Discrimination of two-qubit unitaries via local operations and classical communication

Distinguishability is a fundamental and operational measure generally connected to information applications. In quantum information theory, from the postulates of quantum mechanics it often has an intrinsic limitation, which then dictates and also characterises capabilities of related information tasks. In this work, we consider discrimination between bipartite two-qubit unitary transformations by local operations and classical communication (LOCC) and its relations to entangling capabilities of given unitaries. We show that a pair of entangling unitaries which do not contain local parts, if they are perfectly distinguishable by global operations, can also be perfectly distinguishable by LOCC. There also exist non-entangling unitaries, e.g. local unitaries, that are perfectly discriminated by global operations but not by LOCC. The results show that capabilities of LOCC are strictly restricted than global operations in distinguishing bipartite unitaries for a finite number of repetitions, contrast to discrimination of a pair of bipartite states and also to asymptotic discrimination of unitaries.

We here approach to characterising distinguishability of bipartite unitaries by LOCC in the single-shot scenario and investigate its relation to entangling capabilities. A pair of two-qubit unitaries, both entangling and non-entangling cases, are considered. We show that, on the one hand, any pair of entangling unitaries which do not contain local unitaries (see Eqs. (3) and (4) for the precise form) are perfectly distinguishable by LOCC whenever they are perfectly distinguishable by global operations. This compares to the single-shot scenario of two-state discrimination where any pair of orthogonal bipartite states, i.e. globally distinguishable, are perfectly distinguishable by LOCC 7,8 . On the other hand, we show non-entangling unitaries, i.e. local unitaries, that are perfectly distinguishable by global operations but not by LOCC. This contrasts to the asymptotic case where there exists a finite number of repeated applications such that multipartite unitaries are perfectly distinguishable by LOCC 9,10 . Finally, we consider minimum-error discrimination of entangling unitaries and show that LOCC protocols can achieve optimal discrimination by global operations.

Results
Discrimination of unitary transformations. Let U 1 and U 2 denote two unitary transformations we consider throughout. A general framework of distinguishing unitary transformations works as follows. (Note that, in order to compare to the results in refs 9, 10, we here do not consider ancillary systems in discrimination between unitaries). Suppose that there is a box in which one of two unitaries either U 1 or U 2 is applied with probabilities q 1 and q 2 respectively, once a quantum state arrives at the box. After application of unitaries, the resulting state is returned. Let ρ denote the input state to the box, and then the resulting state must be either ρ † U U 1 1 or ρ † U U 2 2 . The optimal discrimination between these states concludes which unitary transformation has been applied in the box.
For arbitrary two states, minimum-error discrimination has been completely analysed and the success probability is given by, depending on the choice of an input state, where ⋅ 1 denotes the trace norm, = † A tr A A for hermitian operators A. For optimal discrimination between unitaries, an input state should be found such that the success probability is maximised. This introduces optimisation of the distance over input states, which we call distinguishability of unitaries. Then, the success probability for unitaries is simplified as To compute distinguishability of unitaries, one can in fact restrict the consideration to pure states. This is due to the convexity of distinguishability, as follows. Suppose a pure-state decomposition of an input state ρ ρ = ∑ p Hence, distinguishability of unitaries is obtained by taking pure states as an input to unitary transformations. In addition, the trace distance for pure states has the reciprocal relation as From this, we introduce an equivalent quantity, the fidelity of unitaries F(U 1 , U 2 ), such that We then have the following relation between distinguishability and fidelity of unitaries, This shows the reciprocal relation between fidelity and distinguishability of unitaries. Or, to maximise the success probability, the task is to find state ψ that finds the fidelity F(U 1 , U 2 ) of unitaries.
In fact, discrimination of unitaries with N repetitions 4 can be rephrased in terms of the fidelity of unitaries. It has considered N repeated applications of unitaries, U N 1 and U N 2 . It has shown that, if applications of unitaries can be repeated so that the task becomes to discriminating between × × ×  U U U Two-qubit unitary transformations and LOCC scenario. Let us also recall a useful decomposition of two-qubit unitary transformations into entangling and non-entangling parts, and then summarise a discrimination task with LOCC.
