Worst-case Quantum Hypothesis Testing with Separable Measurements

For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_\text{I}$ of rejecting the null hypothesis and $p_\text{II}$ of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes $p_\text{II}$ under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separable measurements, in order to describe tests that are performed locally. The case $p_\text{I}=0$ has been recently studied under the name of"quantum state verification". We show that the problem can be cast as a semi-definite program (SDP). Then we study in detail the two-qubit case. A comprehensive study in parameter space is done by solving the SDP numerically. We also obtain analytical solutions in the case of commuting hypotheses, and in the case where the two hypotheses can be orthogonal (in the latter case, we prove that the restriction to separable measurements generically prevents perfect distinguishability).


I. INTRODUCTION
The task of quantum hypothesis testing [1] is to optimally identify, according to some given payoff function, an unknown quantum state given as a black box. The strategy consists of performing a quantum measurement, whose optimality depends upon the payoff function and the prior information about the state, available in the form of a probability distribution over the state space. Several discrimination problems [2][3][4][5][6][7] are based on quantum hypothesis testing.
In the simplest non-trivial instance of the problem, the prior distribution has support over two states only, the null and the alternative hypotheses. Hence, binary quantum hypothesis testing corresponds to the derivation of the tradeoff relation between two error probabilities: I) the probability of rejecting the null hypothesis, and II) the probability of accepting the alternative hypothesis. The case when such hypotheses are given explicitly was solved analytically by Helstrom [1].
Here, instead, we consider the case in which only one state (the null hypothesis) is explicitly given. For the other state (the alternative hypothesis), we consider a constrained worst-case scenario. The worst-case alternative hypothesis is the one that maximizes the type-II error probability, under a given lower bound on the distinguishability of the two hypotheses. Additionally, we consider multipartite hypotheses, and we restrict the optimization to separable measurements only. This setup generalizes the so-called "quantum state verification" [8][9][10][11][12][13], by relaxing the assumption that the type-I error probability is null.
Our first contribution is a formulation of the aforementioned problem as a semi-definite program, that can be efficiently solved with readily available numerical tools. Then, we specify to the case in which the hypotheses are two-qubit states, and we derive an analytical form for the worst-case hypothesis. Finally, we analytically derive the tradeoff relation between type-I and type-II error probabilities in the case when the hypotheses commute.
The structure of this paper is as follows. In Section II we recall the general problem of quantum hypothesis testing and we introduce the specific problem addressed here. In Section III we reformulate our problem as a semidefinite program, and we analytically derive the worstcase alternative hypothesis in the two-qubit case. In Section IV we analytically derive the tradeoff relation between type-I and type-II error probabilities for commuting hypotheses. Section V summarizes our results.

II. HYPOTHESIS TESTING OF QUANTUM STATES
The simplest scenario of hypothesis testing is binary quantum state discrimination between ρ 0 and ρ 1 . In other words, one is asked to decide which is more likely between two hypotheses H 0 -the null hypothesis -representing the fact that the unknown state is ρ 0 , and H 1 -the alternative hypothesis -corresponding to the unknown state being ρ 1 . The decision process can be formalized by a POVM {Ω, 1 1 − Ω} where the element Ω accepts H 0 and 1 1 − Ω accepts H 1 . This naturally gives rise to two errors, type I or false positive p I = tr(ρ 0 (1 1 − Ω)) arXiv:1910.10954v1 [quant-ph] 24 Oct 2019 and type II or false negative p II = tr(ρ 1 Ω). False positive probability captures the situation that the decision process accepts H 1 when hypothesis H 0 is true. False negative probability corresponds to the other situation where one accepts H 0 when H 1 is true. Therefore, in this language, the problem is to design an optimal measurement Ω that optimizes certain figure-of-merit. For example, Helstrom strategy minimizes the average probability of error p 0 p I + p 1 p II where p 0 is the a priori probability of occurrence of hypothesis H 0 and ditto for p 1 .
Several problems in quantum information such as quantum channel coding [14] and quantum illumination [7] can be seen as hypothesis testing problems by assigning appropriate sets to hypothesis and choosing appropriate figures-of-merit (see e.g. [15,16]). Here we look at the task that has been called, maybe confusingly, quantum state verification [8]. In this task, H 0 is the state |ψ ψ|, and H 1 is the set of states ψ| σ |ψ ≤ 1 − . Previous works [8,10] considered strategies that have no false positive, i.e. p I = 0, and set out to minimize the worst-case probability of false negative The set to which Ω belongs can be that of all effects, or a restricted one. When dealing with composite systems, a particularly relevant set is the set SEP of separable effects. In this work, we shall focus on this one.
Here we relax the condition p I = 0 to p I ≤ δ, leading to the optimisation This generalisation is relevant, as it allows the study of the tradeoff between p I and p worst II (henceforth denoted simply as p II ). From the technical point of view, this study does not constitute a straightforward extension of previously employed mathematical tools for the following reason. The condition ψ| Ω |ψ = 1 forces Ω to commute with |ψ ψ|, which provides a significant simplification in the number of parameters and structure of the problem. When that condition is relaxed to ψ| Ω |ψ ≥ 1 − δ, commutativity can no longer be assumed a priori (and we shall show that, for some values of δ and the other parameters, the optimal strategy is indeed not the commuting one).

