A combined-probability space and (un)certainty relations for a finite-level quantum system
AA combined-probability space and (un)certainty relations for a finite-level quantumsystem
Arun Sehrawat ˚ Department of Physical Sciences, Indian Institute of Science Education & Research (IISER) Mohali,Sector 81 SAS Nagar, Manauli PO 140306, Punjab, India
The Born rule provides a probability vector (distribution) with a quantum state for a measurementsetting. For two settings, we have a pair of vectors from the same quantum state. Each pairforms a combined-probability vector that obeys certain quantum constraints, which are triangleinequalities in our case. Such a restricted set of combined vectors, titled combined-probabilityspace, is presented here for a d -level quantum system (qudit). The combined space turns out acompact convex subset of a Euclidean space, and all its extreme points come from a family ofparametric curves. Considering a suitable concave function on the combined space to estimate theuncertainty, we deliver an uncertainty relation by finding its global minimum at the curves for aqudit. If one chooses an appropriate concave (or convex) function, then there is no need to searchfor the absolute minimum (maximum) on the whole space, it will be at the parametric curves. Sothese curves are quite useful for establishing an uncertainty (or a certainty) relation for a generalpair of settings. In the paper, we also demonstrate that many known tight (un)certainty relationsfor a qubit can be obtained with the triangle inequalities. I. INTRODUCTION
Every setting for a measurement on a quantum sys-tem can be completely specified by an orthonormal basisof the system’s Hilbert space. Identical systems can beindependently prepared in a (pure) state ρ such that, ev-ery time, we get a definite outcome when a system ismeasured in a setting a . If we change a to a physicallydistinct setting b , then we observe—sometimes one andsometimes other—multiple outcomes. In other words,there the probability is one for an outcome in a -setting,whereas none of the probabilities is one in b -setting. Ofcourse, in any setting, all the probabilities are nonnega-tive numbers that sum up to one. Apart from that, theprobability vectors (distributions) (cid:126)p and (cid:126)q —associatedwith the two settings a and b , respectively—must followcertain constraints, called quantum constraints (QCs),together.Historically, such QCs are expressed in terms of un-certainty relations (URs) by taking Hermitian operatorsrather than orthonormal bases. An UR is an inequal-ity, c p a, b, ρ q ď u p a, b, ρ q , between two real-valued func-tions: uncertainty measure u and its lower bound c . In1927, Heisenberg introduced the first UR [1, 2] (derivedby Weyl in [3]) for the position and momentum operators.Different aspects of his seminal work are reviewed in [4].Robertson [5] generalized the Heisenberg’s relation foran arbitrary pair of operators by employing the standarddeviation as a measure of uncertainty. In Robertson’sUR, the lower bound c is a function of state ρ . Deutschcriticized it and introduced a new UR [6] for a finite-dimensional state space by taking entropy as a measureof uncertainty. He achieved a state independent c p a, b q .Later, a better lower bound was conjectured by Kraus [7] ˚ email: [email protected] and then proved by Maassen and Uffink [8]. Such URsare—known as entropy URs—reviewed in [9–11].Throughout the article, we are considering d -levelquantum systems (qudits) and projective measurements.Our primary objective is to study a set of combined-probability vectors p (cid:126)p, (cid:126)q q , called combined-probabilityspace , where every vector respects certain, if not all, QCs.Here the elemental QCs are the triangle inequalities (TIs)between quantum angles , and the (un)certainty relationsemerge from them. As an angle between a pair of kets—called quantum angle—is a metric over the set of all purestates [12], we own TIs. Landau and Pollak obtained asingle TI [13] of this kind for continuous-time signals andprovided a classical UR (see also Sec. 8 in [14]).In Sec. II, we present the combined space that is acompact convex subset of the d -dimensional real vectorspace R d . Thanks to the Krein-Milman theorem (seeTheorem . . and Appendix A.3 in [38]), every compactconvex subset of R d can be generated by the convexcombinations of its extreme points. As a principal result,we provide a family of parametric curves in Sec. II, whichrepresents all the extreme points of the combined space.In the case of d “ , all the parametric curves form anellipse, and the same ellipse also appears in [15–17] as aspecial case.An uncertainty measure u p a, b, ρ q ” u p (cid:126)p, (cid:126)q q should bea concave function on the combined-probability space,argued in the beginning of Sec. III. The concavity of u ensures that its global minimum c will occur at the para-metric curves (extreme points) of the space (see Theo-rem . . and Appendix A.3 in [38]). Hence, one canexploit these curves to obtain an UR, rather easily, forher or his liking of u and, of course, for general measure-ment settings a and b .In Sec. III, we choose a concave, thus uncertainty, mea-sure u p (cid:126)p, (cid:126)q q . A significance of our choice lies in the factthat u is again a concave function on every parametriccurve (that is, as a function of the parameter). There- a r X i v : . [ qu a n t - ph ] A p r fore its absolute minimum c will occur nowhere but atthe endpoint(s) of these curves. A simple three-step pro-cedure is delivered to find the lower bound c ď u for anarbitrary pair t a, b u of settings and for a finite d . One canemploy an ordinary computer to run the procedure. Be-sides, c is presented in analytic forms for d “ , , and inthe case of mutually unbiased bases (MUBs) [18]. Refer-ences [7, 16, 19–23] contains URs particularly for MUBs.At the end of Sec. III, we provide another uncertaintymeasure that is also concave on all the parametric curves,so the whole analysis given before for u can be straight-forwardly applied to this measure.If a suitable concave function can be a measure of theuncertainty, then an appropriate convex function will bea measure of certainty. In Sec. IV, we pick some otherconcave and convex functions and exhibit that the tight(un)certainty relations given in [6, 8, 16, 24–31] for aqubit can be achieved with the TIs that specifies the el-lipse. We conclude the article with Sec. V.The appendices are kept for certain technical detailsand proofs: the TIs are derived in Appendix A. It ismanifested in Appendix B that the combined space is acompact convex set. The parametric curves are explicitlyobtained in Appendix D with the help of Appendix C. II. QUANTUM CONSTRAINTS ANDCOMBINED-PROBABILITY SPACE
