Povms Are Equivalent to Projections for Perfect State Exclusion of Three Pure States in Three Dimensions

POVMs are equivalent to projections for perfect state exclusion of three pure states in three dimensions

Abel Molina

Institute for Quantum Computing and School for Computer Science, University of Waterloo January 25, 2019

Performing perfect/conclusive quantum state able to give an index j such that the state was not pre- exclusion means to be able to discard with cer- pared in the state ρj. When Alice can achieve this with tainty at least one out of n possible quantum probability 1, we will say that we have perfect state ex- state preparations by performing a measure- clusion. This task of state exclusion has recently been ment of the resulting state. This task of state studied at length in [5], and is at the heart of the cel- exclusion has recently been studied at length in ebrated PBR thought experiment [31], where [11] (the [5], and it is at the heart of the celebrated PBR article from where we take the problem we solve) is cred- thought experiment [31]. When all the prepara- ited as the original source for the concept. The concept tions correspond to pure states and there are no of this task has also been used for proving results in the more of them than their common dimension, it is context of quantum communication complexity [24, 30], an open problem whether POVMs give any ad- as well as for designing quantum signature schemes [2]. ditional power for this task with respect to pro- Formalizing further this concept of state exclusion, [5] jective measurements. This is the case even for obtains the following semidefinite programming (SDP) the simple case of three states in three dimen- formulation: sions, which is mentioned in [11] as unsuccess- fully tackled. In this paper, we give an analyt- X ical proof that in this case considering POVMs minimize: pi hMi, ρii does indeed not give any additional power with i X respect to projective measurements. To do so, subject to: M = I (1) we first make without loss of generality some i i assumptions about the structure of an optimal M ≥ 0. POVM. The justification of these assumptions i involves arguments based on convexity, rank and where M ≥ 0 means that M is positive semi-definite. symmetry properties. We show then that any i i Being able to perform perfect state exclusion corre- pure states perfectly excluded by such a POVM sponds to the optimal value of this SDP being equal to meet the conditions identified in [11] for perfect 0. Similarly, any optimal solution to the semidefinite exclusion by a projective measurement of three program corresponds to an optimal positive-operator pure states in three dimensions. We also dis- valued measure (POVM) for state exclusion. Note that cuss possible generalizations of our work, includ- since we are only concerned with perfect state exclusion, ing an application of Quadratically Constrained we can just ignore the p in the rest of this presentation, Quadratic Programming that might be of special i since whether the value of the SDP is 0 or not does not interest. arXiv:1702.06449v4 [quant-ph] 23 Jan 2019 depend on them. Perfect exclusion of quantum states is also a meaning- 1 Context and Motivation ful concept in the context of the foundations of quantum mechanics, in particular when considering the topic of The task of quantum state exclusion corresponds to a quantum state compatibility. In that framework, one setting where an agent Alice is given a quantum sys- considers several quantum states {ρ1, . . . , ρn} as differ- tem. The state of this system is chosen at random be- ent beliefs about the same system. Then, one can ask tween n options {ρ1, . . . , ρn}, with corresponding non- whether the outcome of a measurement on the system zero probabilities {p1, . . . , pn}. It is unknown to Alice will disprove some of these beliefs, or they will all still which of the ρi was chosen, but she does know the {ρi} be possible. In the latter case, we say that the states and {pi} values characterizing the corresponding distri- are compatible with each other. Different ways of for- bution. Alice’s goal in the state exclusion task is to be malizing this idea will lead to different definitions of

