {1}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }| \otimes {Z}_{P},V$ = $| {0}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| \otimes {{\mathbb{I}}}_{P}+| {1}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }| \otimes X$ and $H=| {+}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| +| {-}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }|$ and $Z=2{E}_{0| 0}-{{\mathbb{I}}}_{P},X=2{E}_{0| 1}-{{\mathbb{I}}}_{P}$ and ${X}^{2}={Z}^{2}={{\mathbb{I}}}_{P}$. After applying this isometry to the physical state $| \psi \rangle$ we obtain the state

$$\begin{eqnarray*}&&| \psi ^{\prime} \rangle ={E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle .\end{eqnarray*}$$

The desired result of this isometry to establish an ebit in the Hilbert space ${{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P^{\prime} }={{ \mathcal H }}_{C}\otimes {{ \mathcal H }}_{P}^{\prime }$ in addition to the measurements ${\tilde{E}}_{a| x}$ acting on the Hilbert space ${{ \mathcal H }}_{P^{\prime} }$. Therefore we wish to give an upper bound to

$$\begin{eqnarray}&&| | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}.\end{eqnarray} \tag{ 5 }$$

At this point we can now apply a combination of lemmas 1 and 2 to bound this norm. Firstly, we observe that by virtue of lemma 2 we have that

$$\begin{eqnarray*}| | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )-({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )| | & \leqslant & 2\sqrt{\epsilon },\\ | | ({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )-({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{\tilde{E}}_{1| 0}\otimes X| \psi \rangle | {1}_{P^{\prime} }\rangle )| | & \leqslant & 2\sqrt{\epsilon },\end{eqnarray*}$$

where, for the sake of brevity, we do not write identities ${{\mathbb{I}}}_{C}$, e.g. ${E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle ={{\mathbb{I}}}_{C}\otimes {E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle$.

We can apply these observations in conjunction with lemma 1 (and noticing that $| | {E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle | | \quad =\quad 1$) to equation (5) to obtain

$$\begin{eqnarray*} & & | | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ & \leqslant & 2\sqrt{\epsilon }+| | ({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ & \leqslant & 4\sqrt{\epsilon }+| | ({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{\tilde{E}}_{1| 0}\otimes X| \psi \rangle | {1}_{P^{\prime} }\rangle )\\ & & \times (\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}.\end{eqnarray*}$$

Since $X=2{E}_{0| 1}-{{\mathbb{I}}}_{P}$ and, for the Pauli-X matrix ${\tau }_{x}=2| +\rangle \langle +| -{\mathbb{I}}$, we obtain the following result that

$$\begin{eqnarray*}| | {{\mathbb{I}}}_{C}\otimes X| \psi \rangle -{\tau }_{x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | | & \leqslant & 2| | {{\mathbb{I}}}_{C}\otimes {E}_{0| 1}| \psi \rangle -{\tilde{E}}_{0| 1}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | | -| | | \psi \rangle -| \psi \rangle | | \\ & \leqslant & 4\sqrt{\epsilon }.\end{eqnarray*}$$

We then obtain

$$\begin{eqnarray*}&&| | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ &&\quad \leqslant 8\sqrt{\epsilon }+| | ({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{\tilde{E}}_{1| 0}{\tau }_{x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {1}_{P^{\prime} }\rangle )\\ &&\quad \quad \times (\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}.\end{eqnarray*}$$

We will now apply the same reasoning to $(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )$ but we need the fact that

$$\begin{eqnarray*}&&| | {\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{\tilde{E}}_{1| 0}{\tau }_{x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {1}_{P^{\prime} }\rangle | | \quad =\quad \sqrt{2\langle \psi | {\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle }\leqslant \sqrt{1+2\epsilon }\leqslant 1+\epsilon ,\end{eqnarray*}$$

which follows from the condition on the reduced state ${\rho }_{C}$ and ${\tilde{E}}_{1| 0}{\tau }_{x}={\tau }_{x}{\tilde{E}}_{0| 0}$. Using these observations and lemma 2 we arrive at

$$\begin{eqnarray*}&&| | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ &&\quad \leqslant 16\sqrt{\epsilon }+8\epsilon \sqrt{\epsilon }\\ &&\quad \quad +| | ({\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {0}_{P^{\prime} }\rangle +{\tilde{E}}_{1| 0}{\tau }_{x}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | {1}_{P^{\prime} }\rangle )\\ &&\quad \quad \times (\langle \psi | {\tilde{E}}_{0| 0}\otimes {{\mathbb{I}}}_{P}\langle {0}_{P^{\prime} }| \;+\;\langle \psi | {\tau }_{x}{\tilde{E}}_{1| 0}\otimes {{\mathbb{I}}}_{P}\langle {1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ &&\quad =16\sqrt{\epsilon }+8\epsilon \sqrt{\epsilon }+| | (\langle {0}_{C}| \psi \rangle | {0}_{C}{0}_{P^{\prime} }\rangle \\ &&\quad \quad +\langle {0}_{C}| \psi \rangle | {1}_{C}{1}_{P^{\prime} }\rangle )(\langle \psi | {0}_{C}\rangle \langle {0}_{C}{0}_{P^{\prime} }| +\langle \psi | {0}_{C}\rangle \langle {1}_{C}{1}_{P^{\prime} }| )-| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ &&\quad =16\sqrt{\epsilon }+8\epsilon \sqrt{\epsilon }+| | 2\langle {0}_{C}| \psi \rangle | \tilde{\psi }\rangle \langle \psi | {0}_{C}\rangle \langle \tilde{\psi }| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\\ &&\quad \leqslant 16\sqrt{\epsilon }+8\epsilon \sqrt{\epsilon }+| | 2\langle {0}_{C}| \psi \rangle \langle \psi | {0}_{C}\rangle -| { \mathcal A }\rangle \langle { \mathcal A }| | {| }_{1}\\ &&\quad \leqslant 16\sqrt{\epsilon }+8\epsilon \sqrt{\epsilon }+2\epsilon ,\end{eqnarray*}$$