In quantum theory, observables are represented byHermitian operators. If such an operator is degenerate,then it possesses more than one eigenbases, where someof them can represent physically different measurementsetups. Hence, ‘measurement in an orthonormal basis’ ofthe underlying Hilbert space is rather well defined than ‘ameasurement of an operator’ (see Chapter 7 in [39]). Infact, measurement in a basis B a measure all the operatorswhose eigenbasis is B a . Moreover, Deutsch pointed outthat a measure of uncertainty for a discrete observablemust not depend on its eigenvalues, but on its eigenbasis[6]. With all these considerations, we choose orthonormalbases instead of Hermitian operators to specify differentprojective measurements for a qudit.We begin with two orthonormal bases B a : “ (cid:32) | a i y ( di “ and B b : “ (cid:32) | b j y ( dj “ (1)of a d -dimensional Hilbert space H d to depict the twomeasurement settings a and b , respectively. In this paper,all (un)certainty relations are preparation (un)certaintyrelations that are applicable in the following experimentalscheme. N number of independent qudits are identi-cally prepared in a quantum state ρ . Thenhalf of them are measured in the basis B a andthe rest in B b , one by one. (2)A similar scenario Peres used in his book [39] at page 93to interpret the position-momentum UR. In proposal (2), clearly, the two measurements have no influence whatso-ever on each other.Throughout the text, we assume ρ is a pure quan-tum state | ψ yx ψ | so that we can associate angles (4)and TIs (12) with the state vector | ψ y . Although ev-ery (un)certainty relation presented in this paper as itis applicable for every qudit’s state [see the text around(39)].The state ρ “ | ψ yx ψ | provides two probability distri-butions for the two measurement settings [given in (1)]by the Born rule: p i “ |x a i | ψ y| and q j “ |x b j | ψ y| (3)are the probabilities of getting outcome a i in the a -settingand outcome b j in the b -setting, respectively. Next, wepresent quantum angles: α i “ arccos |x a i | ψ y| and β j “ arccos |x b j | ψ y| (4)are the angles between | ψ y and | a i y and between | ψ y and | b j y , respectively. In the entire article, we consideronly the principal values r , π s of the (multivalued) arccos function. With (3) and (4), one can recognize that theabsolute value of the inner product establishes a one-to-one correspondence between the angles—that belong to r , π s —and the probabilities—that lie in r , s .Related to the a -setting, every probability vector (cid:126)p : “ p p , ¨ ¨ ¨ , p d q satisfies ř di “ p i “ and (5) ď p i for all ď i ď d , (6)and the collection of all such vectors constitutes a prob-ability space Ω a . Similarly, Ω b is—related to the basis B b —defined be the constraints ř dj “ q j “ and (7) ď q j for all ď j ď d . (8)Equations (5) and (7) state that all the probabilities addup to one, and inequalities (6) and (8) tell that probabil-ities are nonnegative numbers. Both Ω a and Ω b are—thestandard p d ´ q -simplices—compact convex subsets ofthe d -dimensional real vector space R d , and their Carte-sian product Ω : “ Ω a ˆ Ω b is a compact convex subsetof R d [see Appendix B]. Basically, Ω is determined bythe conditions (5)–(8).Performing measurement on every qudit using a sin-gle setting, say a , looks like throwing a d -sided dice, ev-ery time. The vector (cid:126)p alone is limited by (5) and (6)that specify Ω a , which is also the probability space ofa d -sided dice. Whereas the experimental scheme (2) isnot similar to throwing one out of two d -sided dices ata time, although Ω is the probability space of two dices:every pure or mixed state of a qudit gives a unique pair p (cid:126)p, (cid:126)q q P Ω by the Born rule [see (3) and (39)], but not ev-ery pair p (cid:126)p, (cid:126)q q P Ω has a quantum state. For example, if |x a i | b j y| ‰ for some i, j , then one cannot get always thesame outcome: a i in the a -setting and b j in the b -setting.In other words, it is impossible to prepare [47] a quantumsystem in a state (in this case, there exists no quantumstate) that can provide p (cid:126)p, (cid:126)q q , where p i “ “ q j , whichidentifies an extreme point of Ω .So, other than (5)–(8), there are certain constraintsthat are purely quantum mechanical in nature and mustbe obeyed by (cid:126)p and (cid:126)q together . In our case, QCs are theTIs given in (12), which arise naturally from the structureof Hilbert space on which quantum theory is based. Towrite the TIs, we need r ij “ |x a i | b j y| ` ď i, j ď d ˘ (9)that is the probability of getting outcome a i if | b j yx b j | (or b j if | a i yx a i | ) is our state for the system. Like α i and β j in (4), θ ij “ arccos |x a i | b j y| (10)is the angle between the pure states | a i yx a i | and | b j yx b j | .In the subscripts of r ij and θ ij , from left, the first andsecond indices are reserved for B a and B b , respectively.Therefore, note that r ji “ |x a j | b i y| is different from r ij ,and likewise for θ .After choosing the measurement settings, B a and B b in (1), the entries in R : “ ¨˚˝ r ¨ ¨ ¨ r d ... . . . ... r d ¨ ¨ ¨ r dd ˛‹‚ and Θ : “ ¨˚˝ θ ¨ ¨ ¨ θ d ... . . . ... θ d ¨ ¨ ¨ θ dd ˛‹‚ (11)get fixed by (9) and (10). Each entry in R and in Θ belong to r , s and r , π s , respectively. Sum of all theentries in each row and every column of R is one, thusit is a doubly stochastic matrix. If the two measurementsettings described by (1) are physically the same, then R will be a permutation matrix. For every state vector | ψ y P H d , there are three TIs | θ ij ´ β j | ď α i ď θ ij ` β j (12)attached to each entry in Θ . These TIs [see (A20)] arederived in Appendix A.For simplicity, out the three TIs (12), here we chooseonly one θ ij ď α i ` β j for every ď i, j ď d . (13)Angles α i and β j vary, whereas θ ij is fixed, as we changethe state vector | ψ y . The kets that saturates TI (13) forcertain i, j lie in the linear span of t| a i y , | b j yu [consider(A14) and (A15) with ď β ď θ from Appendix A]. Inthe triangle equality (TE) θ ij “ α i ` β j , α i and β j arereminiscent of complementary angles from planar geom-etry, and ď α i , β j ď θ ij . Identifying f , D , and B in[13] by our | ψ y , | a yx a | , and | b yx b | , respectively, one cansee that the TI θ ď α ` β is obtained by Landau and Pol-lak for continuous-time signals (see also Sec. 8 in [14]).They also plotted elliptic curves (for different θ s) one of this kind is shown in Fig. 1 between the point E and E (see also [15]). The results in [13, 15] are more gen-eral than here, but they are only for a pair of projectors.Whereas, we take every possible pair | a i yx a i | and | b j yx b j | and present three TIs [see (12)], not just one, for eachpair.The cosine function is strictly decreasing on r , π s , soapplying it on both sides of TI (13) and using (3), (4),(9), and (10), we attain ? p i q j ď ? r ij ` a p ´ p i qp ´ q j q (14)after a rearrangement of terms. As both sides in (14) arenonnegative functions of the probabilities, squaring andfurther simplification lead to p i ` q j ď r ij ` ` b r ij p ´ p i qp ´ q j q (15)for every ď i, j ď d .All those pairs p (cid:126)p, (cid:126)q q P Ω that obey QC (15) for ev-ery ď i, j ď d build the combined-probability space ω for the two measurement bases in (1). In the case of d ą , even if we consider all TIs given in (12) for each ď i, j ď d , they do not capture the full QCs for a gen-eral pair of settings. Therefore, one can still find some p (cid:126)p, (cid:126)q q P ω that corresponds to no quantum state. Nev-ertheless, our analysis relies on the following fact: every p (cid:126)p, (cid:126)q q that does not belong to ω cannot be obtained froma quantum state, thus it is discarded. To investigate aspace ω q —that contains all those, and only those, pairs p (cid:126)p, (cid:126)q q that originate from the quantum states—is not theaim of this paper. However, it is not tough to realize that ω q “ ω for d “ ; in general, ω q Ď ω .Note that ω is a proper subset of Ω . To prove thisone can show: only one out of the two extreme points—specified by p i “ “ q j and p i “ “ q l , where j ‰ l —of Ω can belong to ω . Recall that if and only if r ij “ thenthe point described by p i “ “ q j belongs to ω , other-wise θ ij ď α i ` β j will be violated. Secondly, if r ij “ then r il “ , and θ il ď α i ` β l cannot be obeyed by theother point; hence that stays outside of ω .The space ω is—held by the conditions (5)–(8) and(15)—a compact and convex subset of R d [for a proof,see Appendix B]. Every point of such a set can be writ-ten as a convex combination of its extreme points dueto the Krein-Milman theorem (see Theorem . . andAppendix A.3 in [38]). We begin our journey from aninterior point of ω in Appendix D 1 and arrive at its