Accepted in Quantum 2019-01-14, click title to verify 1 quantum state compatibility. [11] proposes several for- When this result was introduced in [11], it was men- malizations, one of which corresponds to the impossi- tioned that the authors were not able to prove that bility of performing perfect state exclusion. Since this PP-ODOP = PP-POVM in the context of 3 pure states formalization is a generalization of previous work by in 3 dimensions, despite having numerical evidence that Peierls [28], they refer to it as post-Peierls (PP) com- this is the case. The authors also present results es- patibility. tablishing that this is the first open case – they cite In more detail, the post-Peierls compatibility of sev- previous work [25] showing that for two pure states in eral quantum states {ρ1, . . . , ρn} (relative to a subset any dimension PP-ODOP = PP-POVM, and establish S of all POVMs) means that for all measurements in that for k > 2 pure states in 2 dimensions this will not S, there will be at least one outcome that can be ob- necessarily be the case. tained with non-zero probability for all of the possible We will give now an analytical proof which an- states/beliefs {ρ1, . . . , ρn}. If we consider the negation swers the corresponding question, by showing that of this definition, we obtain that this negation corre- PP-ODOP = PP-POVM in the context of 3 pure states sponds with the existence of a measurement in S such in 3 dimensions. that each outcome of the measurement excludes at least Our work can be seen as part of the line of work that one of the quantum states, which corresponds to an studies POVMs in the context of low-dimensional sys- agent being able to perform perfect state exclusion given tems of a fixed dimension. For example, [36] and [38] re- a mixture of the quantum states {ρ1, . . . , ρn} and ac- cently examined 2-dimensional POVMs in the contexts cess to measurements in S. When the set S of allowed of nonlocal games and quantum state discrimination, re- measurements corresponds to the set of all POVMs, spectively, while [40] looked into 4-dimensional POVMs the corresponding compatibility criteria is called PP- in the context of imposing symmetry conditions. POVM compatibility. When S is restricted to the set We conclude with a discussion about different ways in of projective measurements (or more precisely, the set which our work might be generalized. Of special interest of measurements defined by one-dimensional orthogo- here might be our discussion on the usage of Quadrati- nal projectors), [11] names the corresponding criteria cally Constrained Quadratic Programming (QCQP) to as PP-ODOP compatibility. model the n-dimensional variant of the question we One can consider the case where all of the n solve. This is a type of mathematical optimization for- states/beliefs {ρ , . . . , ρ } are known to belong to a par- 1 n malism that has seen a large number of applications ticular subset A of all quantum states, and ask whether in recent years, but only limited usage so far within for all such tuples of n beliefs in A the PP-ODOP the context of quantum information processing. To our and PP-POVM criteria will coincide with each other. knowledge, this is the first time that state exclusion of When this happens, we will say that in that context PP- pure states through projections is expressed through a ODOP=PP-POVM. Note that this is equivalent to pro- problem in a standard form of a mathematical optimiza- jections being optimal for perfect state exclusion within tion framework. the context of input states in A. In [11], the authors identify a necessary and sufficient In our derivation, we will use standard quantum in- condition for PP-ODOP incompatibility of 3 pure states formation theory notation and vocabulary – standard {a, b, c} in 3 dimensions (i.e. they establish a condition texts on the topic (e.g. [37, 39]) can be consulted for for the states to be perfectly excludable via a projec- definitions of the corresponding terms. tive measurement). This condition can be expressed in terms of the magnitudes of their inner products, given by |ha, bi|, |ha, ci|, |hb, ci|, and which we will denote as 2 Main derivation j1, j2 and j3, respectively. In particular, the condition obtained in [11] is that 3 pure states will be PP-ODOP 2.1 Restrictions that can be imposed without incompatible whenever loss of generality on POVMs that achieve perfect 2 2 2 j1 + j2 + j3 + 2j1j2j3 ≤ 1. (2) exclusion We will refer to this formula as the Caves-Fuchs- Our goal is to prove that for any set of 3 pure states in Schack inequality, after the authors of [11]. Note 3 dimensions that are perfectly excluded by a POVM, that we have corrected in our presentation of this for- they are also perfectly excluded by a projective mea- mula the original strict inequality sign that they use, surement. Following Equation (1) and our analysis of following the indications in [6, 33], and we have also it, we can identify the perfect exclusion of three pure merged the two conditions from the original presenta- states a, b and c with obtaining an optimal value of 0 in tion in [11] into one single condition. the following semidefinite program:

Accepted in Quantum 2019-01-14, click title to verify 2 1. The condition M1 + M2 + M3 = I corresponds to ∗ ∗ ∗ the equations minimize: a M1a + b M2b + c M3c X subject to: Mi = I (3) i ∗ ∗ v1v1 + w1w1 = 1 (4) Mi ≥ 0, ∗ ∗ v2v2 + w2w2 = x (5) Note that all the operators involved can be represented ∗ ∗ v3v3 + w3w3 = y (6) as 3 × 3 matrices. It is well-known in convex optimiza- v v∗ = −w w∗ (7) tion that the solution to optimizing a linear function 1 2 1 2 ∗ ∗ over a non-empty compact convex set in a finite di- v1v3 = −w1w3 (8) ∗ ∗ mensional Hilbert space can be assumed without loss of v2v3 = −w2w3. (9) generality to be an extreme point of the set of feasible solutions1 (note that in this case, that feasible set is the 2. We can assume v1 6= 0 and w1 6= 0, as otherwise set of POVMs). Therefore, we can assume that at most {M1,M2,M3} can be trivially transformed into a one out of the Mi has rank greater than 1. Otherwise, projective measurement. To see this, suppose for assume for the sake of contradiction that two of them example that w1 = 0. Then, (4) implies that |v1| = (say M and M ) have rank at least 2, so there is a com- 1 2 1, (7) that v2 = 0, and (8) that v3 = 0. We have mon vector u in the images of M and M . Then, for 1 2 then that M2 is diagonal, and its diagonal is equal small enough both {M + uu∗,M − uu∗,M } and 1 2 3 to (1, 0, 0). This implies that M3 is diagonal as {M −uu∗,M +uu∗,M } are POVMs, which implies 1 2 3 well, while w1 = 0 implies that the first term in {M ,M ,M } is not an extreme point of the feasible 1 2 3 its diagonal is equal to 0, so we can group M1 and set. M3 into a single operator and obtain a projective Without loss of generality, we can permute indices measurement. so that the ranks of M1, M2, and M3 are sorted in non-increasing order. Also, if M1 has rank 3 it cannot 3. Suppose we had v2 6= 0 and v3 6= 0. Then, (9) exclude any quantum state, so it must be the case that implies w2 6= 0 and w3 6= 0. This means we can its rank is at most 2. Note too that if the ranks are divide (7) by (8) and the conjugate of (9), and of the form (1, 1, 1), it is not hard to see that the con- obtain that v1 = v2 = v3 . Let λ be the value w1 w2 w3 dition M1 + M2 + M3 = I implies that {M1,M2,M3} of these ratios. Then, each of equations (7)-(9) form a projective measurement themselves. Similarly, implies that λ = 0, which contradicts v1 6= 0. in the case where the ranks are of the form (2, 1, 0), M1 = I − M2 implies that M1 and M2 form a pro- 4. We have then that either v2 = 0 or v3 = 0, and jective measurement (this is because for the right hand by symmetry we can assume without loss of gen- side I − M2 to have rank 2, M2 must have its non-zero erality that v3 = 0. Then, w3 = 0 as well, since eigenvalue equal to 1, which implies then the same for otherwise (8) would imply w1 = 0, which we know the eigenvalues of I − M2 = M1). not to be the case. (6) implies then that y = 0. We can assume then without loss of generality that 5. If we were now to additionally impose that v2 = there is an optimal POVM {M1,M2,M3} with ranks of the form (2, 1, 1). We can also choose now without loss 0, (7) would imply that w2 = 0, which can only happen when x = 0, by (5). However, in the x = 0 of generality to work in a basis such that M1 is diagonal and it perfectly excludes a = |0i. case we have that M1 is a projection on |1i and |2i, and M2 and M3 can be merged into a projection on We have then that M1 will be determined by the choice of a real diagonal vector (0, 1 − x, 1 − y), and M |0i, so there trivially is an optimal projection for 2 state exclusion, and the case is not of interest to and M3 by a choice of complex vectors v = (v1, v2, v3) ∗ us. We can assume then that v2 6= 0, and similarly and w = (w1, w2, w3) such that M2 = vv and M3 = ∗ that w2 6= 0. ww . We claim now that we can assume y = 0, v1 6= 0, w 6= 0, v 6= 0, w 6= 0, v = 0, w = 0 To see why, 1 2 2 3 3 We can introduce now a parameter r, which deter- consider the following five observations: mines the distribution of the weight x between M2 and 2 1 M3, and let |v2| be equal to x r+1 (r ∈ (0, ∞)). We 2 r have then that (5) implies |w2| = x r+1 , and that (7) 2 r 2 1 1 This fact follows from applications of the Krein-Milman and and (4) imply then that |v1| = r+1 , |w1| = r+1 . The Extreme Value theorems, which in their most general versions magnitudes of each element of v and w are then com- require in fact constraints less strong than the ones we have here. pletely characterized by the values of r and x.