where to obtain the last inequality we chose $| { \mathcal A }\rangle$ to be the pure state that is proportional to $| {0}_{C}\rangle \langle {0}_{C}| \psi \rangle$, i.e. $| { \mathcal A }\rangle ={\beta }^{-\tfrac{1}{2}}| {0}_{C}\rangle \langle {0}_{C}| \psi \rangle$ where $\beta =\langle \psi | {0}_{C}\rangle \langle {0}_{C}| \psi \rangle$ thus $| {\rm{tr}}(| {0}_{C}\rangle \langle {0}_{C}| {\rho }_{C})-{\rm{tr}}(| {0}_{C}\rangle \langle {0}_{C}| {\tilde{\rho }}_{C})| \leqslant | \beta -\tfrac{1}{2}| \leqslant \epsilon$.

We have shown that $D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )\leqslant 8\sqrt{\epsilon }+4\epsilon \sqrt{\epsilon }+\epsilon$. Now we consider the case of self-testing where measurements are made. That is, establishing an upper bound on the expressions of the form in equation (4) where ${Q}_{a| x}\ne {{\mathbb{I}}}_{P}$ and ${\tilde{Q}}_{a| x}\ne {\mathbb{I}}$ and after applying the SWAP isometry described above, the projector acting on the physical state ${E}_{a| x}| \psi \rangle$ gets mapped to

$$\begin{eqnarray*}&&{E}_{0| 0}{E}_{a| x}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}{E}_{a| x}| \psi \rangle | {1}_{P^{\prime} }\rangle .\end{eqnarray*}$$

In the case that x = 0, utilising the fact that ${E}_{a| x}{E}_{a^{\prime} | x}={\delta }_{a^{\prime} }^{a}{E}_{a| x}$, for equation (4) we obtain:

$$\begin{eqnarray*}| | {E}_{0| 0}| \psi \rangle \langle \psi | {E}_{0| 0}\otimes | {0}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| -\displaystyle \frac{1}{2}| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | {0}_{C}{0}_{P^{\prime} }\rangle \langle {0}_{C}{0}_{P^{\prime} }| | {| }_{1}{\rm{}}\;{\rm{for}}\;{\rm{}}a & = & 0,\\ | | {{XE}}_{1| 0}| \psi \rangle \langle \psi | {E}_{1| 0}X\otimes | {1}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }| -\displaystyle \frac{1}{2}| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | {1}_{C}{1}_{P^{\prime} }\rangle \langle {1}_{C}{1}_{P^{\prime} }| | {| }_{1}{\rm{}}\;{\rm{for}}\;{\rm{}}a & = & 1.\end{eqnarray*}$$

By using the same reasoning as above we obtain the bounds $4\sqrt{\epsilon }+\epsilon$ and $12\sqrt{\epsilon }+\epsilon$ for the a = 0 and a = 1 cases respectively. For the case that x = 1, more work is required in bounding equation (4). However, again by repeatedly applying the observation in lemma 2, as shown in appendix B we obtain the bound of

$$\begin{eqnarray}&&| | | {\rm{\Phi }},{Q}_{a| x}\rangle \langle {\rm{\Phi }},{Q}_{a| x}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes ({{\mathbb{I}}}_{C}\otimes {\tilde{Q}}_{a| x})| \tilde{\psi }\rangle \langle \tilde{\psi }| ({{\mathbb{I}}}_{C}\otimes {\tilde{Q}}_{a| x})| {| }_{1}\leqslant 24\sqrt{\epsilon }+\epsilon ,\end{eqnarray} \tag{ 6 }$$

thus concluding the proof.□

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Central to the proof of this theorem was lemma 2, but it is worth noting that the minimal requirements for proving this lemma were bounds on the probabilities and not necessarily bounds on the elements of the assemblage. We utilised the fact that bounds on the probabilities are obtained from the elements of the assemblage, but if one only bounds the probabilities then our result still follows. We then obtain the following corollary.

Corollary 1. For the EPR experiment, $f(\epsilon )$-robust CST is possible for $f(\epsilon )=24\sqrt{\epsilon }+\epsilon$.

Furthermore, one can also obtain this result using an EPR-steering inequality as we outline in appendix C with some minor alterations to the function $f(\epsilon )$. The fact that the function $f(\epsilon )$ in theorem 1 and corollary 1 are the same suggests at the sub-optimality of our analysis, since AST could utilise more information than CST.

It is now worth commenting on the function $f(\epsilon )$ and contrasting it with results in the standard self-testing literature. In particular, we want to contrast this result with other analytical approaches. This is quite difficult since the measure of closeness to the ideal case is measured in terms of closeness to maximal violation of a Bell inequality and not in terms of elements of an assemblage or individual probabilities. Here we give an indicative comparison between the approach presented here and the current literature. Firstly, McKague, Yang and Scarani developed a means of RST where if the observed violation of the CHSH inequality is -close to the maximal violation then the state is $O({\epsilon }^{(1/4)})$-close to the ebit [5]. This is a less favourable polynomial than our result which demonstrates $O(\sqrt{\epsilon })$-closeness. On the other hand, the work of Reichardt, Unger and Vazirani [6] does demonstrate $O(\sqrt{\epsilon })$-closeness in the state again if -close to the maximal violation of the CHSH inequality. However, the constant factor in front of the $\sqrt{\epsilon }$ term has been calculated in [11] to be of the order 10⁵ and our result is several orders of magnitude better even considering the analysis in appendix C for a fairer comparison. In various other works [9, 31, 32] more general families of self-testing protocols also demonstrate $O(\sqrt{\epsilon })$-closeness of the physical state to the ebit when the violation is -far from Tsirelson's bound. We must emphasise that our analysis could definitely be tightened at several stages to lower the constants in $f(\epsilon )$ but EPR-steering already yields an improvement over analytical methods in standard self-testing.