Decomposition of unitary transformations. Two-qubit unitary transformations have a canonical form of decomposition 11,12 . A two-qubit unitary transformation W AB can be factorised into entangling and local unitaries, where the entangling part corresponds to the diagonal one W AB [d] and non-entangling ones {U A , U B , V A , V B } are local unitaries that only change local basis. The entangling part can be written in a compact way, Note that elements in the diagonal matrix d satisfy the order relation, π/4 ≥ v x ≥ v y ≥ v z ≥ 0. The entangling part can be alternatively expressed in its spectral decomposition with Bell states: From the relations and the order among {v i } i=x,y,z in the above, it also holds that The goal is then to find the relation between distinguishability of unitaries and their entangling capabilities. For the purpose, we here restrict our consideration to two entangling unitaries U 1,AB and U 2,AB which do not contain local parts. That is, both of them are expressed in the diagonal form in Eq. (4). Consequently, their product in Eq. (2) is also in the diagonal form, that is, for some diagonal matrix d. Note that it can also be decomposed into Bell states as it is in Eq. (5). The consideration in Eq. (6) also holds true for a pair of arbitrary two unitaries having their product in the diagonal form.
LOCC discrimination. Having specified the form of unitary transformations to be considered, we now introduce how discrimination between a pair of bipartite unitaries works, together with discrimination of bipartite quantum states, by LOCC. Suppose that there are two parties, Alice and Bob, who want to discriminate between two-qubit unitaries denoted by U 1,AB and U 2,AB , are far in distance. Then, there is a box in the middle such that it is not reached by both parties. Once a bipartite state ρ AB comes to the box, one of two-qubit unitaries either U 1,AB or U 2,AB is applied and then the resulting state, either of the followings returns to the two parties. Since they are far in distance, LOCC are only available in both stages of preparation of an input state and discrimination between two resulting states. As it has been shown in the above, distinguishability of two unitaries is obtained by taking an input state as a pure state, see also Eq. (2). Thus, it suffices for Alice and Bob to prepare product states as follows, Then, the next is to discriminate between resulting states, ψ , U AB AB 1 and ψ , U AB AB 2 , by LOCC. For LOCC discrimination between multipartite states, it has been shown that any pair of orthogonal two-qubit states, i.e., distinguishable by global operations, are also perfectly distinguishable by LOCC 7,8 . An LOCC protocol that perfectly discriminates between orthogonal two-qubit states has also been provided. This means that, as long as the resulting states ψ , U AB AB 1 and ψ , U AB AB 2 are orthogonal, they can be perfectly discriminated by an LOCC protocol. Therefore, for two unitaries that are perfectly distinguishable by global operations i.e. F(U 1,AB , U 2,AB ) = 0, the LOCC protocol for discrimination of unitaries reduces to finding a product state such that resulting states are orthogonal. That is, from the results on LOCC discrimination 7,8 , we conclude that two-qubit unitaries are perfectly distinguishable by LOCC if there exists a product state ψ in Eq. (7) The results for orthogonal bipartite states have been generalised to non-orthogonal states 13,14 . In fact, LOCC protocols can achieve the minimum-error discrimination that is obtained with global operations, regardless of how entangled given a pair of states are 13 . Therefore, we can also restrict to our consideration to preparing local states as the input state ψ which is in fact equal to the case that the minimisation is taken with entangled states 13 . Then, once Alice and Bob receive the resulting states, they can distinguish them by an LOCC protocol that can achieve optimal discrimination with global operations.

Distinguishability of two-qubit unitary transformations.