Accepted in Quantum 2019-01-14, click title to verify 3 As for the phases of the elements of v and w, we can 4. A similar analysis applies to c, and we have that assume that v1, w1 ∈ R without affecting the values of it can be parametrized by a real positive value c1 2 1 M2 and M3. Then, if the phase of v2 is given by θ, (7) such that 0 ≤ c1 ≤ 1/ 1 + rx , together with the implies that the phase of w2 is given by π + θ. phase γ of c3. In this case, the value of c2 is given q We reach then our final form for what a POVM iθ 1 by c1e rx , and the magnitude of c3 is given by {M1,M2,M3} for perfect state exclusion of 3 pure states q 2 1 in 3 dimensions can be assumed to be without loss of 1 − c1 1 + rx . generality. In matrix form, it is given by 5. We can assume now that the phases ϑ and γ of b3 and c3 are selected in order to maximize the 0 0 0 left hand side of the Caves-Fuchs-Schack inequal- M1 = 0 1 − x 0 , (10) ity. The reason we can do this is because we are 0 0 1 interested in proving that the Caves-Fuchs-Schack √  r e−iθ rx 0 inequality holds, so this is a worst-case scenario in 1 √ M = eiθ rx x 0 (11) our situation. 2 r + 1   0 0 0 To do so, note that j1 = b1 and j2 = c1, so they do √  1 −e−iθ rx 0 not depend on the phases of b3 and c3. Therefore, 1 iθ maximizing the left hand side of the Caves-Fuchs- M3 = −e rx rx 0 (12) r + 1 Schack inequality will be equivalent to maximizing 0 0 0 ∗ j3 = |b c|. To do that, we compute first the value ∗ where 0 < x < 1, r ∈ (0, ∞), 0 ≤ θ < 2π. of b c, given by

s 2.2 Verification that any states perfectly ex- r r 1 ei(γ−ϑ) 1 − b2 1 + 1 − c2 1 + cluded by our parametrized optimal POVM satisfy 1 x 1 rx the Caves-Fuchs-Schack inequality 1 + b c − b c . (13) We look first at the structure of the states b and c per- 1 1 x 1 1 fectly excluded by M2 and M3, and obtain that it is enough to consider a one-parameter family for each of The magnitude of this expression will be the largest possible whenever the term in the first line inter- them. Let b be given by (b1, b2, b3), and c by (c1, c2, c3). Then, our conclusion follows from the following five ob- feres constructively with the term in the second servations: line. This will happen whenever the first line term is also real, and has the same sign as b1c1(1 − 1/x) 1. As usual, we can get rid of unphysical global We can in fact achieve this by picking γ = ϑ + π, phases, and assume b1 is a real positive number. since 0 < x < 1. We obtain then that in our worst- This is because multiplying b by a phase will not case situation, affect the value of our semidefinite program (3), and it will not affect either the satisfaction of the s Caves-Fuchs-Schack inequality. r r 1 j = 1 − b2 1 + 1 − c2 1 + 3 1 x 1 rx 2. It can be seen from (11) that the value of b2 is com- pletely determined by the value of b1 by the con- + b1c1(1/x − 1). (14) ∗ straint M2b = 0 (which is equivalent to b M2b = 0, The Caves-Fuchs-Schack inequality is expressed then in since M2 is Hermitian). In particular, one obtains iθp r our case as that b2 = −b1e x . j2 + b2 + c2 + 2j b c ≤ 1, (15) 3. The fact that b has norm 1 (since it represents a 3 1 1 3 1 1 pure state) allows us now to express the magni- where j3 is given in (14), x ∈ (0, 1), r ∈ (0, ∞), tude of b3 as a function of b1. In particular, the q r q 1 q b1 ∈ [0, 1/ 1 + ), c1 ∈ [0, 1/ 1 + ). We magnitude of b is given by 1 − b2 1 + r , while x rx 3 1 x will refer from now on to the left hand side of (15) its phase, which we will denote by ϑ, can take any as f(x, r, b1, c1). If b1 = 0 or c1 = 0, a simple al- value. gebraic manipulation of the value of j3 gives us that 2 Note that this implies an upper bound on b1, given f(x, r, b1, c1) ≤ 1. Expanding the value of j3, we have r by 1/ 1 + x . that f(x, r, b1, c1) is given by