As demonstrated by the general framework in [11, 13], numerical methods can be employed to obtain better bounds for self-testing. For reasons that will become clear we will shift focus from AST to CST instead and, in particular, CST based on violation of an EPR-steering inequality. Also, we will not be considering CST in full generality and only seek to establish a bound on the trace distance between the physical and reference states (up to isometries). This will facilitate a direct general comparison with previous works.

We begin by constructing the same SWAP isometry as used in the proof of theorem 1. As before, it is applied to the physical state $| \psi \rangle$ and again we wish to upper bound the norm in equation (5). Since this is the trace distance between the pure states, ${E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle$ and $| { \mathcal A }\rangle | \tilde{\psi }\rangle$, we have that [29]

$$\begin{eqnarray*}&&\displaystyle \frac{1}{2}| | ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle \;+\;{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| \;+\;\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )\\ &&\qquad \quad -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| | {| }_{1}\leqslant \sqrt{1-{({F}^{* })}^{2}},\end{eqnarray*}$$

where ${F}^{* }={\rm{max}}\;{\rm{}}F$ such that

$$\begin{eqnarray*}F & = & \sqrt{\langle { \mathcal A }| \langle \tilde{\psi }| ({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| \;+\;\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )| { \mathcal A }\rangle | \tilde{\psi }\rangle }\\ & = & \displaystyle \frac{1}{\sqrt{2}}\sqrt{\langle { \mathcal A }| (\langle {0}_{C}| {E}_{0| 0}| \psi \rangle +\langle {1}_{C}| {{XE}}_{1| 0}| \psi \rangle )(\langle \psi | {E}_{0| 0}| {0}_{C}\rangle +\langle \psi | {E}_{1| 0}X| {1}_{C}\rangle )| { \mathcal A }\rangle }.\end{eqnarray*}$$

Inspired by the work in [11, 13], instead of bounding the quantity F, we wish to bound another quantity G which is the singlet fidelity. For $| \tilde{\psi }\rangle =\tfrac{1}{\sqrt{2}}(| {0}_{C}{0}_{P^{\prime} }\rangle +| {1}_{C}{1}_{P^{\prime} }\rangle )$, this quantity is defined as

$$\begin{eqnarray*}G & = & \langle \tilde{\psi }| {{\rm{tr}}}_{P}[({E}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle )(\langle \psi | {E}_{0| 0}\langle {0}_{P^{\prime} }| +\langle \psi | {E}_{1| 0}X\langle {1}_{P^{\prime} }| )]| \tilde{\psi }\rangle \\ & = & \displaystyle \frac{1}{2}(\langle {0}_{C}| {\sigma }_{0| 0}| {0}_{C}\rangle +2\langle {0}_{C}| ({\sigma }_{0| \mathrm{1,0}| 0}-{\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0})| {1}_{C}\rangle \\ & & +2\langle {1}_{C}| ({\sigma }_{0| \mathrm{0,0}| 1}-{\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0})| {0}_{C}\rangle +\langle {1}_{C}| ({\rho }_{C}-{\sigma }_{0| 0})| {1}_{C}\rangle )\end{eqnarray*}$$

such that ${\sigma }_{0| \mathrm{1,0}| 0}={\sigma }_{0| 0,0| 1}^{\dagger }={{\rm{tr}}}_{P}({E}_{0| 1}{E}_{0| 0}| \psi \rangle \langle \psi | )$ and ${\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0}={{\rm{tr}}}_{P}({E}_{0| 0}{E}_{0| 1}{E}_{0| 0}| \psi \rangle \langle \psi | )$. The above two quantities are related through ${({F}^{* })}^{2}\geqslant 2G-1$ as shown in [11].

The goal is now to give a lower bound to G given constraints on the assemblage. In fact, to facilitate comparison with previous work, we will use the violation of the CHSH inequality to impose these constraints. Every Bell inequality gives an EPR steering inequality when assuming the form of the measurements on the trusted side. If on the client's side we assume the measurements that give the maximal violation of the CHSH inequality for the assemblage generated in the EPR experiment the CHSH expression, denoted by ${\rm{tr}}S$, can be written as

$$\begin{eqnarray*}{\rm{tr}}S & = & {\rm{tr}}\displaystyle \frac{1}{\sqrt{2}}(({\tau }_{z}+{\tau }_{x})({\sigma }_{0| 0}-{\sigma }_{1| 0})+({\tau }_{z}+{\tau }_{x})({\sigma }_{0| 1}-{\sigma }_{1| 1})\\ & & +({\tau }_{z}-{\tau }_{x})({\sigma }_{0| 0}-{\sigma }_{1| 0})-({\tau }_{z}-{\tau }_{x})({\sigma }_{0| 1}-{\sigma }_{1| 1}))\\ & = & {\rm{tr}}(\sqrt{2}{\tau }_{z}({\sigma }_{0| 0}-{\sigma }_{1| 0})+\sqrt{2}{\tau }_{x}({\sigma }_{0| 1}-{\sigma }_{1| 1}))\\ & = & {\rm{tr}}(\sqrt{2}{\tau }_{z}(2{\sigma }_{0| 0}-{\rho }_{C})+\sqrt{2}{\tau }_{x}(2{\sigma }_{0| 1}-{\rho }_{C}))=2\sqrt{2},\end{eqnarray*}$$

where the last bound is Tsirelson's bound. The measurements that the client makes are measurements of the observables in the set $\{1/\sqrt{2}({\tau }_{z}\pm {\tau }_{x})\}$. We then have the constraint that ${\rm{tr}}S\geqslant 2\sqrt{2}-\eta$ for a near-maximal violation.

