Geometrical self-testing of partially entangled two-qubit states

Quantum nonlocality has recently been intensively studied in connection to device-independent quantum information processing, where the extremal points of the set of quantum correlations play a crucial role through self-testing. In most protocols, the proofs for self-testing rely on the maximal violation of the Bell inequalities, but there is another known proof based on the geometry of state vectors to self-test a maximally entangled state. We present a geometrical proof in the case of partially entangled states. We show that, when a set of correlators in the simplest Bell scenario satisfies a condition, the geometry of the state vectors is uniquely determined. The realization becomes self-testable when another unitary observable exists on the geometry. Applying this proven fact, we propose self-testing protocols by intentionally adding one more measurement. This geometrical scheme for self-testing is superior in that, by using this as a building block and repeatedly adding measurements, a realization with an arbitrary number of measurements can be self-tested. Besides the application, we also attempt to describe nonlocal correlations by guessing probabilities of distant measurement outcomes. In this description, the quantum set is also convex, and a large class of extremal points is identified by the uniqueness of the geometry.


Introduction
It was shown by Bell that the nonlocal correlations predicted by quantum mechanics are inconsistent with local realism [1]. Bell nonlocality, or quantum nonlocality, has attracted many research interests over the years (see [2] for a review). Recently, it has been intensively studied in connection to device-independent quantum information processing (see [3,4] for reviews), where the extremal points of the convex set of quantum correlations plays a crucial role through self-testing.
The correlation that attains the maximal quantum violation of 2 2 [5] in the Clauser-Horne-Shimony-Holt inequality [6] is an extremal point of the quantum set, for which the quantum realization (state and measurements) is unique up to unavoidable local isometry. This implies that attaining the value of 2 2 can selftest the state and the measurements in the Bell experiment [7]. When a realization is a unique maximizer of a Bell inequality, the realized correlation is a self-testable extremal point. Although there exist non-exposed extremal points that cannot be a unique maximizer of any Bell inequality, a correlation is extremal when the realization is self-testable [8]. In this way, self-testability and extremality are intimately connected. In most protocols, the proofs for self-testing rely on the maximal violation of the Bell inequalities. However, even in the simplest Bell scenario (two parties and two binary measurements on each party), the maximal violation by a partially entangled state is known for only a few Bell inequalities [9][10][11][12][13], and not many protocols are proposed for selftesting partially entangled states [14][15][16][17][18].
On the other hand, the proof for self-testing in [19] is fascinating, because no Bell inequality is used directly. In the simplest Bell scenario, when marginal probabilities of outcomes are unbiased, the boundaries of the quantum set are identified by the Tsirelson-Landau-Masanes (TLM) criterion [20][21][22]. The proof in [19] relies on the fact that the geometry of the state vectors is uniquely determined when the TLM criterion is satisfied (and the anti-commutation relation between observables is proven on the geometry). However, this geometrical proof is restricted to the case of a maximally entangled state by the restriction of the TLM criterion. In a general case, where an extremal correlation may be realized by a partially entangled state, the criterion for the identification has been only conjectured, based on the probabilities of guessing outcomes of a distant party (referred as 'guessing probability' hereafter) [23].
In this paper, we present a geometrical proof in the case of partially entangled states. We show that, when a set of correlators in the simplest Bell scenario satisfies a condition, the geometry of state vectors is uniquely determined. The realization becomes self-testable when another unitary observable exists on the geometry to prove anti-commutation relation. Applying this proven fact, we propose self-testing protocols by intentionally adding one more measurement to prove the anti-commutation relation. This geometrical scheme for selftesting is superior in that, by using this as a building block and repeatedly adding measurements, a realization with an arbitrary number of measurements can be self-tested.
Beside applications, efforts have been made to describe the quantum set having a complicated structure [24-26, 8, 27] in a more tractable way; some descriptions exist such as covariance [28] and entropy [29]. For this purpose, we attempt to describe nonlocal correlations by guessing probabilities. We show that the quantum realizable set is also convex in this description, and a large class of extremal points is identified by the uniqueness of the geometry of state vectors. Moreover, with the help of this extremality, we show that the sufficiency of the extremal criterion conjectured in [23] can be reduced to certifiability of guessing probabilities.
This paper is organized as follows: in section 2, we briefly summarize the preliminaries. For details, see [2][3][4] and the references therein. For clarity, we first introduce the description of correlations by guessing probabilities in section 3, and discuss the properties of the quantum set, such as the extremality and self-testability. In section 4, we investigate the geometrical properties of realizations in the standard description of correlations. Finally, as an application, we propose self-testing protocols for partially entangled states in section 5, whose selftestability is geometrically proven, regardless of the validity of the conjectured extremal criterion. A summary is given in section 6.