Accepted in Quantum 2019-01-14, click title to verify 4 3 Perspectives for generalization

2 2 2 b1c1(1 + 1/x − 2/x) r 1 3.1 Usage of Quadratically Constrained + 1 − b2 1 + 1 − c2 1 + 1 x 1 rx Quadratic Programs (QCQPs) r s 2 r 2 1 We will now discuss how to study the perfect exclu- + 2b1c1(1/x − 1) 1 − b 1 + 1 − c 1 + 1 x 1 rx sion of n pure states by a projection through a collec- 2 2 2 2 tion of Quadratically Constrained Quadratic Programs + b1 + c1 + 2b1c1(1/x − 1) s (QCQPs). For a situation with a n-dimensional com- r r 1 plex variable x and m constraints, the standard form + 2b c 1 − b2 1 + 1 − c2 1 + 1 1 1 x 1 rx for such a program can be taken to be r 1 2 1 1 = 1 − b2 − c2 + c2b2 + r + 1 x 1 rx 1 1 x2 x r s r ∗ 1 2 r 2 1 minimize: x Gx + 2b1c1 1 − b 1 + 1 − c 1 + . x 1 x 1 rx ∗ (19) subject to: x Ckx ≥ lk, ∀k ∈ {1, . . . , m}, To prove that this is less or equal than 1, one can act similarly to the standard proof for x + 1 ≥ 2, moving x where the lk take real values, and G and the Ck are everything to one side of the inequality and writing as n × n Hermitian matrices. a square what one obtains. In more detail, multiplying by x and dividing by b2c2 our last expression, we have This is a type of mathematical optimization formal- 1 1 ism that has received considerable attention in recent that f(x, r, b1, c1) will be less or equal than 1 whenever years, with wide-ranging applications in science and en- gineering (see [1,9, 19, 22] for just a few amongst many s s 1 r 1 1 relevant examples). There has also been a considerable 2 2 − 1 + 2 − 1 + number of results about the theoretical structure of the b1 x c1 rx corresponding problems and the design of algorithms 1 1 1 2 1 that can solve them (see e.g. [20, 21, 26]). However, ≤ r 2 + 2 − + r + (16) c1 r b1 x r there have only been a handful of applications so far Observe now that both sides of this inequality are [4, 14, 23, 34] of the QCQP model to quantum informa- positive. This is trivial for the left hand side, and fol- tion processing. lows for the right hand side from the previous obtained In our collection of QCQPs, there will be one program upper bounds on b1 and c1. If we square both sides of for every n-combination with repetition {s1, . . . , sn} out this inequality and simplify the resulting expression, we of the set {w1, . . . , wn} of states to be excluded. Each obtain choice represents a possibility for how the states excluded after obtaining different outcomes of the projec- 2 1 2 1 1 2 r 4 − 2 + 1 + 2 4 − 2 + 1 tion relate to each other, and the reason why we need c1 c1 r b1 b1 to consider those choices is that two different outcomes 1 1 1 of the projection could plausibly lead to excluding the +2 2 + 2 − 2 2 − 1 ≥ 0. (17) c1 b1 b1c1 same state (which in the POVM case would be handled This can be rewritten as by grouping those two outcomes into the same one). In particular, each of the corresponding QCQPs for perfect state exclusion via projections formalizes the following 2 1 1 1 two ideas: r 2 − 1 − 2 − 1 ≥ 0, (18) c1 r b1 which is true, so we have successfully proved that a, • A projection in n dimensions corresponds to a b and c satisfy the Caves-Fuchs-Schack inequality, and choice of n unit vectors {v1, . . . , vn} that are pair- therefore can be excluded by a projective measurement. wise orthogonal. Note that x is not involved at all in (17), although one can verify computationally that the difference between the left hand side and the right hand side of (16) does • We would like for every vi to be orthogonal to the depend on x. corresponding si.