We now want a numerical method of minimising the singlet fidelity G (so as to give a lower bound) such that ${\rm{tr}}S\geqslant 2\sqrt{2}-\eta$. This method is given by the following semi-definite program (SDP):

$$\begin{eqnarray}&&\quad {\rm{minimise}}\;{\rm{}}\ {\rm{tr}}({M}^{T}{\rm{\Gamma }})=G\\ &&{\rm{subject}}\;{\rm{to:}}\;{\rm{}}{\rm{\Gamma }}\geqslant 0,\\ &&\qquad \qquad \quad {\rm{tr}}({N}^{T}{\rm{\Gamma }})={\rm{tr}}B\geqslant 2\sqrt{2}-\eta ,\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray*}{\rm{\Gamma }} & = & \left(\begin{array}{cccc}{\rho }_{C} & {\sigma }_{0| 0} & {\sigma }_{0| 1} & {\sigma }_{0| \mathrm{0,0}| 1}\\ {\sigma }_{0| 0} & {\sigma }_{0| 0} & {\sigma }_{0| \mathrm{1,0}| 0} & {\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0}\\ {\sigma }_{0| 1} & {\sigma }_{0| \mathrm{0,0}| 1} & {\sigma }_{0| 1} & {\sigma }_{0| \mathrm{0,0}| 1}\\ {\sigma }_{0| \mathrm{1,0}| 0} & {\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0} & {\sigma }_{0| \mathrm{1,0}| 0} & {\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0}\end{array}\right),M=\displaystyle \frac{1}{2}\left(\begin{array}{cccc}W & {\bf{0}} & {\bf{0}} & Y\\ {\bf{0}} & {\tau }_{z} & {\bf{0}} & {\bf{0}}\\ {\bf{0}} & {\bf{0}} & {\bf{0}} & {\bf{0}}\\ {Y}^{T} & {\bf{0}} & {\bf{0}} & -2{\tau }_{x}\end{array}\right),\\ N & = & 2\sqrt{2}\left(\begin{array}{cccc}\tfrac{-{\tau }_{x}-{\tau }_{z}}{2} & {\bf{0}} & {\bf{0}} & {\bf{0}}\\ {\bf{0}} & {\tau }_{z} & {\bf{0}} & {\bf{0}}\\ {\bf{0}} & {\bf{0}} & {\tau }_{x} & {\bf{0}}\\ {\bf{0}} & {\bf{0}} & {\bf{0}} & {\bf{0}}\end{array}\right),\end{eqnarray*}$$

such that $W=\left(\begin{array}{cc}0 & 0\\ 0 & 1\end{array}\right),Y=\left(\begin{array}{cc}0 & 0\\ 2 & 0\end{array}\right)$ and ${\bf{0}}$ is a 2-by-2 matrix of all zeroes. We constrain Γ in the optimisation to be positive semi-definite and not that each sub-matrix of Γ corresponding to something like an element of an assemblage is a valid quantum object. It actually turns out that all assemblages that satisfy no-signalling can be realised in quantum theory [33, 34]. Discussion of this point is beyond the scope of this paper as all we wish to do is give a lower bound on the value of G therefore just imposing ${\rm{\Gamma }}\geqslant 0$ gives such a bound.

Before giving an indication of the results of the above SDP, we still need to show that ${\rm{\Gamma }}\geqslant 0$. We do this by showing that Γ is a Gramian matrix and all Gramian matrices are positive semi-definite. First observe that entries of Γ are of the form ${{\rm{\Gamma }}}_{{lm}}=\langle {i}_{C}| \sigma | {j}_{C}\rangle$ for $\sigma \in \{{\rho }_{C},{\sigma }_{0| 0},{\sigma }_{0| 1},{\sigma }_{0| \mathrm{1,0}| 0},{\sigma }_{0| \mathrm{0,0}| 1},{\sigma }_{0| \mathrm{0,0}| \mathrm{1,0}| 0}\}$. By cyclicity of the partial trace we can also write $\sigma ={{\rm{tr}}}_{P}(F| \psi \rangle \langle \psi | {G}^{\dagger })$ for $F,G\in \{{{\mathbb{I}}}_{P},{E}_{0| 0},{E}_{0| 1},{E}_{0| 1}{E}_{0| 0}\}$. We now note that

$$\begin{eqnarray*}\langle {i}_{C}| \sigma | {j}_{C}\rangle & = & \displaystyle \sum _{| y\rangle \in {{ \mathcal H }}_{P}}\langle {i}_{C}| \langle y| F| \psi \rangle \langle \psi | {G}^{\dagger }| y\rangle | {j}_{C}\rangle \\ & = & \left(\displaystyle \sum _{| y\rangle \in {{ \mathcal H }}_{P}}\langle {i}_{C}| \langle y| F| \psi \rangle \langle y| \right)\left(\displaystyle \sum _{| y^{\prime} \rangle \in {{ \mathcal H }}_{P}}\langle \psi | {G}^{\dagger }| y^{\prime} \rangle | {j}_{C}\rangle | y^{\prime} \rangle \right)\\ & = & \displaystyle \sum _{y}{\alpha }_{y}\langle y| \displaystyle \sum _{y^{\prime} }{\alpha }_{y^{\prime} }^{* }| y^{\prime} \rangle \\ & = & \langle u| v\rangle ,\end{eqnarray*}$$

where $\{| y\rangle \}$ is an orthonormal basis in ${{ \mathcal H }}_{P}$ such that $\langle y^{\prime} | y\rangle ={\delta }_{y^{\prime} }^{y}$ and ${\alpha }_{y}=\langle {i}_{C}| \langle y| F| \psi \rangle$ is some scalar. Since the elements of Γ are all the inner product of vectors associated with a row and column, ${\rm{\Gamma }}={V}^{\dagger }V$ where V has column vectors associated with the vectors v. Therefore, Γ is Gramian. This then makes the above optimisation problem a completely valid problem for lower bounding G. We further note that matrix Γ represents the EPR-steering analogue of the moment matrix in the Navascués–Pironio–Acín (NPA) hierarchy [35] which is useful for approximating the set of quantum correlations⁶ .