ex-treme points at the end of Appendix D 3. There it isconcluded that the set of all extreme points of ω comesfrom a family of parametric curves.One can skip all those technical details and start con-structing the parametric curves straight from the conclu-sion (D56): the first step is to pick a set of m anglesfrom a single column or row of the matrix Θ given in(11). Such a set is called m -set, and ď m ď d ´ . Forinstance, we pick the top m angles t θ i u mi “ from the firstcolumn. Then we associate m TEs with the m -set as α i “ θ i ´ β for all i “ , ¨ ¨ ¨ , m (16)by taking β , where the subscript 1 reflects the selectedcolumn.Next, with (3) and (4), we assign m ` probabilitiesto the angles: p i “ cos α i and q “ cos β . They createthe probability vectors (cid:126)p p β q “ ` cos α , ¨ ¨ ¨ , cos α m , , p s , ˘ , (17) (cid:126)q p β q “ ` cos β , , q t , ˘ , where (18) p s “ ´ ř mi “ cos α i p m ` ď s ď d q , (19) q t “ ´ cos β p ď t ď d q , and (20) ” , ¨ ¨ ¨ , . (21)One can observe that ` (cid:126)p p β q , (cid:126)q p β q ˘ serves as a vector-valued function of a single real parameter β , thus it ex-hibits a parametric curve. Since the curve is associatedwith an m -set and all its points obey m TEs (16), we callit an m -parametric curve.A part of the curve, identified by the upper and lowerlimits β ď β ď β , lies in ω and represents its extremepoints because ` (cid:126)p p β q , (cid:126)q p β q ˘ cannot be written into a con-vex combination of other points of ω . In Appendix D 4,we realize that the two limits are fixed by p s p β q “ cos p θ s ´ β q when ď m ď d ´ , (22) β “ θ ´ θ t ` π when “ m , and (23) p s p β q “ when ă m ď d ´ (24)[see (D74)]. Equations (22) and (24) are like Eq. (D73),whose roots are stated in (D80). Always the root with+ sign delivers the correct limit [for justifications, see thelast paragraph in Appendix D 4].If one chooses an m -set from a row of Θ , say t θ j u mj “ ,then the m -parametric curve is constructed as β j “ θ j ´ α for all j “ , ¨ ¨ ¨ , m , (25) (cid:126)p p α q “ ` cos α , , p s , ˘ , (26) (cid:126)q p α q “ ` cos β , ¨ ¨ ¨ , cos β m , , q t , ˘ , (27) p s “ ´ cos α p ď s ď d q , and (28) q t “ ´ ř mj “ cos β j p m ` ď t ď d q . (29)Now the parameter is α P r α , α s , and the limits aredetermined by q t p α q “ cos p θ t ´ α q when ď m ď d ´ , (30) α “ θ ´ θ s ` π when “ m , and (31) q t p α q “ when ă m ď d ´ . (32)One can check that, for m “ , both (16)–(23) and (25)–(31) describe the same thing, provided s and t are identi-cal in both the cases. So an m -parametric curve is iden-tified by an m -set and the positions of p s and q t (that is, s and t ) in (cid:126)p and (cid:126)q , respectively.Let us count the total number of curves such as de-scribe by (16)–(20). One can harvest d ! m ! p d ´ m q ! distinct m -sets from a single column of Θ , and there are total d columns. The probability p s can take d ´ m separateplaces in (cid:126)p of (17) for distinct s , and q t can take d ´ separate places in (cid:126)q of (18) for distinct t . Thus we have p d ´ m qp d ´ q individual m -parametric curves with asingle m -set. Since ď m ď d ´ , we collect d ř d ´ m “ d ! m ! p d ´ m q ! p d ´ qp d ´ m q (33)number of curves, where each m -set is made of anglesfrom a column of Θ .We secure the same number if we consider rows, ratherthan columns, to build an m -set and then a curve suchas given by (25)–(29). For m “ , every m -set is a partof a row as well as a part of a column. So, to avoid dou-ble counting errors, we take the cases m “ and m ą separately. In total, there are d p d ´ q ` d ř d ´ m “ d ! m ! p d ´ m q ! p d ´ qp d ´ m q“ d p d ´ qr d ´ p d ` qs (34)number of parametric curves for a qudit.If one adopts a suitable concave function u p (cid:126)p, (cid:126)q q onthe combined space ω to estimate the uncertainty, thenits absolute minimum will occur only at the parametriccurves (see Theorem . . and Appendix A.3 in [38]). Soultimately one needs to find absolute minima of, at most, d p d ´ qr d ´ p d ` qs functions, each of a single variable[for example, see (42)]. Then the smallest minimum willbe the lower bound c ď u in an UR. This task can beeasily completed with a regular computer. In the nexttwo sections, we discuss certain concave as well as convexfunctions on ω . III. UNCERTAINTY MEASURES ANDRELATIONS If u quantifies the uncertainty—about the outcomes a i when a qudit is measured in the basis B a of (1)—then u should be a concave function of (cid:126)p P Ω a . It isbecause mixing probability distributions, (cid:126)p and (cid:126)p as λ (cid:126)p ` p ´ λ q (cid:126)p “ (cid:126)p with λ P r , s , can only increaseuncertainty λ u p (cid:126)p q ` p ´ λ q u p (cid:126)p q ď u p (cid:126)p q (see Chap-ter 9 in [39]). In this regard, every mixed state,say λ | ψ yx ψ | ` p ´ λ q| ψ yx ψ | “ ρ mix , has more uncer-tainty.So, here, we adopt a real-valued smooth concave func-tion u p (cid:126)p q : “ ř di “ ? p i (35)as an uncertainty measure. It is associated with theTsallis entropy [40] S { p (cid:126)p q “ K p u p (cid:126)p q ´ q , where K the Boltzmann constant. To prove u p (cid:126)p q is a concavefunction on Ω a , it is sufficient to demonstrate that the p d ´ q ˆ p d ´ q Hessian matrix—that is a symmetricmatrix of second-order partial derivatives of u —is anegative semidefinite matrix at every point in Ω a (seeTheorem . in [41]). At an interior point (where all p i ą ) of Ω a , the entry in the k th row and l th column p ď l, k ď d ´ q in the Hessian matrix is B u B p k B p l “ ´ ˜ p { l δ lk ` p { d ¸ “ B u B p l B p k , (36)where p d “ ´ ř d ´ i “ p i and δ lk is the Kronecker deltafunction. These entries indeed provide a negative definitematrix, thus u p (cid:126)p q is strictly concave in the interior of Ω a .At a boundary point (where one or more p i “ ), all thepartial derivatives in a certain row(s) and column(s) ofthe Hessian matrix become zero, thus the matrix turnsout to be a negative semidefinite and u p (cid:126)p q to be a con-cave function. By the way, u p (cid:126)p q can be employed for theentanglement detection (see Remark 2 in [42]).If the state vector | ψ y is an equal superposition of allthe kets in B a or the state is completely mixed, then allthe outcomes a i will be equally probable: p i “ d for ev-ery ď i ď d is the center of Ω a , where u p (cid:126)p q reaches itsmaximum value ? d . Whereas, only in the case of a def-inite outcome—that is when | ψ yx ψ | “ | a i yx a i | , and then p i “ for a particular i —we have the minimum uncer-tainty u p (cid:126)p q “ as it should be. Note that p i “ char-acterizes an extreme point of Ω a .To establish a measure of combined uncertainty for theexperimental proposal (2), we take the same function, u p (cid:126)q q “ ř dj “ ? q j , (37)for the b -setting. Like u p (cid:126)p q of (35), u p (cid:126)q q is a concavefunction on Ω b with the range r , ? d s . Now we defineour combined uncertainty measure u p (cid:126)p, (cid:126)q q : “ u p (cid:126)p q ` u p (cid:126)q q “ ř dl “ ` ? p l ` ? q l ˘ (38)on the convex set ω , rather than Ω . Sum of two concavefunctions is concave, so u is also a concave function.A mixed quantum state is a convex combination ofpure states, the probabilities p i “ tr ppp (cid:37) | a i yx a i |qqq and q j “ tr ppp (cid:37) | b j yx b j |qqq (39)are linear functions of the state (cid:37) ( ď (cid:37) , tr p (cid:37) q “ ), and ω is a compact and convex set. As a result, every p (cid:126)p, (cid:126)q q associated with any (pure or mixed) quantum state liesin ω . And, because u is a concave function on ω , our URgiven in (40) applies to every state for a qudit. This isalso true in the case of other (un)certainty relations pre-sented in Sec. IV, because mostly there also we have ei-ther a concave or a convex function. In (93) and (94), thefunctions are neither concave nor convex on ω , but therelations are followed by every qubit’s state. By the way,one can check that if (cid:37) “ | ψ yx ψ | then the Born rule (39)reduces to (3).The range of u p (cid:126)p, (cid:126)q q and our UR are presented as ď c ď u p (cid:126)p, (cid:126)q q ď ? d , where (40) c : “ min p (cid:126)p,(cid:126)q q P ω u p (cid:126)p, (cid:126)q q (41) is the global minimum that will occur at the m -parametric curves [given in Sec. II]. Whereas, u gainsits absolute maximum ? d only at the point identifiedby p i “ d “ q j for all ď i, j ď d . It is called the center of ω , which represents the uniform distribution for boththe settings. Now recall from Sec. II that an extremepoint of Ω , describe by p i “ “ q j , belongs to ω if andonly if | a i yx a i | “ | b j yx b j | . Only in such a situation—thatdoes not necessarily require both the bases B a and B b to be the same in any way—we have the trivial lowerbound c “ and thus the UR ď u . A similar state-ment is made by Deutsch in [6]. For d “ , the trivialcase is possible if and only if the two measurement set-tings are (physically) the same. A nontrivial lower bound c ą materializes when the settings are completely dif-ferent, that is when r ij ă for every ď i, j ď d . So thefollowing analysis is obviously for the nontrivial cases.To find the lower bound (41) and to establish the UR c ď u , we write the functional form u p β q “ ř mi “ cos α i ` ? p s ` cos β ` sin β , (42)which u p (cid:126)p, (cid:126)q q of (38) acquires on an m -parametric curvespecified by (16)–(21). To show that u of (42) is a concavefunction of β , we present B u B β “ ´ r ř mi “ cos α i ` cos β ` sin β s ` B ? p s B β , (43) B ? p s B β “ ´ p { s ˆ B p s B β ˙ ` ? p s B p s B β , and (44) B p s B β “ ´ r p s ` p m ´ q s . (45)With these derivatives, one can clearly see B u B β ă for ă m ď d ´ . Whereas, for m “ , one can directlyrealize B u B β “ ´ u ă . This proves that u is a (strictly)concave function on every parametric curve. Therefore,its global minimum c will always be at the endpointsof the curves. Endpoints of an m -parametric curve areidentified by the two limits on a parameter [see (22)–(24)as well as (30)–(32)].It is manifested in Appendix D 4 that, to compute alimit, we always have to solve an equation such as (D73);which carries m number of angles from a column or arow of Θ [given in (11)]. Note that we use small let-ter ‘ m ’ p ď m ď d ´ q when we construct a parametriccurve with an m -set [see Sec. II] and use capital letter‘ m ’ p ď m ď d q when we compute a limit with an m -set.Essentially, one needs to follow a three-step procedure tocompute a limit and then the value of u [defined in (38),see also (42)] at the corresponding endpoint of a curve:1. Pick an m -set from a column or a row of Θ ,say t θ , ¨ ¨ ¨ , θ m u , here only one index of θ isshown.2. Solve ř m l “ cos p θ l ´ χ q “ for χ that repre-sents a limit.3. Compute c m : “ ř m l “ cos p θ l ´ χ q ` cos χ ` sin χ that is the value of u at an endpoint. (46)The equation in Step 2 is like Eq. (D73) that is solved inAppendix D 4, and every time we take the solution (D80)with + sign. One can observe that χ and therefore c m are solely determined by the m -set picked in Step 1.After repeating the three-step procedure for every m -set and for each ď m ď d , we collect a set of values t c m u for all the endpoints. Then, the smallest value in this setwill be c [defined by (41)], and thus we own our UR c ď u [presented in (40)]. Since every c m is determined by theentries in Θ -matrix, the lower bound c —depends onlyon the measurement bases in (1)—is independent of aquantum state. Besides, to compute c , we can employ anordinary computer, which repeats the three steps of (46)by taking d ř d m “ d ! m ! p d ´ m q ! “ d r d ´ p d ` qs (47)number of m -sets one by one. In fact, d r d ´ p d ` qs is the total number of endpoints for a qudit.Although we have the solution (D80) for Step 2, itis easy to calculate χ and c m for m “ , d . For a 2-set t θ , θ u , one can directly realize χ “ θ ` θ ´ π , and then (48) c p θ , θ q “ ? “ cos ` θ ´ θ ˘ ` sin ` θ ` θ ˘‰ (49) “ ? ` ? `? r ` ? ´? r ˘` ? `? r ` ? ´? r ˘ . (50)Every endpoint of a m “ parametric curve is deter-mined by a set of m “ angles [see (22), (23), (30), and(31)]. For a d -set t θ , ¨ ¨ ¨ , θ d u , that is an entire column orrow of Θ , we have the total probability ř dl “ cos θ l “ .Therefore, we obtain the solution χ “ , and then (51) c d p θ , ¨ ¨ ¨ , θ d q “ ř dl “ cos θ l ` “ ř dl “ ? r l ` . (52)For general measurement settings, it is—easy to com-pute but—difficult to express c in an analytic form. Nev-ertheless, we present it for d “ , , and when the mea-surement bases in (1) are MUBs [18].In the case of a qubit, d “ , a (un)certainty relationcan be stated with the three probabilities p , q , and r , hence we drop their subscripts here and in the nextsection. Furthermore, all the TIs (13) can now be puttogether as θ ď α ` β ď π ´ θ and | α ´ β | ď θ , (53)where α , β , and θ are associated with p , q , and r , re-spectively [through (3), (4), (9), and (10)]. Here only m “ parametric curves exist, which are four in total[see with (34)]. To draw an endpoint of a curve, wecan use either (48) or (51); both are equal (because θ ` θ “ π ). There are only four [see (47)] endpoints E , ¨ ¨ ¨ , E . Next, one can realize that (49) and (52) arealso the same for a qubit. Furthermore, c d is even iden-tical for every m “ set. It implies that our combined pq E ‚ ‚ E E ‚ E ‚ ‹ FIG. 1. For d “ and r “ , contour plot of u p p, q q on ω ,where a darker shade represents a smaller value of u . Thesquare-shaped and elliptical regions are Ω and ω , respec-tively. Note that ω Ă Ω Ă R and the unseen coordinates are p “ ´ p and q “ ´ q for each point. For every r P r , s , u hits its global minimum c [given in (54)] on ω at all the fourpoints E , ¨ ¨ ¨ , E , which are marked by the bullets p‚q . And, u achieves its global maximum ? [stated in (40)] always atthe center, p “ “ q indicated by the star p‹q , of ω . uncertainty function (38) takes the same value at all thefour endpoints, thus c “ c d “ c and ? r ` ? ´ r ` looooooooomooooooooon c p r q ď ? p ` a ´ p ` ? q ` a ´ q loooooooooooooooooomoooooooooooooooooon u p p, q q (54)is an UR for d “ . It is also given in [24].Together all the parametric curves—that represent allthe extreme points of the combined-probability space ω —can be expressed by an ellipse p p p ϑ q , q p ϑ q q “ ` cos p θ ´ ϑ q , cos ϑ ˘ with ϑ P r , π q (55)in the case of a qubit. As a special case, the same ellipsealso appears in [15–17] through different routes [48], al-though our approach is closer to [15]. One can observethat the ellipse turns into a circle for θ “ π and into cer-tain line segments for θ “ , π . In Fig. 1, we present acontour plot of u p p, q q on ω by taking r “ . So θ “ π ,and one can see that ω is bounded by the ellipse (55).Furthermore, by putting ϑ “ , θ, π , π ` θ in p p p ϑ q , q p ϑ q q ,we can have the four endpoints E , ¨ ¨ ¨ , E , respectively.In the case of d “ , there always exist a quantum statefor each point in ω , thus ω “ ω q . For instance, the ketssuch as (A14) and (A15) correspond to points on theellipse (55) by the Born rule (3). In particular, the ketsof basis B a correspond to the points t E , E u , and thekets of B b are related with t E , E u . So the lower bound c p r q in the UR (54) is achieved—hence, it is a tight UR—only by those state vectors | ψ y that (up to a phase factor)belong to one of the bases in (1). The lower bound will bethe largest ? ` when, r “ , the measurement basesare MUBs [see also (58)].An UR is called tight if there exists a quantum statethat saturates the UR. In the case of a qubit, all