III. REFORMULATIONS OF THE OPTIMISATION
In this section, we first show that the optimisation (1) can be cast as a semidefinite program (SDP), which allows for reliable numerical solutions. Then, for the case of two-qubit states, we solve the optimisation of the inner problem, thus casting the optimisation in a form which will allow deriving some analytical results in Section IV.
A. Reformulation as a SDP Lemma 1. The optimisation (1) can be reformulated as a semidefinite program p II = min Ω,y1,y2 Proof. The constraints on Ω (outer optimisation) remain the same, while we replace the inner optimisation max{tr(Ωσ) : σ 0, tr σ = 1, ψ| σ |ψ ≤ 1 − } by its dual, which is the semidefinite program Moreover, strong duality holds because the primal is feasible, and thanks to Ω † = Ω the dual is strictly feasible (choose y 2 > 0 such that (y 1 1 1 − Ω + y 2 |ψ ψ| 0). This means that the primal and dual optimum are the same, and also the primal optimum is attained. Hence, our minimax problem becomes (2). Note that separability is an SDP constraint albeit exponential in size [17] and not just a hierarchy of SDP constraints [18].

Lemma 2.
For two-qubit pure states |ψ = cos θ |00 + sin θ |11 , the optimisation (2) reduces to an optimisation over real variables p II = min t,z,x,ω,y1,y2 Proof. We first spend the symmetry present in the state. Define with U φ = |0 0| + e iφ |1 1|. For any feasible (Ω, y), the pair (Ω a , y) remains feasible with the same value of the objective function. Moreover, the state is also invariant under swapping S of two qubits, so that (Ω a , y) with Ω a := (Ω a + SΩ a S † ))/2 is feasible as well. Lastly,Ω a can be taken to be real symmetric because the feasible region is preserved under taking entrywise complex conjugate, and the objective value is unchanged. We note that this same symmetrisation was carried out in [10] on the primal inner problem, thanks to the assumption that Ω commutes with |ψ ψ|. It's by looking at the dual that we noticed that the symmetry is independent of the commutation assumption. This observation simplifies the number of variables in our optimisation. Specifically, let ψ ⊥ = − cos θ |00 + sin θ |11 , it suffices to optimize over real symmetric ma-tricesΩ in the ordered basis {|ψ , ψ ⊥ , |01 , |10 }. Writing out the separability constraint, which for qubits is equivalent to positive partial transpose, we arrive at the final form given by the Lemma.
Remarkably, what was the inner optimisation (now optimisation over y 1 and y 2 ) can be further solved analytically; besides, one can set t = 1 − δ − z and ω = |x cos 2θ + z sin 2θ| without loss of generality. The lengthy proof of these steps is presented in Appendix A. The farthest version of the optimisation that we can reach analytically reads: Lemma 3. For two-qubit pure states |ψ = cos θ |00 + sin θ |11 , the optimisation (2) reduces to .

IV. RESULTS
For our results, we keep focusing on the case of two qubits, although we recall that the SDP (2) is valid in general and one could therefore set out to solve it in any other case.