In figure 3 we plot the lower bound on G achieved through this method and then compare it to the value obtained through the method of Bancal et al in [11]. In both cases the violation of the CHSH inequality is lower-bounded by $2\sqrt{2}-\eta$, and we clearly see that the lower-bound is more favourable for our optimisation through EPR-steering as compared to full device-independence. For the case of EPR-steering we observed that the plot can be lower-bounded by the function $1-\eta /\sqrt{2}$ whereas the plot for device-independence is lower-bounded by $1-5\eta /4$. Respectively, these functions give an upper bound on $D(| {\rm{\Phi }}\rangle \langle {\rm{\Phi }}| ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | \tilde{\psi }\rangle \langle \tilde{\psi }| )$ of ${2}^{\tfrac{1}{4}}\sqrt{\eta }\leqslant 1.19\sqrt{\eta }$ and $\sqrt{10}/2\sqrt{\eta }\leqslant 1.59\sqrt{\eta }$. The difference between these two approaches is not as dramatic as the difference in the analytical approaches. However, these results just highlight that the analytical approaches are quite sub-optimal for both EPR-steering and device-independent self-testing.

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Both the analytical and numerical approaches have utilised the same SWAP isometry. While constructing this isometry demonstrates in a clear and simple manner that self-testing is possible, it is natural to ask if there may be more useful isometries that give a different error scaling for our particular scenario? In particular, can we do better than the $\sqrt{\epsilon }$ in the function $f(\epsilon )$ for $f(\epsilon )$-AST? As we have already shown in section 2, in general this is not possible but the example demonstrating this is somewhat contrived. That is, we are trying to self-test a two-qubit state but assume that the Hilbert space of the client is three-dimensional. We wish to ask if $O(\epsilon )$-AST is possible in the particular example of the EPR experiment? In this section we will show that this is not possible and the best we can hope for is $O(\sqrt{\epsilon })$-AST which we have already established is possible.

As a side note, in appendix D we show that the trace distance between the physical and reference states in the EPR experiment can be $O(\epsilon )$ for some isometries. We emphasise that this trace distance between physical and reference states (condition given in the first line of equation (3)) only amounts to part of the criteria for AST. The other part of the criteria (the second line of equation (3)) rules out many isometries that might give the optimal trace distance between physical and reference states only. With this in mind we want to bound the expression in equation (4) for all possible isometries given -closeness between the elements of the physical and reference assemblages. In particular, we give an example of a physical experiment where -closeness for the assemblages is satisfied but for all isometries, the smallest value of equation (4) is $O(\sqrt{\epsilon })$.

Example 3. The physical state is

$$\begin{eqnarray*}&&| \psi \rangle =\displaystyle \frac{1}{\sqrt{2}}(\sqrt{1-\epsilon }| {0}_{C}{0}_{P}\rangle +\sqrt{\epsilon }| {1}_{C}{1}_{P}\rangle )| {0}_{P^{\prime} }\rangle +\displaystyle \frac{1}{\sqrt{2}}(\sqrt{\epsilon }| {0}_{C}{0}_{P}\rangle +\sqrt{1-\epsilon }| {1}_{C}{1}_{P}\rangle )| {1}_{P^{\prime} }\rangle ,\end{eqnarray*}$$

where $P$ and $P^{\prime}$ denote two qubits that the provider has in their possession, thus ${\rho }_{C}=\tfrac{1}{2}{{\mathbb{I}}}_{C}$. The physical measurements are ${E}_{0| 0}={{\mathbb{I}}}_{P}\otimes | {0}_{P^{\prime} }\rangle \langle {0}_{P^{\prime} }| ,{E}_{1| 0}={{\mathbb{I}}}_{P}\;\otimes \;| {1}_{P^{\prime} }\rangle \langle {1}_{P^{\prime} }| ,$ ${E}_{0| 1}=| {+}_{P}\rangle \langle {+}_{P}| \;\otimes \;| {+}_{P^{\prime} }\rangle \langle {+}_{P^{\prime} }| +| {-}_{P}\rangle \langle {-}_{P}| \otimes | {-}_{P^{\prime} }\rangle \langle {-}_{P^{\prime} }|$ and ${E}_{1| 1}=| {+}_{P}\rangle \langle {+}_{P}| \otimes | {-}_{P^{\prime} }\rangle \langle {-}_{P^{\prime} }| +| {-}_{P}\rangle \langle {-}_{P}| \otimes | {+}_{P^{\prime} }\rangle \langle {+}_{P^{\prime} }|$. These physical measurements on the state produce the following assemblage elements:

$$\begin{eqnarray*}{\sigma }_{0| 0} & = & \displaystyle \frac{(1-\epsilon )}{2}| {0}_{C}\rangle \langle {0}_{C}| +\displaystyle \frac{\epsilon }{2}| {1}_{C}\rangle \langle {1}_{C}| ,\quad \quad {\sigma }_{1| 0}=\displaystyle \frac{(1-\epsilon )}{2}| {1}_{C}\rangle \langle {1}_{C}| +\displaystyle \frac{\epsilon }{2}| {0}_{C}\rangle \langle {0}_{C}| ,\\ {\sigma }_{0| 1} & = & \displaystyle \frac{1}{2}| {+}_{C}\rangle \langle {+}_{C}| ,\qquad \qquad \quad \quad \quad {\sigma }_{1| 1}=\displaystyle \frac{1}{2}| {-}_{C}\rangle \langle {-}_{C}| .\end{eqnarray*}$$

We see then that $D({\rho }_{C},{\tilde{\rho }}_{C})=0$ and $| | {\sigma }_{a| x}-{\tilde{\sigma }}_{a| x}| | \leqslant \epsilon$ for all $a,x$.