therelations mentioned in this and the next section are tightbecause ω “ ω q . For d ě , ω q Ď ω , hence our UR c ď u is not tight in general.In the case of d “ (qutrit), there are only two kindsof parametric curves (for m “ , ), and two types of end-points (for m “ , ). So (48) and (51) can specify anyendpoint for a qutrit. To compute the lower bound c ,we have to evaluate the function c of (49) for every 2-set and c d of (52) every d -set drawn from the Θ -matrix.For d “ , there are 18 2-sets and 6 d -sets [see the totalin (47)]. Then, the smallest out of the ` “ valueswill be our c . Now let us consider a pair of MUBs [18]for a finite dimension d .If the two bases given in (1) are such that r ij “ d forevery ď i, j ď d [for r ij , see (9)], then they are calledMUBs and the measurement settings a and b are des-ignated as complementary [7]. In the case of MUBs, θ ij “ arccos ? d for every i, j , so one can straightforwardrealize χ “ arccos ? d ´ arccos ? m , and (56) c m “ ? m ` ` ? p d ´ qp m ´ q`? d ´ ´? m ´ ? d m (57)in Step 2 and 3 of the three-step procedure (46). One canacknowledge that here χ and c m depend on m “ , ¨ ¨ ¨ , d ,not on a particular m -set, because every θ is the same.Furthermore, χ decreases, whereas c m increases, with m .Hence the lower bound is c p d q mub “ c “ ? ´ ` ? d ´ ? d ¯ , (58)which does not deliver a tight UR when d ą , whereastight URs [7, 8, 21] are known for MUBs in a finite d .We close this section with the following remarks. Remark 1:
By the Born rule (3), | ψ y “ | a i y providesan extreme point, given by p i “ and (cid:126)q “ p r i , ¨ ¨ ¨ , r id q ,of ω [see (D33) and (D32) in Appendix D 3]. At thispoint the combined uncertainty function (38) has thevalue ` ř dj “ ? r ij [see also (52)]. Likewise, | ψ y “ | b j y gives the combined uncertainty ` ř di “ ? r ij . Now wetake the minimum value c bases : “ min t u a , u b u , where (59) u a : “ min ď i ď d (cid:32) ` ř dj “ ? r ij ( and (60) u b : “ min ď j ď d (cid:32) ` ř di “ ? r ij ( . (61)Next, one can easily establish ď c ď c q ď c bases ď ` ? d , where (62) c q : “ min | ψ y P H d u p (cid:126)p, (cid:126)q q . (63) The first inequality in (62) comes from (40). The lastinequality is due to ř di “ ? r ij ď ? d and the similar re-lation where the summation is over index j instead of i . c q is the largest lower bound that defines the tight UR c q ď u p (cid:126)p, (cid:126)q q . For d “ , our lower bound c “ c q “ c bases ,and the UR (54) is tight. Whereas, if the two basesin (1) share a ket then c turns out to be the trivialbound: “ c “ c q “ c bases . One can use (62) to avoiderrors while calculating c . Remark 2:
The function H { p (cid:126)p q “ u p (cid:126)p q is theRényi entropy [44] of order . Using (36), one can realizethat H { p (cid:126)p q is a concave function on Ω a , hence the sum H { p (cid:126)p q ` H { p (cid:126)q q “ “ u p (cid:126)p q u p (cid:126)q q ‰ (64)is concave on ω . Taking (43)–(45), one can confirm thatthe sum is also concave on each of the parametric curves,therefore its absolute minimum will be on the endpoints.By repeating the three-step procedure (46)—where in thethird step now we need to compute h m : “ “ ppp ř m l “ cos p θ l ´ χ qqqqppp cos χ ` sin χ qqq ‰ (65)instead of c m —for every m -set, we can own an UR basedon the combined entropy (64) for any pair of measure-ment settings. Analogues to (49), (52), and (57), here wehave h p θ , θ q “ “ ` θ ´ θ ˘ sin ` θ ` θ ˘‰ (66) “ “ ? ´ r ` ? ´ r ‰ ,h d p θ , ¨ ¨ ¨ , θ d q “ ř dl “ cos θ l “ ř dl “ ? r l , (67)and h m “ „ ` ? p d ´ qp m ´ q`? d ´ ´? m ´ ? d , (68)respectively, with these one can directly get URs forqubit, qutrit, and for a pair of MUBs just like above.For a qubit, we express the corresponding tight UR (alsoobtained in [24]) ? r ` ? ´ r ď ` ? p ` a ´ p ˘` ? q ` a ´ q ˘ (69)in terms of the product u p p q u p q q . In this case, the prod-uct turns out not only a concave function on ω but alsoon each of the four parametric curves. And, its abso-lute minimum—given in left-hand side of (69)—occursat all the four endpoints E , ¨ ¨ ¨ , E , and the absolutemaximum at the center [denoted by ‹ in Fig. 1] of ω . IV. OTHER (UN)CERTAINTY MEASURESAND RELATIONS
The negative of a concave function is a convex func-tion, hence a suitable convex function can be taken as ameasure of certainty, rather than uncertainty. Here wepresent other popular measures of (un)certainty and ob-tain the associated (un)certainty relations for d “ byfinding the absolute minimum (for concave) and maxi-mum (for convex) on the ellipse (55). We want to empha-size that all the relations given in this paper for a qubitare already known, thanks to [6, 8, 16, 24–31], throughdifferent methods. The following analysis merely showsthat they all can be obtained from the TIs (53) that char-acterize the ellipse. Recall that one can have the sameellipse from [15–17].One can always construct Hermitian operators, for ex-ample A “ ř di “ a i | a i yx a i | and B “ ř dj “ b j | b j yx b j | , (70)by assigning real numbers to the measurement outcomes a i and b j for the two settings specified by (1). Then a : “ t a i u di “ and b : “ t b j u dj “ are the sets of eigenvaluesof A and B , respectively. With (3) and (70), one canperceive that the squared standard deviations ∆ p A, ρ q “ x ψ | A | ψ y ´ x ψ | A | ψ y “ ř di “ a i p i ´ ` ř di “ a i p i ˘ “ ∆ p a , (cid:126)p q , (71) ∆ p B, ρ q “ ř dj “ b j q j ´ ` ř dj “ b j q j ˘ “ ∆ p b , (cid:126)q q (72)are functions of the probabilities as well as the eigenval-ues.Taking p d “ ´ ř d ´ i “ p i , like the derivatives (36) of u p (cid:126)p q , we get the second-order partial derivatives B ∆ B p k B p l “ ´ p a k ´ a d qp a l ´ a d q “ B ∆ B p l B p k (73)of the function (71) for ď k, l ď d ´ . One can vali-date that the Hessian matrix—made of the derivatives(73)—is a negative semidefinite matrix for any set a ofeigenvalues. Thus, ∆ p a , (cid:126)p q is a concave function on Ω a (see Theorem . in [41]). Likewise, ∆ p b , ρ q is a concavefunction on Ω b . Hence, analogues to u p (cid:126)p, (cid:126)q q of (38), thesum ∆ sq p a , (cid:126)p, b , (cid:126)q q : “ ∆ p a , (cid:126)p q ` ∆ p b , (cid:126)q q (74)establishes a concave, thus uncertainty, measure on thecombined space ω . In [32], URs are presented by taking asum such as (74), however, here the approach is different.In the case of a qubit ( d “ ), every measurement set-ting can also be described by a three-component realvector. So, we designate the two settings [see (1)] bycertain unit vectors p a and p b and then construct the Her-mitian operators A “ p a ¨ (cid:126)σ and B “ p b ¨ (cid:126)σ with the dotproduct, where (cid:126)σ is the Pauli vector operator. One canverify that A “ I “ B , therefore the eigenvalues are: a “ t˘ u “ b . Suppose the kets | a y and | b y of the twobases [in (1)] are associated with the eigenvalue ` of A and B , respectively. Now one can easily derive therelation tr p A † B q “ |x a | b y| ´ “ p a ¨ p b (75)between the three kinds of inner products. From Sec. III,let us recall that we only require three probabilities p , q , and r to express a (un)certainty relation for d “ .So, there is no further need for the subscripts. With allthe above considerations, ∆ sq of (74) turns out to be thefunction ∆ sq ppp˘ , p, ˘ , q qqq “ ´ p p ´ q ` ´ p q ´ q (76)of p and q . pq ‚ F F ‚ F ‚ F ‚ ‹ FIG. 2. A contour plot of ∆ sq p p, q q of (76) on ω , where adarker shade illustrates a smaller value of ∆ sq . Here r “ ,therefore ∆ sq reaches its global minimum r [see the UR (77)and (78)] at the two points F and F . Whereas, ∆ sq gains itsglobal maximum 2 always at the center, p “ “ q denotedby the star p‹q , of ω . Like Fig. 1, ω is the region bounded bythe ellipse (55); while θ “ π here. We plot ∆ sq of (76) on ω in Fig. 2 by taking r “ .Since ∆ sq is a concave function on ω , its absolute min-imum will be at the four parametric curves, which arejointly described by the ellipse (55) and by their end-points E , ¨ ¨ ¨ , E . To compute the minimum, first, weneed to represent ∆ sq as a function of a parameter,like u in (42), on each curve. Then, we have to findthe critical points of ∆ sq . Here we obtain four criti-cal points F , ¨ ¨ ¨ , F —one on each curve—that are de-picted by the bullets p‚q in Fig. 2. By putting ϑ “ θ , θ ` π , θ ` π , θ ` π in p p p ϑ q , q p ϑ q q of (55), one canhave F , ¨ ¨ ¨ , F , in that order. Record that the F -pointsare not the endpoints E , ¨ ¨ ¨ , E that are only shown inFig. 1, not in Fig. 2.The function ∆ sq of (76) takes the value r at both thepoints t F , F u and takes the value p ´ r q at t F , F u .So the global minimum is min (cid:32) r , p ´ r q ( ď ∆ sq ppp˘ , p, ˘ , q qqq , (77)and thus we obtain a tight UR, like (54). One can confirmthat the lower bound is r if r ď p at F , F in Fig. 2 q p ´ r q if r ě p at F , F in Fig. 2 q . (78) Remark 3:
The standard deviation ∆ ppp˘ , p qqq is a con-cave function of p , hence the sum ∆ ppp˘ , p qqq ` ∆ ppp˘ , q qqq is a concave function on ω . As a result, we have anothertight uncertainty relation a ´ p r ´ q ď ∆ ppp˘ , p qqq ` ∆ ppp˘ , q qqq . (79)One can check that the sum reaches its absolute mini-mum value at all the endpoints E , ¨ ¨ ¨ , E , and has itsmaximum value at the center of ω . Both the tight URs(77) and (79) are known due to [25]. A quantum statethat saturates a tight UR is called its minimum uncer-tainty state . Since the E -points and the F -points are notthe same, in general, the set—of minimum uncertaintystates—is different for the two URs (77) and (79) basedon the standard deviation. Note that we always get thetrivial lower bound ď ∆ p a , (cid:126)p q ∆ p b , (cid:126)q q for the productof standard deviations, and this bound can be reachedby any ket belongs to either of the bases given in (1).Next, the Shannon entropy [43] H p (cid:126)p q “ ´ ř di “ p i log p i (80)is arguably the most famous measure of uncertaintyat present. It is superior than the standard deviation ∆ p a , (cid:126)p q [10, 11] because it only depends on (cid:126)p , not on theeigenvalues. One can show that H p (cid:126)p q P r , log d s , andit is a concave function on Ω a with the Hassian matrixcomposed of the second-order derivatives B H B p k B p l “ ´ ˆ p l δ lk ` p d ˙ “ B H B p l B p k , (81)where p d “ ´ ř d ´ i “ p i . Considering the same functionfor the b -setting, that is H p (cid:126)q q , one can formulate a com-bined uncertainty measure by the sum H p (cid:126)p q ` H p (cid:126)q q andthen produce an entropy UR [6–8]. Such URs are re-viewed in [9–11]. For d “ , the tight entropy UR isachieved in [26, 28] (see also [27]), and we can directlyimport all their results here. In fact, Eq. (7) in [26] andEq. (2.4) in [28] are H p p q ` H p q q on the ellipse (55), andthey found the absolute minimum of H p p q ` H p q q on theellipse. In [28], all the results are given in terms of anglesbetween the real unit vectors, which are related to theangles between kets through (75).We can choose u γ p (cid:126)p q “ ř di “ p p i q γ with ă γ ă 8 (82)as another (un)certainty measure, which is closely relatedto the Tsallis [40] and Rényi [44] entropies of order γ . Onecan prove that the Hassian matrix with entries B u γ B p k B p l “ γ p γ ´ q “ p lγ ´ δ lk ` p dγ ´ ‰ “ B u γ B p l B p k , (83) ď k, l ď d ´ , is a negative and positive semidefinitematrix for ă γ ď and ď γ ă 8 , respectively. It con-firms that u γ p (cid:126)p q is a concave (uncertainty) and convex (certainty) measure when ă γ ď and ď γ ă 8 , re-spectively. A similar observation is made in [24, 33]. Infact, our uncertainty measure u p (cid:126)p q of (35) is u γ p (cid:126)p q withthe exponent γ “ . Furthermore, the range of u γ p (cid:126)p q is r , d ´ γ s if γ ď and is r d ´ γ , s if ď γ . When γ “ , u γ p (cid:126)p q “ for every (cid:126)p P Ω a due to Eq. (5), thus u is nota genuine (un)certainty measure.Like before, one can establish a (un)certainty relationwith the sum u γ p (cid:126)p q ` u γ p (cid:126)q q . For γ “ , in the case of d “ , we obtain u p p q ` u p q q “ ´ ∆ sq ppp˘ , p, ˘ , q qqq , and then (84) u p p q ` u p q q ď ´ min t r , ´ r u looooooooooomooooooooooon max t ´ r, ` r u (85)as a tight certainty relation; which is also given in [16]for ď r . Due to (84), one can immediately derive (85)from the UR (77). Where ∆ sq of (76) reaches its ab-solute minimum (uncertainty) on ω , there the function(84) achieves its global maximum (certainty) max t ´ r, ` r u “ ´ r if r ď p at F , F in Fig. 2 q ` r if r ě p at F , F in Fig. 2 q . (86)The certainty measure (84) hits its absolute minimum 1at the center of ω [depicted by the star p‹q in Figs. 1and 2]. Remark 4:
One can have another tight certainty re-lation u p p q u p q q ď max (cid:32) p ´ r q , p ` r q ( , (87)where product of certainty measures is used. The rela-tion (87) is presented in [16] for ď r . One can verifythat u p p q u p q q is a convex functions on ω . Therefore,its absolute maximum [given in (87)] will be on the ellipse[specified by (55)], and the global minimum will be atthe center of ω . The product-function reaches its upperbound on the F -points. By applying the negative of thelogarithm on both sides of the inequality (87), we getthe corresponding tight UR—achieved in [29]—in termsof the collision entropy (that is, the Rényi entropy [44] oforder ).Lastly, we pick the function u max p (cid:126)p q “ max ď i ď d t p i u (88)that defines a norm on R d if we replace p i with | p i | .Since every p i follows (6), the modulus sign is notshown in (88). Every norm is a convex function, so u max can be considered as a certainty measure on Ω a ; u max p (cid:126)p q P “ d , ‰ for every (cid:126)p P Ω a . Note that u max p (cid:126)p q is not differentiable everywhere in Ω a . Nevertheless, wecan assemble a combined certainty measure with the sum u max p (cid:126)p q ` u max p (cid:126)q q on ω .In the case of d “ , the function u max p p q ` u max p q q is0equal to $’’’&’’’% p ´ p q ` p ´ q q if ď p ď and ď q ď p ´ p q ` q if ď p ď and ď q ď p ` p ´ q q if ď p ď and ď q ď p ` q if ď p ď and ď q ď . (89)The limits on p, q stated in (89) divide ω —that is an el-liptical region [see Figs. 1 and 2]—into four quadrants.The function u max p p q ` u max p q q is differentiable in eachof the quadrants. Furthermore, since it is a convex func-tion on ω , its global maximum will be at the ellipse (55).Here we discover four critical points, one in each quad-rant on the ellipse, where the combined function takes amaximum value. In fact, these four points are the same F , ¨ ¨ ¨ , F exhibited in Fig. 2.The combined measure acquires the value ` ? ´ r at both F , F and reaches the value `? r at both F , F .Thus, like (85), we get the tight certainty relation u max p p q ` u max p q q ď max (cid:32) ` ? ´ r , ` ? r ( , (90)for a qubit. And, the absolute maximum (upper bound)is given by ` ? ´ r if r ď p at F , F in Fig. 2 q ` ? r if r ě p at F , F in Fig. 2 q (91)analogues to (86). Besides, u max p p q ` u max p q q has itsglobal minimum 1 at the center of ω [exhibited by thestar p‹q in Figs. 1 and 2].The certainty relation (90) is captured in [30] using theinequality arccos p max ij ? r ij q ď arccos p max i ? p i q ` arccos p max j ? q j q . (92) Instead of TIs (53), for a qubit, all the tight re-lation (54), (69), (77), (79), (85), (87), (90), (93),(94), and the entropy UR given in [26–28] can be ob-tained with (92). In fact, inequality (92), that is min ij θ ij ď min i α i ` min j β j , can be produced from d TIs (13), and it is weaker than the TIs: all those p (cid:126)p, (cid:126)q q P Ω that are bounded by (92) rather than (13) con-stitute a bigger combined-probability space. Remark 5:
One can confirm that the product u max p p q u max p q q is neither a concave nor a convex func-tion on ω (for a similar observation, see [8]), so it notclear to us whether or not we can take it as a goodcombined-(un)certainty measure for every qubit’s state.It also shows that product of two convex (concave) func-tions is not necessarily a convex (concave) function. Bycomputing the gradient of u max p p q u max p q q in each of thefour quadrants, one can realize: the function reaches itsglobal minimum at the center of ω and reaches itsglobal maximum (on the ellipse) at the F -points. Hence,we have the tight relation u max p p q u max p q q ď max (cid:32) p `? ´ r q , p `? r q ( , (93) which is reported in [8] (and implicitly appear in [6]). Infact, for d “ , the ket given by Eq. (11) in [6] is the ket(A14) with β “ θ and ν “ , and the ket corresponds tothe point F . By applying the negative of the logarithmon both sides of the inequality (93), one can turn thisrelation in the min-entropy terms [23]. The min-entropy H min p q q : “ ´ log ppp u max p q qqqq is the smallest in the family ofRényi entropies [44], and it is neither concave nor convexfunction on the interval r , s . Like above, using the min-entropy, one can have another tight relation ´ log ppp max t r , ´ r uqqq ď H { p p q ` H min p q q , (94)that is also given in [8], recall that H { p p q “ ppp u p p qqqq .The function H { p p q ` H min p q q always takes its globalminimum at the endpoints E and E and takes its ab-solute maximum at the center [shown in Fig. 1] of ω . In [31], a general expression for the tight lower boundof a sum of Rényi entropies is given, which is basicallythe minimization of the sum on the ellipse. V. CONCLUSION AND OUTLOOK
Taking a pure quantum state for a qudit, we presentTIs (13) and then the combined-probability space ω fora general pair of measurement settings. The combinedspace is a compact and convex set in R d , and all its ex-treme points are represented by the m -parametric curves, ď m ď d ´ . These curves are determined by the twosettings ( Θ -matrix) and are sufficient to generate thewhole ω as well as to provide a (un)certainty relation.One can pick some suitable concave and convex func-tions on ω to quantify the uncertainty and certainty,respectively. Subsequently, one can establish an un-certainty (a certainty) relation by finding the absoluteminimum (maximum) of a function at the parametriccurves. Due to the parametric curves, formulation of a(un)certainty relation become a single-parameter opti-mization problem.Particularly for the uncertainty measures (38) and(64), the absolute minima can always be easily computedby repeating the three-step procedure given in Sec. IIIwith every m -set, ď m ď d , built with entries in the Θ -matrix. And, thus, one can enjoy the corresponding URsfor any pair of measurement settings. For the other func-tions, one needs to find all the critical points on the curvesfirst and then the absolute extremum at those points.That is, still, much easier than searching the extremumon the whole space. In each case, the extremum—thatis a lower (upper) bound on an uncertainty (certainty)measure—only depends on the measurement settings, noton a quantum state. Every (pure or mixed) state of a qu-dit provides a point in ω by the Born rule and respectsevery (un)certainty relation presented in this write-up.In the case of a qubit, d “ , we show that many knowntight (un)certainty relations, owing to [6, 8, 16, 24–31],can be derived from the TIs (53). These TIs define an el-lipse that represents all the parametric curves, and each1point on the ellipse (and in ω ) corresponds to a qubit’sstate, thus we have tight relations. The same ellipse alsoemerges in [15–17] as a special case. For a pair of mea-surement setting on a qubit, it seems that the TIs (13)and the results in [13, 15–17] provide more fundamentalQCs than the tight (un)certainty relations.TIs (13) do not provide all possible QCs when thedimension d ą , hence there are still some points in ω that correspond to no quantum state, and our URsgiven in Sec. III are not tight in general. However,all our (un)certainty relations are built on the factthat ‘every point outside of ω is, surely, not associatedwith any quantum state’. One can include other QCs,namely TIs (12), then the domain ω of a (un)certaintyfunction will be smaller. Consequently, better boundsand finer (un)certainty relations can be achieved. Toget a tight bound, in the case of general settings and d ą , is a challenging task. Tight URs are only knownin some special cases: position-momentum [3], MUBs[7, 8, 16, 20, 21, 23], and a qubit [6, 8, 16, 24–31].URs have numerous applications in different strands ofphysics. Recently, these are employed for certain quan-tum information processing tasks such as the cryptog-raphy [23] and the entanglement detection [30, 34–37].As our (un)certainty relations arise solely from TIs, onecan directly appoint TIs (12) as genuine QCs for such ajob. Furthermore, in quantum state estimation [45], onecollects data by applying different measurement settings,thus realizes scheme (2) in a laboratory. Then, ρ est isconstructed with the data. There one needs to confirmthat the estimated ρ est represents a legitimate quantumstate. Again TIs (12) could be utilized for such a test,for instance, one can firstly check whether the estimated p (cid:126)p est , (cid:126)q est q follows all the TIs or not. ACKNOWLEDGMENTS
I am very grateful to Arvind for stimulating discussionsand helpful comments on the manuscript. I thank ArunKumar Pati for bringing Ref. [13] to my attention andJędrzej Kaniewski for explaining and making me awareabout their work [17].
Appendix A: Derivation of the triangle inequalities
Landau and Pollak obtained a single TI of the kindgiven in (13) for continuous-time signals. One can spotseveral similarities between their work [13] and the fol-lowing derivation. In this paper, the primary QCs are theTIs (12). To derive such TIs, we consider three kets | ψ y , | a y , and | b y of a d -dimensional Hilbert space H d . Theirinner products are expressed in the polar form as x a | ψ y : “ ? p e i µ “ cos α e i µ , (A1) x b | ψ y : “ ? q e i ν “ cos β e i ν , and (A2) x a | b y : “ ? r e i δ “ cos θ e i δ , (A3) where the phases µ, ν, δ P r , π q . In the main text, | ψ y is associated with a quantum state, and | a y and | b y arewith the two measurement settings [see (1)]. Throughthe inner products, the quantum angles α , β , and θ arerelated with the probabilities p , q , and r [see also (3),(4), (9), and (10)], and i “ ?