We now show discrimination of two-qubit unitary transformations in the following cases: i) when global operations are available, ii) when only LOCC are applied in state preparation and measurement, and iii) when state preparation is performed by LOCC and later global operations are applied in measurement. An input state to unitaries, denoted by ψ AB , can be written with Bell basis in Eq. (5), The state is product if the coefficients satisfy the relations:  4 such that F(U 1,AB , U 2,AB ) = 0, that is, perfect discrimination is achieved. see Fig. 1. If the convex hull does not contain the origin, perfect discrimination cannot be achieved and one has to find optimal input state to find optimal discrimination. In this case, distinguishability is then equivalent to the distance between the origin O and the convex hull, where ⋅ 2 denotes the Euclidean norm in the complex plane. Finally, it is worth to mention the radical difference between quantum states and unitary transformations in the discrimination scenario. In fact, if unitary transformations can be repeatedly applied, one can always find an input state such that the resulting convex hull contains the origin 4,5 . This means the perfect discrimination between two unitaries, which however does not happen in minimum-error discrimination of states.
Perfect distinguishability of entangling unitaries: LOCC are as powerful as global operations. Recall that a pair of two-qubit states that are orthogonal can be perfectly discriminated not only by global operations but also by LOCC 7,8 , where LOCC protocols for the task have been provided. Note also that the result holds true independently to entanglement contained in given two-qubit states.
Let us now consider a pair of two-qubit unitary transformations that are perfectly distinguishable by global operations, in which two resulting states after application of unitaries are orthogonal. Then, for these unitaries, the problem of distinguishing unitaries reduces to finding an input state prepared by LOCC, i.e. a product state in Eq. (9), such that the resulting states are orthogonal. The cases that resulting states are not orthogonal are also to be discussed.
Similarly to the convex hull in Eq. (11) introduced with global operations, let conv L denote the local convex hull constrained by the condition in Eq. (9), that is, constructed by state preparation with LOCC: , it means there exists an input state to unitaries such that the resulting states are orthogonal: hence, two unitaries are perfectly distinguishable by LOCC 7,8 .
In what follows, we show that a pair of two-qubit unitaries that are perfectly distinguishable, i.e. ∈ ( ), O U conv AB there always exists an input state prepared by LOCC such that resulting states are orthogonal, i.e., ∈ ( ), O U conv AB L and thus the two unitaries are perfectly distinguishable by LOCC. We mainly construct the local convex hull conv L within a convex hull in Eq.  Fig. 1(a), the convex hull is given as □ ABCD where {A, B, C, D} correspond to , , , 3 , respectively. Then, as it is shown in the above, the local convex hull is constructed as the set of midpoints of ( , ) p L 1 3 , convex combinations of B and D, and ( , ) p L 2 4 , convex combinations of A and C. The local convex hull thus corresponds to □ PQRS where {P, Q, R, S} are midpoints of {DA, AB, BC, CD}, respectively. This can be constructed as follows, see also Fig. 2(a). First, note that ( , ) p L 1 3 corresponds to any point on the line BD and ( , ) p L 2 4 on the line AC. Then, local convex hull is the collection of all midpoints of points on AC and BD. For instance, taking one point A and points on BD, one can find the midpoints on PQ. Or, taking C and BD, midpoints RS are found. In this way, one can see that □ PQRS is the local convex hull.
Finally, having constructed the local convex hull within the convex hull, it remains to show that if the origin O is in the convex hull, then it is also in the local convex hull. This follows from geometric properties of a circle: if a line is drawn within a circle where both end points touch the circle, then another line from the midpoint such that it is orthogonal to the original line passes through the origin. In this case, see Fig. 2(b), let us consider a line CD and S its midpoint. Applying the property here, it holds true that if a line is drawn from S such that it is orthogonal to CD then it passes the origin. This proves that if the origin is in the convex hull, it is also in the local convex hull. Thus, we have shown that a pair of entangling unitaries that are perfectly distinguishable by global operations can also be perfectly discriminated by LOCC.
Distinguishability of non-entangling unitaries: global operations are strictly more powerful than LOCC. While it has been shown so far that distinguishing entangling unitaries LOCC are as powerful as global operations, we here show that it cannot be generalised to arbitrary unitaries. We provide a pair of non-entangling unitaries, i.e. local unitary transformations, that are perfectly distinguishable by global operations but not by LOCC.
Let us consider two local unitaries, the product of which is The This shows that, unless either of the products U A or U B is perfectly distinguishable, it is not possible to perfectly discriminate between a pair of local unitaries having product ⊗ U U A B . More precisely, the convex hull of U A does not contain the origin, neither does the convex hull of U B . Hence, they cannot be perfectly discriminated by LOCC.