We now show that $| | | {\rm{\Phi }},{E}_{0| 0}\rangle \langle {\rm{\Phi }},{E}_{0| 0}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes {\tilde{E}}_{0| 0}| \tilde{\psi }\rangle \langle \tilde{\psi }| {\tilde{E}}_{0| 0}| {| }_{1}\geqslant \sqrt{\epsilon }$ for all possible isometries Φ. By considering all possible isometries we have

$$\begin{eqnarray*}&&| {\rm{\Phi }},{E}_{0| 0}\rangle =U{E}_{0| 0}| \psi \rangle | \hat{0}\rangle =\displaystyle \frac{1}{\sqrt{2}}U(\sqrt{1-\epsilon }| {0}_{C}{0}_{P}\rangle +\sqrt{\epsilon }| {1}_{C}{1}_{P}\rangle )| {0}_{P^{\prime} }\rangle | \hat{0}\rangle =\displaystyle \frac{1}{\sqrt{2}}| \epsilon \rangle ,\end{eqnarray*}$$

for $| \epsilon \rangle =U(\sqrt{1-\epsilon }| {0}_{C}{0}_{P}\rangle +\sqrt{\epsilon }| {1}_{C}{1}_{P}\rangle )| {0}_{P^{\prime} }\rangle | \hat{0}\rangle$ and U being a unitary applied jointly to the provider's qubits and the ancillae $| \hat{0}\rangle$. This then allows us to observe that

$$\begin{eqnarray*}&&| | | {\rm{\Phi }},{E}_{0| 0}\rangle \langle {\rm{\Phi }},{E}_{0| 0}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes {\tilde{E}}_{0| 0}| \tilde{\psi }\rangle \langle \tilde{\psi }| {\tilde{E}}_{0| 0}| {| }_{1}\\ &&\qquad =D(| \epsilon \rangle \langle \epsilon | ,| { \mathcal A }\rangle \langle { \mathcal A }| \otimes | 00\rangle \langle 00| )=\sqrt{1-| \langle \epsilon | { \mathcal A }\rangle | 00\rangle {| }^{2}}.\end{eqnarray*}$$

We see that $| \langle \epsilon | { \mathcal A }\rangle | 00\rangle {| }^{2}=(1-\epsilon )| \langle { \mathcal A }| \langle 0| U| {0}_{P}\rangle | \hat{0}\rangle {| }^{2}$ which achieves the maximal value of $(1-\epsilon )$. Therefore $| | | {\rm{\Phi }},{E}_{0| 0}\rangle \langle {\rm{\Phi }},{E}_{0| 0}| -| { \mathcal A }\rangle \langle { \mathcal A }| \otimes {\tilde{E}}_{0| 0}| \tilde{\psi }\rangle \langle \tilde{\psi }| {\tilde{E}}_{0| 0}| {| }_{1}\geqslant \sqrt{\epsilon }$ for all possible isometries Φ.

This example demonstrates that $O(\epsilon )$-AST is impossible for the EPR experiment and our analytical results are essentially optimal (up to constants).

Already for three parties, how to modify the client-provider set-up opens up new and interesting possibilities. For example, the simplest modification is to have the new, third party be a trusted part of the client's laboratory; the total Hilbert space of the client ${{ \mathcal H }}_{C}$ is now the tensor product of the two Hilbert spaces associated with these two parties. The next possible modification, as shown in figure 4, is to have a second untrusted party that after receiving their share of the physical state does not communicate with the initial provider: they only communicate with the client. This restriction establishes a tensor product structure between the two untrusted parties which is useful.

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To illustrate the interesting differences between the bipartite and tri-partite cases, we look at the example of self-testing the GHZ state $| \tilde{\psi }\rangle =1/\sqrt{2}(| {\rm{\Psi }}\rangle | {+}_{3}\rangle +| {\rm{\Psi }}^{\prime} \rangle | {-}_{3}\rangle )$ where $| {\rm{\Psi }}\rangle =1/\sqrt{2}(| {0}_{1}{0}_{2}\rangle -| {1}_{1}{1}_{2}\rangle )$ and $| {\rm{\Psi }}^{\prime} \rangle =1/\sqrt{2}(| {0}_{1}{1}_{2}\rangle +| {1}_{1}{0}_{2}\rangle )$ with subscripts denoting the number of the qubit. In the scenario with two trusted parties (that together form the client), a qubit is sent from the provider to each of these parties (say, qubits 1 and 2 are sent); we will call this scenario the 2-trusted setting. In the other scenario with two non-communicating untrusted providers, a qubit (say, qubit 1) is sent to the client; we will call this scenario the 1-trusted setting. These different scenarios correspond to different types of multipartite EPR-steering introduced in [36].

We now describe the reference experiments for both settings for the state $| \tilde{\psi }\rangle$. In the case of the 2-trusted setting, as in the EPR experiment, the provider claims to make measurements ${\tilde{E}}_{j| 0}=| j\rangle \langle j|$ for $j\in \{0,1\}$ as well as ${\tilde{E}}_{0| 1}=| +\rangle \langle +|$ and ${\tilde{E}}_{1| 1}=| -\rangle \langle -|$. The assemblage for the two trusted parties has elements

$$\begin{eqnarray*}{\tilde{\sigma }}_{0| 0} & = & \displaystyle \frac{1}{4}(| {\rm{\Psi }}\rangle +| {\rm{\Psi }}^{\prime} \rangle )(\langle {\rm{\Psi }}| +\langle {\rm{\Psi }}^{\prime} | ),\qquad \quad {\tilde{\sigma }}_{1| 0}=\displaystyle \frac{1}{4}(| {\rm{\Psi }}\rangle -| {\rm{\Psi }}^{\prime} \rangle )(\langle {\rm{\Psi }}| -\langle {\rm{\Psi }}^{\prime} | ),\\ {\tilde{\sigma }}_{0| 1} & = & \displaystyle \frac{1}{2}| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| ,\qquad \qquad \quad \qquad \qquad \qquad \quad {\tilde{\sigma }}_{1| 1}=\displaystyle \frac{1}{2}| {\rm{\Psi }}^{\prime} \rangle \langle {\rm{\Psi }}^{\prime} | .\end{eqnarray*}$$