Preliminaries
In the simplest Bell scenario, Alice (Bob) performs one of two binary measurements on a shared state depending on a given random bit x (y), and obtains an outcome a=±1 (b=±1). The properties of a nonlocal correlation are described by a set of conditional probabilities p ab xy { ( | )}referred as a 'behavior', which specifies a point in the probability space. , We use á ñ  as the abbreviation of y y á ñ | |  . Any state vector has a real-vector representation [20,30,31]. For example, when yñ | is represented by components as yñ = c c , ,  is a real-vector representation. The realizable behaviors constitute a convex set in the C-space, denoted by  C . In the unbiased case where , a behavior belongs to  C , if and only if the TLM inequality [20][21][22] - where p xy is either '+' or '−'. Letting the value of S xy p xy be equal to c sin 2 2 , the following is also introduced: Then, to identify the nonlocal extremal points of  C , the following criterion has been conjectured in [23].
( )are the Pauli matrices (but there is no s 2 term), and hence also necessary for the extremality of  C (see the supplemental material of [23]). Note further that the definition of q x A and q y B are changed from [32,33,23] for convenience (q Moreover, for a given C C C , , x A y B xy { }, the quantity D x B and D y A (explained later) has a device-independent upper bound, which can be obtained by the Navascués-Pironio-Acín (NPA) hierarchy [35,36], and the following is also implicitly conjectured in [23].

Conjecture 2. When a nonlocal behavior C C
, ,

Quantum set in D-space
As mentioned, C A x is the bias of p a x ( | ), but it is also the bias of Bob's optimal probability of guessing Alice's outcome a, without the use of any side information. In the nonlocality scenario, however, Bob has a half of a shared state; the local state r a x | (conditioned on Alice's outcome a), and by the use of it the guessing probability is generally increased. Therefore, it seems another natural way of describing nonlocal correlations to use the guessing probabilities optimized under r a x | . For this purpose, we focus on the quantities introduced in [ The reason for taking the square of D x B and D y A will become clear soon. Such a behavior specifies a point in an 8-dimensional space, which we denote by the D-space. Note that the behaviors in the C-space and the D-space have no one-to-one correspondence. For example, the completely random correlation is uniquely represented by x y 1 on yñ = ñ 00 | | , and the latter is realized by . Now, let us investigate the properties of the behaviors in the D-space. When the behaviors p i are realized by quantum mechanics, there always exists a realization of the behavior This set, denoted by  D , is then at least enclosed by the hyperplanes in the D-space defined from the following inequalities: The quantum bound of the inequalities is given by This is due to the cryptographic quantum bound shown in [32]. Indeed, 10 11 hence any realization obeys (8) and (9) are respected by any quantum realization, which we denote by quantum Bell inequalities in analogy to the Bell inequalities.
It is convenient to introduce another convex set, which is enclosed by inequalities (8) and (9). As inequality (11) holds whenever the first inequality due to the cryptographic quantum bound holds, the behaviors in this set are those satisfying the TLM inequality (1) for both scaled correlators (together with the obvious constraint of ). This convex set, denoted by  crypt , is a superset of  D .
Let us now search for the extremal points of  D . It is known that each extremal point of  C has a two-qubit realization [5,37]. This is due to the fact that A 0 and A 1 (B 0 and B 1 as well) are simultaneously block-diagonalized by appropriate local bases with the block size of at most 2 [37]. However, this cannot be applied to the case of  D due to the convexity of D x B 2 ( ) and D y A 2 ( ) as in inequality (7). Fortunately, however, we have the following: Lemma 2. A behavior in  D , which simultaneously saturates the quantum Bell inequalities (8) and (9), has a twoqubit realization.
Proof. As the maximization in D x B is rewritten by using the Lagrange multiplier l as , any realization must satisfy y y y y y y ñá = ñá + ñá where F x is an optimal operator attaining the maximum. Let y B y | , and yñ F x | , respectively, which are all unit vectors. Then, equation (12) implies On the other hand, the saturation of inequality [19] as shown in figure 1. Similarly, the saturation of inequality (9) implies that four real vectors A 0 is the real vector optimizing D y A . However, as a high-dimensional vector space is considered, the relationship between the two planes has not been determined yet.
Suppose that From the laws of sines and cosines, y BA 2 | |  is given by where Δ is the angle between A 0  and A 1  . From equation (13),    being a common vector as shown in figure 1. The two-qubit realization of equation (5) can realize the same geometry of real vectors.
The behavior in the D-space realized by such a simple geometry can be realized by equation (5). Similarly, when , As such a behavior saturates inequality (1) for scaled correlators, it is located at a boundary of  crypt . Conversely, a boundary behavior of  crypt generally does not have a realization with the geometry of figure 1 and cannot be realized by quantum mechanics; hence, Hereafter, to describe the geometry of figure 1, we also use the shortcut notations of When such a geometry is given, we can easily construct the quantum Bell inequalities (8) and (9) that are simultaneously saturated by the geometry, as shown in appendix B. Conversely, let us investigate the realizations to maximize such a given pair of the quantum Bell inequalities. Note that there exists unavoidable ambiguity of the realizations, which is referred as obvious symmetries hereafter, as the four geometries with the parameters q q c , , {¯} realize the same behavior in the D-space. In general, the realization that saturates either inequality (8) or (9) (8) and (9), q D B and q D A are constrained to satisfy respectively. The two planes intersect at the angle of p c -2 2 , with y¢  being a common vector. The angle between y¢ (1) is saturated by scaled correlators, inequality (17) holds in this geometry. We assume, without loss of generality, c p   0 4 throughout this paper.    (8) and (9), is unique up to obvious symmetries when equation (18) only has a trivial solution; hence such a behavior is an extremal point of  D .
For a given pair of quantum Bell inequalities, no pair of a c and b c is identical in general and equation (18) only has a trivial solution. This implies that the behaviors realized by two-qubit realizations equation (5) with the parameters satisfying inequality (17) are generally extremal for  D , constituting a large class of extremal points. Note that the uniqueness of the realization is not necessarily required for the extremality, and hence lemma 4 does not exclude the possibility that the behaviors realized by equation (5) with inequality (17) are all extremal.
In any case, for an extremal behavior of  D proven by lemma 4, the geometry of real vectors is unique up to the obvious symmetry. Is such a behavior self-testable? The answer is negative by two reasons (apart from the problem of how D x B and D y A is determined by experiments). The first is that y¢ ( ) . As shown in appendix C, there exists an example in which the correlation P, despite being an extremal point of  D , may have two different realizations due to the strict convexity. However, in some cases, we can exclude the possibility of such strict convexity, that is, the certifiability of D x B and D y A .
Suppose that Conjecture 2 holds true. As inequality (1)  . This correlation, denoted by p, is then found to be an extremal point of  D by lemma 4. When a realization of p is decomposed into two-qubit realizations of p i , based on the block-diagonalization [37], D x