A. Commuting strategy
As we mentioned earlier, when δ = 0, which is the case considered in [8,10], the condition ψ| Ω |ψ = 1 immediately implies that Ω is diagonal in the same basis as |ψ ψ|, that is x = 0 in our notation. When δ = 0, there is no a priori guarantee that the optimal solution will be a commuting one; but we can obtain an upper bound p c II by enforcing x = 0. In this case, the optimisation (6) becomes trivial: This result could have been derived at an earlier stage than Lemma 3 (see Lemma 6 in the Appendix). In fact, it can also be derived without any reliance on the SDP formulation, by adapting the steps made in Ref. [10] to the case δ = 0.
B. Analytical solution for = 1 Next, we present the analytical solution of (6) for the special case = 1. The optimisation now reads p II = min z,x |x cos 2θ + z sin 2θ| where we have spelled out the two matrix constraints in (6). Even for this simple case, the study is heavy, though without intrinsic difficulties.
First we notice that for the maximally entangled state (cos 2θ = 0, sin 2θ = 1) the figure of merit is simply z, and the last constraint is z ≥ 1−δ 3 . The two quadratic constraints are both feasible for z = 1−δ 3 , for a variety of values of x including x = 0. Thus, for θ = π 4 we find p II = p c II = 1−δ 3 ; both the commuting strategy and several non-commuting ones achieve this bound.
(10) Since both x 0 and x 1 are non-positive, p II ≤ p c II as expected. Notice that (9) captures also the case θ = π 4 (only, x * is not unique in that case). Besides, x 0 = 0 holds only for cos 2θ = 1 i.e. for the product state; and x 1 = 0 holds only for δ = 0. In summary, for = 1, 0 < δ < 1, and 0 < θ < π 4 , the optimal strategy is not the commuting one.
Since we have set = 1, which means that σ can be orthogonal to |ψ , it is natural to check also when p II = 0, which would be obviously the case if we had not added the constraint that Ω must be separable. By inspection, we see that this is the case only for θ = 0, i.e. when the state itself is product (in which case it is trivial: one can find an orthogonal product state and check the orthogonality locally).

C. Numerical solutions of the SDP
We have seen that, even in the case = 1 when the figure of merit is at its simplest, the analytical solution requires some work and yields a not-so-transparent result. In view of this, for arbitrary values of we leave aside any attempt of solving the optimisation (6) analytically, and resort rather to numerical solutions of the SDP (2).
The results are presented in Fig. IV C. We see that, in a large portion of parameter space, a commuting strategy is very close to being optimal (if not exactly so). A significant difference is seen only for 0.8, that is, when the state σ is allowed to be almost orthogonal to |ψ .

V. CONCLUSIONS
We have presented a scenario of quantum hypothesis testing that generalises Helstrom's original one by allowing the alternative hypothesis to be a worst-case state for fixed distinguishability. It also generalises the task that has been called "quantum state verification" by removing the constraint that the probability of rejecting the null hypothesis would be p I = 0. After reformulating the problem as a SDP, we studied it in detail for the case of two qubits and a pure state as null hypothesis. We found values of the parameters of the problem, for which the POVM describing the optimal measurement strategy does not commute with the null hypothesis state.