For the 1-trusted setting, in addition to the provider claiming to making the above measurements, the second untrusted party, or second provider claims also to make the same measurements, which we denote by ${\tilde{E}}_{c| z}$ for $c,z\in \{0,1\}$. The assemblage will be $\{{\tilde{\sigma }}_{a,c| x,z}\}{}_{a,c,x,z}$ where each element is ${\tilde{\sigma }}_{a,c| x,z}={{\rm{tr}}}_{P}({{\mathbb{I}}}_{C}\otimes {\tilde{E}}_{c| z}\otimes {\tilde{E}}_{a| x}| \tilde{\psi }\rangle \langle \tilde{\psi }| )$. The assemblage for the one trusted party will have 16 elements but for the sake of brevity we will not write out the elements.

We then wish to self-test this reference experiment when the elements of the physical assemblage are close to the elements of the ideal, reference experiment. Instead of doing this, we will mimic the numerical approach in section 3.2 by considering the GHZ–Mermin inequality [37] adapted to the 1-trusted and 2-trusted scenarios. Utilising the notation of ${\tau }_{x}$ and ${\tau }_{z}$ for the Pauli-X and Pauli-Z matrices respectively, for the 2-trusted and 1-trusted settings, the inequalities respectively are:

$$\begin{eqnarray*}{\rm{tr}}{B}_{2} & = & 2{\rm{tr}}(({\tau }_{z}\otimes {\tau }_{z})(2{\sigma }_{0| 1}-{\rho }_{C})+({\tau }_{x}\otimes {\tau }_{z})(2{\sigma }_{0| 0}-{\rho }_{c})\\ & & +({\tau }_{z}\otimes {\tau }_{x})(2{\sigma }_{0| 0}-{\rho }_{C})-({\tau }_{x}\otimes {\tau }_{x})(2{\sigma }_{0| 1}-{\rho }_{C}))\leqslant 2,\\ {\rm{tr}}{B}_{1} & = & 2{\rm{tr}}({\tau }_{z}({\sigma }_{00| 01}-{\sigma }_{01| 01}-{\sigma }_{10| 01}+{\sigma }_{11| 01})+{\tau }_{x}({\sigma }_{00| 00}+{\sigma }_{11| 00}-{\sigma }_{01| 00}-{\sigma }_{10| 00}))\\ & & +2{\rm{tr}}({\tau }_{z}({\sigma }_{00| 10}+{\sigma }_{11| 10}-{\sigma }_{01| 10}-{\sigma }_{10| 10})-{\tau }_{x}({\sigma }_{00| 11}+{\sigma }_{11| 11}-{\sigma }_{01| 11}-{\sigma }_{10| 11}))\leqslant 2.\end{eqnarray*}$$

The maximal quantum violation of these inequalities is 4. We now aim to carry out self-testing if the physical experiment achieves a violation of $4-\eta$. For the untrusted parties, we implement the SWAP isometry to each of their systems as outlined in section 3.1. For the 2-trusted setting, the physical state $| \psi \rangle$ gets mapped to $| \psi ^{\prime} \rangle ={E}_{0| 0}| \psi \rangle | {0}_{P}^{\prime }\rangle +{{XE}}_{1| 0}| \psi \rangle | {1}_{P}^{\prime }\rangle$. In the 1-trusted setting, the physical state $| \psi \rangle$ gets mapped to

$$\begin{eqnarray*}| \psi ^{\prime\prime} \rangle & = & {E}_{0| 0}{F}_{0| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle | {0}_{P^{\prime\prime} }\rangle +{{XE}}_{1| 0}{F}_{0| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle | {0}_{P^{\prime\prime} }\rangle \\ & & +{E}_{0| 0}X^{\prime} {F}_{1| 0}| \psi \rangle | {0}_{P^{\prime} }\rangle | {1}_{P^{\prime\prime} }\rangle +{{XE}}_{1| 0}X^{\prime} {F}_{1| 0}| \psi \rangle | {1}_{P^{\prime} }\rangle | {1}_{P^{\prime\prime} }\rangle ,\end{eqnarray*}$$

where ${F}_{c| z}$ is the physical measurement made by the second untrusted party, $X^{\prime} =2{F}_{0| 1}-{\mathbb{I}}$ and $P^{\prime}$ denotes the ancilla qubit introduced for one party and $P^{\prime\prime}$ for the other party.

Our figure of merit for closeness between the physical and reference states is the GHZ fidelity which for the 2-trusted and 1-trusted settings is G₂ and G₁ respectively where

$$\begin{eqnarray*}{G}_{2} & = & \langle \tilde{\psi }| {{\rm{tr}}}_{P}(| \psi ^{\prime} \rangle \langle \psi ^{\prime} | )| \tilde{\psi }\rangle ,\\ {G}_{1} & = & \langle \tilde{\psi }| {{\rm{tr}}}_{P}(| \psi ^{\prime\prime} \rangle \langle \psi ^{\prime\prime} | )| \tilde{\psi }\rangle ,\end{eqnarray*}$$

where in both cases we trace out the provider's (providers') Hilbert space(s) ${{ \mathcal H }}_{P}$. Now we minimise G₂ while ${\rm{tr}}{B}_{2}\geqslant 4-\eta$ and minimise G₁ such that ${\rm{tr}}{B}_{2}\geqslant 4-\eta$. These problems again can be lower-bounded by an SDP and in figure 5 we give numerical values obtained with these minimisation problems. This case is numerically more expensive than the simple self-testing of the EPR experiment and for tackling it we used the SDP procedures described in [38]. We also compare our results to those obtained in the device-independent setting where all three parties are not trusted but the violation of the GHZ–Mermin inequality is $4-\eta$. We see that the GHZ fidelity increases when we trust more parties. Interestingly, we can see that the curve for 1-trusted scenario is obviously closer to the curve of 2-trusted scenario than to the device-independent one. This may hint that multi-partite EPR-steering behaves quite differently to quantum non-locality. However, to draw this conclusion from self-testing one would have to pursue more rigorous research, since we have only obtained numerical lower bounds on the GHZ fidelity using only one specific isometry.