Accepted in Quantum 2019-01-14, click title to verify 5 These ideas are then reflected in the following QCQP: ical optimization packages. While the work on solver software supporting QCQP is not yet at a stage giving X ∗ ∗ minimize: vi (sisi )vi a simple path for an implementation of the programs i described by (20), it seems reasonable to expect that subject to: such a stage will be reached in the near term. Then, ∗ ∗ such a piece of software could be compared with an- vi vj + vj vi = 0 (20) other one that implements the program in (1). From ∀i, j ∈ {1, . . . , n} s.t. i < j, this, one would obtain a numerical study through stan- ∗ vi vi = 1, ∀i ∈ {1, . . . , n}, dard solvers of the difference between POVMs and pro- n vi ∈ C , ∀i ∈ {1, . . . , n}. jections for perfect state exclusion of n pure states in n ∗ ∗ dimensions. Note that we have written vi vj +vj vi = 0 rather than ∗ vi vj = 0, in order to have the matrix representing the constraint be Hermitian, as required in (19) (one can 3.2 Direct generalizations of our proof then go as usual from an equality with 0 constraint to two constraints of inequality with respect to 0). We can A naive approach for generalizing our result would ∗ ∗ start by considering conditions equivalent to the Caves- also write vi vi ≥ 1 rather than vi vi = 1, making usage of the fact that such a change does not alter whether Fuchs-Schack inequality in the 4-dimensional case. the value of the program is 0 or not. Also, while for However, this seems far from trivial, since the original ease of presentation we have stated the problem with derivation in [11] presents obstacles to such a general- n variables, they can be easily combined into one sin- ization. In particular, it relies on the fact that when 2 gle variable taking values in Cn in order to obtain a using the basis determined by an excluding projection, program of the exact same form as (19). the sums corresponding to the inner products between The number of such programs in dimension n that two of the perfectly excluded states {a, b, c} will have ex- we need to consider is given by the number of n- actly one non-zero term. This makes it relatively easy to combinations with repetition out of a set of length n, obtain formulas for the coefficients of a, b and c in that 2n−1 basis as a function of the inner products between the equal to n . While asymptotically this will scale very quickly as a function of n, it will still be com- states. However, solving the corresponding equations putationally tractable for values like n = 5 or n = 6, in 4 dimensions seems like a significantly more com- which goes beyond the theoretically understood range plicated task, as each inner product between excluded of up to n = 3. To compute the final answer, one will states involves not 2 but 4 non-zero coefficients. take the minimum value out of all the programs. If this It could also be fruitful to take a geometrical perspec- value is equal to zero, then the states {w1, . . . , wn} can tive in order to better understand the situation at hand, be perfectly excluded with a projection, while if it is following the approach in [7]. To see at an intuitive level a non-zero value then perfect state exclusion of the set what this might be like, one can start by observing that {w1, . . . , wn} will not be possible. the space of density matrices is a section of the con- As for its applications to future results, there are two vex cone of positive semidefinite matrices. Also, the main consequences of the formalism we just described, space of probability distributions with 3 outcomes can beyond the indirect consequence of our work possibly be seen as an equilateral triangle, with each vertex of inspiring further usage of the QCQP framework within the triangle corresponding to a different deterministic quantum information processing. distribution. Then, as one can see in Chapter 10 of [7], The first of these consequences correspond to our for any fixed 3-outcome POVM the map which takes newfound ability to use results about QCQPs in or- a density matrix to the probability distribution associ- der to obtain new structural results about the perfect ated with applying the POVM to the density matrix exclusion of pure states through projections. One can will be an affine map from the convex cone section to straightforwardly check that basic weak duality results the equilateral triangle. will not help, since the value of the Lagrangian dual In light of these facts, we can interpret any limits programs will always be zero. However, as we discussed to state exclusion via projectors as saying that three earlier there is an ongoing stream of non-trivial theo- points close to each other in the section of the convex retical results about QCQPs, and it seems reasonable cone cannot be sent to 3 different faces of the trian- to conjecture that some of those results will eventually gle by an affine map corresponding to a projection, as apply to the highly structured programs that we con- otherwise some points in the section would be sent out- sider. side the triangle, which is not allowed. Then, our result The second of these consequences corresponds to the that projections are equivalent to POVMs can be seen increased potential for the usage of standard mathemat- as saying that in the case of pure states this does not