ACKNOWLEDGMENTS
We thank Masahito Hayashi for discussions about our modifications of the hypotheses, and more generally the connection between quantum verification and hypothesis testing. This work is supported by the National Research Foundation and the Ministry of Education, Singapore, under the Research Centres of Excellence programme.
Proof. With the notation introduced in the proof of Lemma 2, for any feasible (Ω a , y 1 , y 2 ) with ψ|Ω a |ψ > 1−δ ≥ 0 for δ ∈ [0, 1] there is another feasible (Ω a , y 1 , y 2 ), whereΩ ensures ψ|Ω a |ψ = 1 − δ, achieving the same objective value, we can without loss of generality assume that t + z = 1 − δ. It is clear that by reducing ω, we increase the size of the feasible region of the inner optimisation over variables y 1 , y 2 . Therefore since |x cos 2θ + z sin 2θ| ≤ 1 follows from other constraints, we have that 0 ≤ |x cos 2θ + z sin 2θ| ≤ ω ≤ 1, which means it suffices to take ω equal the lower bound.
We now solve the inner optimisation, that is the optimisation over y 1 and y 2 . It is natural to split our consideration into commuting strategy x = 0 and noncommuting strategy x = 0, as the commuting case is a simpler linear programming problem.
Lemma 5. For the commuting strategy x = 0, the solution of the inner optimisation is where y * 1 := max{t − z, |z sin 2θ|}.
Proof. By Sylvester's criterion for psd, the inner minimization becomes min y1,y2 ) ≥ x 2 y 1 ≥ |x cos 2θ + z sin 2θ|, y 2 ≥ 0 When x = 0 the quadratic constraint trivially follows from the inequality constraints so we can drop it and the optimisation is linear. The constraints are y 1 ≥ max{t − z, |x cos 2θ + z sin 2θ|} y 2 ≥ max{0, (t + z) − y 1 } so that the minimum is reached at the lower bounds.
Lemma 6. For the commuting strategy x = 0, the optimal error probability is given by Proof. Since x = 0, we are left with the program The objective function can be rewritten as from which we consider two cases. If y * 1 ≥ 1 − δ then Here y * 1 ≥ 0 always, so the minimum is at least 1 − δ. If It is straightforward to see that the minimum is achieved when corresponding to an optimal solution z * = 1−δ 2+sin 2θ with optimum value (1 − δ) 1 − 1 + sin θ cos θ .
Since the global minimum is the smaller value of these two cases, the proof of the Lemma is complete.
We remark that the structure of the optimal verification operator among all commuting strategies is rather simple. Explicitly we have that This can be seen as a generalization of the optimal commuting strategy that Wang and Hayashi found for δ = 0 case [10] to the δ ∈ [0, 1] case. We now consider the noncommuting case, which is no longer a linear optimisation problem.
Proof. When x = 0 (noncommuting strategy), the feasible region excludes the points (y 1 , y 2 ) where y 1 − (t − z) = 0, or y 1 + y 2 − (t + z) = 0 and so the optimisation becomes min y1,y2 Here the optimisation splits into two branches. Firstly, consider the branch and explicitly in terms of the roots y 1 ≤ λ min := t − x 2 + z 2 or , But then t − √ x 2 + z 2 < t − z < t + √ x 2 + z 2 implies that the feasible region is y 1 ≥ max{ω, λ max } leading to the optimum value y * 1 = max{ω, λ max } which is always at least λ max . Secondly, the remaining branch which is equivalent to could be infeasible depending on ω. However, whenever feasible, i.e. ω ≤ λ max , the minimum is upper bounded by the value of the objective function at the feasible point y 1 = λ max , i.e. for which the objective value is λ max + (1 − ) * 0. Therefore, without loss of generality we consider this latter branch whenever feasible.
The inner optimisation becomes min y1 from which is is clear that the minimum is reached at the stationary pointŷ 1 which is the largest solution of (ŷ 1 − (t − z)) 2 = (1 − )x 2 whenever this point is feasible, or at the endpoints ω if y * 1 < ω and λ max if y * 1 > λ max . (Note that x = 0 ensureŝ y 1 > t − z if exists so that the smallest solution is always infeasible.) Finally, we present the proof of Lemma 3: Proof. With y * 1 given before, we have to solve This becomes an optimisation over two real variables z, x after eliminating t. To see the branch reduction, we consider feasible (z, x) that satisfies and show that objective value (abuse of notation and redefine the function f eliminating variable t) is smaller in the region I defined by 1 − δ − 2z + 1 − |x| ≤ |x cos 2θ + z sin 2θ| .
In the region III defined by the inequality the objective function takes the value which is a function of two independent variables z, x, and is increasing in terms of |x| for a fixed value of z.
Likewise, in the region II defined by the inequality the objective function take the value which is also increasing in |x|. Moreover, at the boundary between two regions, the objective functions agree.
The argument now goes as follows: for each feasible z, we look at the set of feasible x that is defined by (A8). For any feasible x 1 , x 2 in region III (if exist), since the objective function is increasing, the point with smaller |x j | achieves a lower objective value. Hence for minimization, it suffices to consider feasible x in the boundary of region III. Since this boundary is also contained in region II, we have shown that without loss of generality it suffices to consider the feasible x that belong to regions I and II. Now the argument can be repeated: points x 3 , x 4 in region II with smaller |x j | achieve small objective value. This reduces the feasible region to region I only. lower-bounded by a broken line with kink at x k = − 1−δ 2 tan(2θ) the intersection with x cos 2θ + z sin 2θ = 0.
The figure of merit to be minimised is |x cos 2θ +z sin 2θ|: this means that p II is given by the smallest distance between the line z = − cot 2θx and a point of the feasible region. A graphical inspection (see Figs 2 and 3) shows that this minimal distance is always achieved by the point that is the intersection of either P 0 or P 1 with the line z ≥ 1−δ−x cos 2θ 2+sin 2θ . These are the points whose x coordinates have been called x 0 and x 1 , given in Eq. (10) in the main text. A more fully analytical proof of this result would not bring further clarity (and for good measure, the correctness of the result has been double-checked numerically with the solution of the corresponding SDP).
FIG. 3. When θ is reduced, the slope of the left segment of K increases and also cuts P0, whence the feasible region consists of two disjoint sets. Nonetheless, the closest point to the line z = − cot 2θx remains the one on the right segment of K.