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The previous section hints at what might be the most useful aspect of self-testing through EPR-steering: establishing a tensor product structure in the provider's Hilbert space. In the work of Reichardt, Unger and Vazirani, a method is presented for self-testing many copies of the ebit between two untrusted parties [6]. This testing is achieved through measurements made in sequence. Recent work has established the same feat but now with measurements being made at the same time, thus giving a more general result [39]. The difficulty in establishing that the two untrusted parties have multiple copies of the ebit is to establish that (up to isometries) the Hilbert spaces of the parties decompose as a tensor product of several two-dimensional Hilbert spaces: in each sub-space there is one-half of an ebit.

We now remark that EPR-steering offers a useful simplification in achieving the same task of identifying a tensor product structure. Note that in the trusted laboratory a tensor product structure is known: the client knows they have, say, two qubits. If the assemblage for each qubit is close to the ideal case of being one half of an ebit, then we may use lemma 2 to 'transfer' the physical operations on the untrusted side to one of the qubits on the trusted side. We also note that this observation forms part of the basis of the work presented in [40], in the context of verification of quantum computation.

To be more exact, we now have the client's Hilbert space being constructed from a tensor product of N two-dimensional Hilbert spaces, i.e. ${{ \mathcal H }}_{C}={\displaystyle \bigotimes }_{i=1}^{N}{{ \mathcal H }}_{{C}_{i}}$ where ${{ \mathcal H }}_{{C}_{i}}={{\mathbb{C}}}^{2}$. We now have a modified form of the EPR experiment with the reference state being $| \tilde{\psi }\rangle ={\displaystyle \bigotimes }_{i=1}^{N}| {\tilde{\psi }}_{i}\rangle \in {\displaystyle \bigotimes }_{i=1}^{N}{{ \mathcal H }}_{{P}_{i}}\otimes {{ \mathcal H }}_{{C}_{i}}$ for each $| {\tilde{\psi }}_{i}\rangle =\tfrac{1}{\sqrt{2}}(| 00\rangle +| 11\rangle )\in {{ \mathcal H }}_{{P}_{i}}\otimes {{ \mathcal H }}_{{C}_{i}}$. That is, in the reference experiment, the provider's Hilbert space has a tensor product structure. For each Hilbert space ${{ \mathcal H }}_{{P}_{i}}$, there is a projective measurement with projectors ${\tilde{E}}_{{a}_{i}| {x}_{i}}$ acting on that space where ${a}_{i},{x}_{i}\in \{0,1\}$ and these projectors are the qubit projectors in the EPR experiment. Therefore, the total reference projector is of the form ${\displaystyle \bigotimes }_{i=1}^{N}{\tilde{E}}_{{a}_{i}| {x}_{i}}$ which act on the Hilbert space ${\displaystyle \bigotimes }_{i=1}^{N}{{ \mathcal H }}_{{P}_{i}}$. In this case, the measurement choices and outcomes are bit-strings ${\bf{x}}:= ({x}_{1},{x}_{2},\ldots ,{x}_{N})$ and ${\bf{a}}:= ({a}_{1},{a}_{2},\ldots ,{a}_{N})$ respectively. We call this reference experiment the N-pair EPR experiment and we are now in a position to generalise lemma 2.

Lemma 3. For the N-pair EPR experiment, if for all $i,| | {\sigma }_{{\bf{a}}| {\bf{x}}}-{\tilde{\sigma }}_{{\bf{a}}| {\bf{x}}}| {| }_{1}\leqslant \epsilon$ and $D({\rho }_{C},{\tilde{\rho }}_{C})\leqslant \epsilon$ where ${\tilde{\sigma }}_{{\bf{a}}| {\bf{x}}}={\displaystyle \bigotimes }_{i=1}^{N}{\tilde{\sigma }}_{{a}_{i}| {x}_{i}}$ and ${\tilde{\rho }}_{C}={\displaystyle \bigotimes }_{i=1}^{N}\tfrac{{{\mathbb{I}}}_{{\mathbb{C}}}}{2}$ then

$$\begin{eqnarray}&&| | {{\mathbb{I}}}_{C}\otimes {E}_{{\bf{a}}| {\bf{x}}}| \psi \rangle -\underset{i=1}{\overset{N}{\displaystyle \bigotimes }}{\tilde{E}}_{{a}_{i}| {x}_{i}}\otimes {{\mathbb{I}}}_{P}| \psi \rangle | | \quad \leqslant \quad 2\sqrt{\epsilon }.\end{eqnarray} \tag{ 8 }$$

The proof of this lemma is almost identical to the proof of lemma 2 and so we will leave it out from our discussion. A nice relaxation of the conditions of the above lemma is to insist that each observed element of an assemblage ${\sigma }_{{a}_{i}| {x}_{i}}$ is -close to ${\tilde{\sigma }}_{{a}_{i}| {x}_{i}}$ and still recover a similar result. This requires a little bit more work since we have not been specific in how we model the provider's measurements. For example, we have not stipulated whether the probability distribution $p({\bf{a}}| {\bf{x}})={\rm{tr}}({\sigma }_{{\bf{a}}| {\bf{x}}})$ satisfies the no-signalling principle. Furthermore, even if these probabilities satisfy this principle, it does not immediately enforce a constraint on the behaviour of the measurements. For the sake of brevity we will not address this issue in this work. It remains to point out that lemma 3 can be used to develop a result for self-testing (see [40]).