Accepted in Quantum 2019-01-14, click title to verify 6 change when we also allow the affine maps correspond- 3.3 Other considerations ing to non-projection POVMs. It might be interesting It might also be of interest to find relations between the to fully formalize this thought, mathematically prove in optimality of projections for tasks involving POVMs, this framework the known results about limits to state and the optimality of unitaries (without the use of an- exclusion, and see if it is now easier to extend them to cillas) for certain tasks involving channels, discussed for the case of 4 pure states, where the space of outcomes example in [3,8], specially considering the numerical ev- of a 4-outcome POVM can be seen as a regular tetra- idence suggestive of such kind of connection identified hedron. in [3,5]. When doing so, it might also be of interest to consider results (such as those in [15]) that charac- Another way in which a geometric perspective might terize from a computational complexity point of view useful would be for obtaining a constructive algorithm the power of computing with unitaries as opposed to that transforms an excluding POVM into an excluding general quantum channels. projection for the case we analyze in this paper (3 pure Note too that for the case of mixed states, state ex- states in 3 dimensions). It seems plausible that obtain- clusion is mathematically equivalent to state discrimina- ing such an algorithm would then give insight about tion (the discriminated states would be those we obtain how to generalize our result. by computing I − ρi). This means one can apply exist- ing results about optimal measurements for mixed state discrimination [13, 16], and also consider the chance for Along the lines of using state exclusion characteriza- generalizing results [12, 35] that look into the optimal- tions alternative to the one given by (1), the work in [18] ity of projections for state discrimination of pure states. considers a generalization of the explicit perfect state Relatedly, as we discussed earlier the work in [11] gives exclusion criteria given in [11] for the 2-dimensional a characterization for perfect exclusion of 2-dimensional case. However, it finds this generalization to be a suf- pure states that makes it clear that for any number of ficient condition for the n-dimensional case but not a k > 2 states in 2 dimensions, projections are not nec- necessary one. This work also observes that if pure essarily equivalent to POVMs. It is the case that they states are perfectly excluded via a POVM, they will be additionally use a reduction to that setting to point out an eigenvector (with eigenvalue 0) of the corresponding that for three mixed states in three dimensions, projec- POVM element, and they can be assumed to be an ele- tions are not equivalent to POVMs within the context ment of its spectral decomposition. Then, one can con- of perfect state exclusion. sider the feasibility of an optimization program where Note as well that if one considers the possibility of one tries to fill in the remaining coefficients and vectors constraining the number of non-zero components of a in the spectral decomposition of the POVM elements. perfectly excluding POVM, the possibility of doing so While one might expect at first glance that the stan- simply corresponds to being able to perfectly exclude a dard SDP framework in (1) would offer a greater chance subset of the set of states under consideration. Another of applying mathematical optimization results, perhaps related variation one might want to study is requiring the fact that this formulation is closer in its shape to the that there are no zero components of a perfectly exclud- QCQPs in (20) could help make non-trivial connections ing POVM, as considered in [18]. between the projection case and the POVM case. One could also look into determining whether the results here carry over to the gradual measure of PP- Note too that the main insight that leads to our result incompatibility defined in [10], which is the value of is the fact that one can take a POVM for perfect state the SDP in (1) when the uniform distribution is as- exclusion to be an extremal one. This does not triv- sumed. This would correspond for example to asking ially lead to an answer, since there are extremal POVMs whether projections are optimal for state exclusion of 3 that are not projections, such as those in the family in pure states in 3 dimensions even when perfect exclusion Equations (10)–(12). However, an analysis of what the cannot be achieved by POVMs. If that was successfully ranks of an extremal POVM in 3 dimensions have to answered, it would be natural to relax assumptions even look like allows us to obtain a parametrization of the further, and consider arbitrary distributions. [29] offers situation that can be algebraically solved. The usage of a partial answer to these questions, by giving for an ar- more sophisticated facts about the structure of extremal bitrary number of pure states a sufficient condition for POVMs (such as those facts derived in [17, 27, 32]) the existence of an optimal excluding POVM that is a could be similarly involved in a generalization of our projection (this is the condition that there is an optimal results to higher dimensionality. In fact, these consid- POVM such that none of the outcomes are perfectly ex- erations seem to us a very likely ingredient of any such cluded, and we also have that the pure states are linearly generalization. independent).