Quantum set in C-space
From now on, let us show some geometrical properties of the realizations for the behaviors in the standard Cspace. Note that these hold true regardless of the validity of Conjectures 1 and 2. To begin with, we show that the geometry of the realization of a behavior in the C-space is uniquely determined when the correlators satisfy a condition: Lemma 6. For a nonlocal behavior C C C , , The unique geometry is the same as figure 1, but the obvious symmetry now refers the ambiguity between q q c , , , is an explicit counter example for extremality. Interestingly, P is located in the strict interior of the quantum set, according to the + AB 1 level of the NPA hierarchy [38]. This also implies that, even though y c ¢ = cos 2 | | is ensured to be the same as the two-qubit realizations, the uniqueness is still insufficient for self-testing. The condition = = = + + + + S S S S 00 01 10 11 is crucial, apart from the unique determination of the geometry, for making the realization self-testable through the certification of D x B and D y A , as shown by lemma 5. However, other than the unproved certification condition, a more general condition that makes the unique geometry self-testable is found as follows: Proof. As the geometry is uniquely determined as figure 1 by lemma 6, the 'only if' part is obvious: when the realization is self-testable, it is a two-qubit realization of equation (5) ). Let us prove the 'if' part. We again use the notation of equation (16). For the operator Z B defined by and so on, and measurements are self-tested. , For self-testability, the proof of the anti-commutation relation between B 0 and B 1 (equation (22)) is crucial. To prove it, lemma 7 implies that the third unitary observable G, whose real vector lies in the same B-plane, is necessary. In the unbiased case where χ=π/4, the four vectors A 0 all lie in the same plane, and A x can be used as the third unitary observable [19]. However, in the other general case of c p lie in a different A-plane, and A x cannot be used anymore. It is not limited, but the optimal operator F x for D x B is a good candidate for G. Interestingly, in the special case that = F B 0 0 , the candidate for G is missing in the B-plane, but the correlation in this case is always local.