Optimal distinguishability of entangling unitaries: LOCC are as powerful as global operations. Coming back to entangling unitaries having their product in the diagonal form in Eq. (6), suppose that for any input state ψ , the are not orthogonal. In this case, the alternative is to find an optimal state such that the resulting states are most distinguishable. And then, if global operations are available, one may apply optimal state discrimination between them 15 . Remarkably, the results for orthogonal states 7,8 have been generalized to non-orthogonal ones 13,14 . Namely, for a pair of non-orthogonal multipartite states, one can always find an LOCC protocol that achieves the minimum-error discrimination by global operations. Therefore, global operations in optimal discrimination of non-orthogonal states can be replaced by LOCC protocols.
Then, what remains for optimal discrimination of unitaries is to compare state preparation, that is, local states and entangled states as input states to unitaries. This means that one has to find the convex hull in Eq. (12) with local states. That is, denoted by F L the fidelity from the local convex hull, we have the following as the fidelity between unitaries, AB L with ⋅ 2 denotes the Euclidean norm in the complex plane. One can then compare fidelities, F in Eq. (12) and F L in the above.
We recall that in the complex plane in Fig. 2, the local convex hull is constructed by connecting midpoints. In Fig. 3, when the convex hull does not contain the origin, the local convex hull is constructed in the same way as □ PQRS where S corresponds to the midpoint of DC. Moreover, OS is orthogonal to DC and thus gives the minimal distance between the local convex hull to the origin. In fact, this shows that the minimal distance is equal to the distance between the convex hull □ ABCD and the orgin, and hence, we have We have shown that, for distinguishing entangling unitaries having their product in the diagonal form in Eq. (6), there exists an LOCC protocol of state preparation and measurement that achieves optimal discrimination by global operations.

Discussion
We have considered distinguishability of bipartite two-qubit unitary transformations in the following cases: i) global operations are available, ii) LOCC are only available, and iii) LOCC are available in state preparation and global operations can be applied in measurement for state discrimination. We have then compared the three cases. We first recall that for a pair of bipartite quantum states, if they are orthogonal, they are perfectly distinguishable not only by global operations but also LOCC 7,8 . The capability of LOCC, however, does not generalise to bipartite unitaries. We have shown a pair of unitaries having their product in the form in Eq. (14) which are perfectly distinguishable by global operation operations but not by LOCC. We also recall the result of asymptotic discrimination of unitaries: any pair of unitaries can be perfectly discriminated in finite repetitions of unitaries by global operations 4 , and also by LOCC for multipartite unitaries 9,10 . We here have shown that the capability of LOCC cannot apply to cases when the number of repetitions is fixed: if unitary transformations can be applied only once, there exist local unitaries that are perfectly distinguishable by global operations but not by LOCC. are not orthogonal and do not lead to perfect discrimination between two unitary transformations. (a) The fidelity is given by the distance between origin and the convex hull, see also Eq. (12). (b) The local convex hull constructed by a product state is found as □ PQRS where {P, Q, R, S} are midpoints of {DA, AB, BC, CD}, respectively. Note that, while an input state is prepared locally, discrimination between resulting states is performed by global operations. Then, the distance between the local convex hull and the origin corresponds to the distance OS, since OS is orthogonal to DC and gives the minimal distance between the local convex hull and the origin.
Scientific RepoRts | 5:18270 | DOI: 10.1038/srep18270 Finally, we have shown that a pair of entangling unitaries having their product in Eq. (6) that contains no local unitaries, if they are distinguishable by global operations, can also be perfectly distinguishable by LOCC. For cases that they are not perfectly distinguishable by global operations, we have shown that there exists an LOCC protocol of state preparation and measurement that can reach the capability of global operations. Our results have found that relations between entangling capabilities and distinguishability of unitaries are highly non-trivial, while entanglement contained in states is closely related to distinguishability and related tasks, e.g. 16 . We envisage deeper relations between distinguishability and entangling capabilities of unitary transformations in characterisations of capabilities of quantum information tasks.