Accepted in Quantum 2019-01-14, click title to verify 7 Note that the QCQP framework discussed in Sec- [3] Srinivasan Arunachalam, Abel Molina, and Vin- tion 3.1 extends without issues to the variants of the cent Russo. Quantum hedging in two-round problem discussed in the previous paragraphs. In par- prover-verifier interactions. In LIPIcs-Leibniz In- ticular, if one wishes to study the exclusion of k 6= n ternational Proceedings in Informatics, volume 73. states, one can simply write {w1, . . . , wk} rather than Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, {w1, . . . , wn} for the set of states to be excluded, giving 2018. DOI: 10.4230/LIPIcs.TQC.2017.5. k+n−1 2n−1 rise to n rather than n programs of the [4] Koenraad MR Audenaert and Stefan Scheel. Quan- form in (20). Similarly, if one wishes to study mixed tum tomographic reconstruction with error bars: states rather than pure states or introduce a probabil- a Kalman filter approach. New Journal of ity distribution on the states, one can simply modify Physics, 11(2):023028, 2009. DOI: 10.1088/1367- the objective function in (20) by replacing the values of 2630/11/2/023028. ∗ sisi with a corresponding ρi and combining them with [5] Somshubhro Bandyopadhyay, Rahul Jain, a multiplicative term pi, respectively. Jonathan Oppenheim, and Christopher Perry. Furthermore, if one wishes to study non-perfect state Conclusive exclusion of quantum states. Physical exclusion, the programs given in Section 3.1 can be used Review A, 89(2):022336, 2014. DOI: 10.1103/phys- towards that purpose without additional modifications. reva.89.022336. As for the variant that limits the number of non-zero [6] Jonathan Barrett, Eric G Cavalcanti, Raymond components of a perfectly excluding POVM, it will Lal, and Owen JE Maroney. No ψ-epistemic correspond to limiting the number of distinct terms model can fully explain the indistinguishabil- that can appear in the n-combinations with repetition ity of quantum states. Physical Review Let- of {w , . . . , w } that characterize the programs in (20). 1 n ters, 112(25):250403, 2014. DOI: 10.1103/phys- Similarly, the requirement that there are no zero com- revlett.112.250403. ponents of the POVM corresponds to requiring that every state is excluded by a measurement outcome, [7] Ingemar Bengtsson and Karol Zyczkowski. Geom- and therefore to only considering the n-combination etry of Quantum States: An Introduction to Quan- given by {s , . . . , s } = {w , . . . , w }. tum Entanglement. Cambridge University Press, 1 n 1 n 2007. DOI: 10.1017/9781139207010. [8] Michael R Beran and Scott M Cohen. Nonoptimal- ity of unitary encoding with quantum channels as- Acknowledgments sisted by entanglement. Physical Review A, 78(6): 062337, 2008. DOI: 10.1103/PhysRevA.78.062337. Thanks are due to John Watrous for numerous help- [9] Subhonmesh Bose, Dennice F Gayme, K Mani ful suggestions, as well as to Juani Bermejo-Vega, Alex Chandy, and Steven H Low. Quadratically con- Bredariol-Grilo, Richard Cleve, Philippe Faist, Nicolás strained quadratic programs on acyclic graphs with Guarín-Zapata, George Knee, Robin Kothari, Deb- application to power flow. IEEE Transactions on bie Leung, Alexandre Nolin, Christopher Perry, Bu- Control of Network Systems, 2(3):278–287, 2015. rak Şahinoğlu, Jamie Sikora and Jon Tyson for insight- DOI: 10.1109/tcns.2015.2401172. ful discussions. This work was partially supported by [10] Todd A Brun, Min-Hsiu Hsieh, and Christopher NSERC, the Canada Graduate Scholarship program, Perry. Compatibility of state assignments and the Mike and Ophelia Lazaridis Fellowship program, pooling of information. Physical Review A, 92(1): and the David R. Chariton Graduate Scholarship pro- 012107, 2015. DOI: 10.1103/physreva.92.012107. gram. [11] Carlton M Caves, Christopher A Fuchs, and Rüdiger Schack. Conditions for compatibility References of quantum-state assignments. Physical Re- view A, 66(6):062111, 2002. DOI: 10.1103/phys- [1] Chris Aholt, Sameer Agarwal, and Rekha Thomas. reva.66.062111. A QCQP approach to triangulation. In European [12] Yonina C Eldar, Alexandre Megretski, and Conference on Computer Vision, pages 654–667. George C Verghese. Designing optimal quantum Springer, 2012. DOI: 10.1007/978-3-642-33718- detectors via semidefinite programming. IEEE 5_47. Transactions on Information Theory, 49(4):1007– [2] Juan Miguel Arrazola, Petros Wallden, and 1012, 2003. DOI: 10.1109/tit.2003.809510. Erika Andersson. Multiparty quantum signature [13] Yonina C Eldar, Mihailo Stojnic, and Babak schemes. arXiv preprint, 2015. URL https: Hassibi. Optimal quantum detectors for unam- //arxiv.org/abs/1505.07509. biguous detection of mixed states. Physical Re-

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Accepted in Quantum 2019-01-14, click title to verify 10