Scheme for self-testing partially entangled state
As shown in section 3, under the conjectured certifiability of D x B and D y A , the realizations are automatically selftestable by lemma 5; however, Conjecture 2 has not been proven. Fortunately, however, lemma 7 tells us how to self-test such realizations irrespective of the validity of the conjecture; it suffices to intentionally introduce a unitary observable by adding one more binary measurement.
The simplest protocol may be to add the measurement of Z B . Let us add a binary measurement to the Bell scenario, such as the Bell 2, 3, 2 ( )-scenario but on Bob's side only, whose observable is B 2 ( = B I 2 2 ). Suppose that the correlators by the original set A A B B , , , The additional measurement is not restricted to Z B . In the second protocol, suppose that the correlators by  A A B B , , , is ensured to lie in the B-plane, and again, B 2 can be used as the third observable for proving the anti-commutation relation between B 0 and B 1 ; the proof of lemma 7 runs similarly, and the realization is self-tested.
Note that B 2 is also self-tested at the end of both protocols. Obviously, the scheme of the second protocol can be repeated to add more measurements on both sides of Alice and Bob. In this way, by using the geometry of figure 1 as a building block, the two-qubit realizations in the form of equation (5) with arbitrary number of measurements (whose basis lies in the X-Z plane) can be self-tested.

Summary
In this paper, we studied the self-testability and extremality from the viewpoint of the geometry of the state vectors of the realizations for quantum correlations, and showed a condition that determines the geometry uniquely. Interestingly, in the case of the realizations using partially entangled states, the condition for the unique determination of the geometry is strictly looser than that for the self-testability.
We first showed that the saturation of the TLM inequality for scaled correlators, together with the existence of a two-qubit realization in the form of equation (5), uniquely determines the geometry of state vectors in both cases of the D-space and the C-space (lemma 4 and 6). The uniqueness of the geometry generally ensures the extremality of  D , because it is a unique simultaneous maximizer of two quantum Bell inequalities in the Dspace. In the case of the C-space, however, such quantum Bell inequalities are lacking, and the uniqueness of the geometry is insufficient for the extremality of  C . Indeed, there exists a two-qubit realization such that, despite being an extremal point of  D , it is not an extremal point of  C due to the convexity of guessing probabilities. This suggests that the structure of  D is simpler than  C . The complete characterization of the extremal points of  D is an intriguing open problem.
We next showed that, when the conjectured certifiability of the guessing probabilities holds true, the selftestability in the C-space (hence the extremality of  C ) comes to be ensured by the extremality of  D (lemma 5). Namely, the sufficiency of the extremality criterion conjectured in [23] was shown to rely on the certifiability of guessing probabilities. The proof of the certifiability (i.e. the proof of the device-independent upper bound of guessing probabilities) seems quite challenging but attractive, because it would also lead to the discovery of the information principles [2,39] behind quantum mechanics, and 'almost quantumness' [40] as well.
Moreover, the realization with a unique geometry becomes self-testable if and only if another unitary observable exists on the geometry (lemma 7). Applying this proven fact, we proposed self-testing protocols for partially entangled two-qubit states, where one more measurement is intentionally added to prove the anticommutation relation between observables. This geometrical scheme provides a building block used for a more complicated geometry. Indeed, repeatedly adding measurements by this scheme, a realization with an arbitrary number of measurements can be self-tested. It is an open problem of how robust this scheme is.
As all the known nonlocal extremal points in the simplest Bell scenario are self-testable, it is natural to expect that the true extremal criterion must be the one that determines the geometry of state vectors as well as the TLM criterion. The conjectured criterion in [23] fulfills this expectation. Interestingly, although the validity of the conjecture has not been proven, the property of determining the geometry proves the self-testability of the realizations in the Bell scenario with more measurement settings as in the above self-testing protocols. r l r r = - are the matrix elements with respect to the eigenstates of r r + - 1 | | with m k and ¢ m k being the eigenvalues, as shown in appendix A of [32]. See also [33].
Appendix B. Uniqueness of geometry I First, we explicitly show how to construct a pair of the quantum Bell inequalities (8) and (9) where the last equation is the saturation condition for the second inequality of inequality (11). It is then sufficient to choose for both c=A, B as follows:  where ò is a small angle ( p < <  0 40) to ensure that equation (18) only has a trivial solution. As P and Q saturate equation (1) for scaled correlators, they are the extremal points of  D . Let us then consider L extrapolated from P and Q as where λ is chosen such that + + -= C C C C 2 00 01 10 11 at L. Suppose that C C C , , , L is a local correlation. This implies that P can also be realized by a convex sum of Q and deterministic correlations, despite that P is an extremal point of  D . On the other hand, when d d C , , ) are ensured to lie in the same B-plane (A-plane) [19]. However, the relationship between the two planes has not been determined yet.  , and y A  lie in the same plane. After all, the geometry of real vectors is determined as figure 1 with y c ¢ = cos 2 | | . The obvious symmetry is q q c , , , but H<0 for this choice.