´ . Recall that the angleslie in r , π s , and the probabilities belong to the interval r , s .It is always feasible to write one ket, say | ψ y , as a sumof its component in the linear span of other two t| a y , | b yu and its component in the orthogonal complement of thespan [see (A6)]. In general, | a y and | b y are not orthogo-nal to each other. In the case of ă |x a | b y| ă , employ-ing the Gram-Schmidt orthogonalization process, one canconvert the linearly independent set t| a y , | b yu into an or-thonormal set t| b y , | b K yu or t| a y , | a K yu , where | b K y “ | a y ´ x b | a y| b y a ´ |x a | b y| and | a K y “ | b y ´ x a | b y| a y a ´ |x a | b y| . (A4)The two sets are related by a unitary transformation: ˆ | b y| b K y ˙ “ ˆ x a | b y a ´ |x a | b y| a ´ |x a | b y| ´x b | a y ˙ ˆ | a y| a K y ˙ . (A5)Now we can resolve | ψ y “ cos β e i ν | b y ` x b K | ψ y| b K y ` x x | ψ y| x y (A6)with a suitable ket | x y that follows x b | x y “ “ x b K | x y .If and only if | ψ y lies in the span of t| a y , | b yu , thelast term in the expansion (A6) vanishes, otherwisenot. With the normalization of | ψ y , one can recognize |x b K | ψ y| ` |x x | ψ y| “ sin β , and subsequently ď |x x | ψ y| ñ |x b K | ψ y| ď sin β . (A7)Taking the transformation (A5) and the polar form(A3), we realize another representation of the ket | ψ y “ ` cos θ cos β e i p ν ` δ q ` sin θ x b K | ψ y ˘ | a y ` ` sin θ cos β e i ν ´ cos θe ´ i δ x b K | ψ y ˘ | a K y `x x | ψ y| x y (A8)from (A6). With the new representation (A8) and thepolar form x b K | ψ y : “ |x b K | ψ y| e i ξ , ξ P r , π q , (A9)we attain p “ |x a | ψ y| “ cos θ cos β ` sin θ |x b K | ψ y| ` θ sin θ cos β |x b K | ψ y| cos p ξ ´ p ν ` δ qq . (A10)Remember that x a | x y “ “ x a K | x y because | x y lies in theorthogonal complement of t| a y , | b yu . Owing to cos p ξ ´ p ν ` δ qq ď , (A11)2first, we obtain the left-hand side inequality in p ď ` cos θ cos β ` sin θ |x b K | ψ y| ˘ ď cos p θ ´ β q , (A12)and afterwards the right-hand side inequality with theaid of (A7). Eventually, from above, we have p “ cos α ď cos p θ ´ β q (A13)[using the polar form (A1)].If there are equalities in (A11) as well as in (A7),then we reach an equality—at the place of inequality—in (A13): ξ “ ν ` δ p mod π q are the solutions of equa-tion cos p ξ ´ p ν ` δ qq “ . And, |x x | ψ y| “ implies that | ψ y is contained in the subspace generated by t| a y , | b yu ,thus |x b K | ψ y| “ sin β . These two conditions turn (A6)and (A8) into | ψ y “ e i ν “ cos β | b y ` sin β e i δ | b K y ‰ (A14) “ e i ν “ cos p θ ´ β q e i δ | a y ` sin p θ ´ β q | a K y ‰ . (A15)These | ψ y kets—where δ is specified by the polarform (A3), provided x a | b y ‰ , and the global phase ν can be any real number—are the only kets that saturatethe inequality (A13). We can not straightforward use theabove analysis for the next two cases |x a | b y| “ , , hencethese are studied individually.In the case of x a | b y “ , | b K y “ | a y and | a K y “ | b y ;in fact, there is no need for the orthogonalization pro-cess, and both the representations (A6) and (A8) of | ψ y become the same. Furthermore, δ is not determinedby the polar form (A3), whereas θ “ π . Now the in-equality (A13) becomes cos α ` cos β ď , which is—directly realized from (A6) due to (A7)—saturated bythe ket (A14) with an arbitrary real phase δ [remember cos α “ |x a | ψ y| due to (A1)].In the case of |x a | b y| “ , θ “ and | b y “ e i δ | a y accord-ing to (A3), and the above orthogonalization process,thus | b K y and | a K y , does not exist. Consequently, theterm x b K | ψ y| b K y will not then appear in the decomposi-tion (A6) of | ψ y . At the places of (A7), (A13), and (A14)we have ď |x x | ψ y| ñ cos β ď , cos α “ cos β , and | ψ y “ e i ν | b y , respectively. In this case, there is no genuineQC, nevertheless cos β ď is saturated by the ket(s) | ψ y “ e i ν | b y [remember cos β “ |x b | ψ y| , see (A2)].One can appreciate that inequality (A13) is a legit-imate QC, and α and β must respect that for every θ P r , π s . Applying square root to both sides of the in-equality, we gain cos α “ | cos α | ď | cos p θ ´ β q| “ cos p θ ´ β q . (A16)Since α P r , π s and p θ ´ β q P r´ π , π s , both cos α and cos p θ ´ β q are nonnegative numbers, hence there is noneed to use the modulus on either side of the above in-equality. As the arccos function is a strictly decreas-ing function and arccos p cos ς q “ | ς | for ς P r´ π , π s , from(A16), we own an equivalent form | θ ´ β | ď α (A17) of (A13). In fact, (A17) carries two TIs: θ ď α ` β and β ď α ` θ . | ψ y of (A14) with ď β ď θ saturates theTI θ ď α ` β and with θ ď β ď π saturates the other TI β ď α ` θ . TIs such as θ ď α ` β [see (13)] are used todefine the combined-probability space ω in Sec. II.Replacing the ordered set t b, β, ν u by t a, α, µ u in (A6)and repeating the above analysis, one will discover q “ cos β ď cos p θ ´ α q and (A18) | θ ´ α | ď β (A19)at the places of (A13) and (A17), respectively. Jointly(A17) and (A19) can be written as | θ ´ β | ď α ď θ ` β , (A20)which displays three TIs associated with the three angles.A TI says: the sum of two quantum angles must be greaterthan or equal to the remaining quantum angle. In fact, the quantum angle “ arccos |x | y| " is a metric(and a distinguishability measure [12]) on the set S pure ofall pure states ( ρ “ ρ ). It is because the four conditions,1. arccos |x a | b y| ě arccos |x a | b y| “ if and only if | a yx a | “ | b yx b | arccos |x a | b y| “ arccos |x b | a y| arccos |x a | b y| ď arccos |x a | ψ y| ` arccos |x ψ | b y| ,are satisfied for every | a yx a | , | b yx b | , and | ψ yx ψ | in S pure ,where |x a | b y| “ a tr ppp| a yx a | | b yx b |qqq . Note that every purestate on H d is made of a ket in H d , and two kets that areequal up to a global phase provide the same pure state.As the arccos function is nonnegative, the first conditionis valid. The second and third are true by the virtue of |x a | b y| “ ô | a yx a | “ | b yx b | and |x a | b y| “ |x b | a y| , respec-tively. The last condition is, the TI θ ď α ` β , alreadyderived above.Returning to the TIs (A20), as α P r , π s , θ ` β willbe a true upper bound on α only if it is smaller than orequal to π . Hence, we can further improve (A20) as | θ ´ β | ď α ď min (cid:32) θ ` β , π ( . (A21)Taking the right-hand side inequality and applying thecosine function—that decreases monotonically on r , π s —to both the terms, we get max t cos p θ ` β q , u ď cos α . (A22)Now, considering the Heaviside’s unit step function η p υ q : “ if υ ă if υ ě , (A23)one can rewrite (A22) as η ppp cos p θ ` β qqqq cos p θ ` β q ď cos α . (A24)3Since the terms on either side of the above inequality arenonnegative, squaring both sides delivers η ppp cos p θ ` β qqqq cos p θ ` β q ď cos α . (A25)Putting (A13) and (A25) side by side, we accomplish η ppp cos p θ ` β qqqq cos p θ ` β q ď cos α ď cos p θ ´ β q . (A26)Furthermore, due to (A1)–(A3), (A26) becomes η p τ ´ q τ ´ ď p ď τ ` , where (A27) τ ´ : “ ? r q ´ a p ´ r qp ´ q q and (A28) τ ` : “ ? r q ` a p ´ r qp ´ q q . (A29)In essence, we obtain QCs (A21) and (A27) that areequivalent to each other, one is in terms of the quantumangles and the other is in terms of the probabilities. Appendix B: Compactness and convexity of ω Ă Ω The real vector space R d is also a metric space withthe Euclidean distance, and both its subsets Ω and ω areclosed as well as bounded, hence they are compact sets(thanks to the Heine-Borel theorem, see in [46]). Sincea convex combination of probability vectors is again aprobability vector, both Ω a and Ω b are convex subsets of R d . Moreover, Ω “ Ω a ˆ Ω b is a convex set because it isa Cartesian product of two such sets.To prove the convexity of ω , we consider two combinedvectors ` (cid:126)p , (cid:126)q ˘ and ` (cid:126)p , (cid:126)q ˘ that belong to ω . It meansthat their components follow the constraints (5)–(8) and(15) that is p i ` q j ď r ij ` ` b r ij p ´ p i qp ´ q j q , (B1) p i ` q j ď r ij ` ` b r ij p ´ p i qp ´ q j q (B2)for every ď i, j ď d . For the proof, we need to showthat a convex combination ` (cid:126)p, (cid:126)q ˘ “ λ ` (cid:126)p , (cid:126)q ˘ ` p ´ λ q ` (cid:126)p , (cid:126)q ˘ (B3)fulfills all the requirements (5)–(8) and (15)—therefore,lies in ω —for every λ P r , s . Thanks to the convexity of Ω , the combination (B3) belongs to Ω and ` (cid:126)p, (cid:126)q ˘ meetsall the demands (5)–(8).Now we demonstrate that the components p i and q j of ` (cid:126)p, (cid:126)q ˘ respect inequality (15): p i ` q j “ λ p p i ` q j q ` p ´ λ qp p i ` q j q (B4) ď r ij ` ` ? r ij ” λ b p ´ p i qp ´ q j q `p ´ λ q b p ´ p i qp ´ q j q ı (B5) ď r ij ` ` ? r ij b ´ λp i ´ p ´ λ q p i b ´ λq j ´ p ´ λ q q j (B6) “ r ij ` ` ? r ij b p ´ p i qp ´ q j q . (B7) We have equality (B4) due to the convex combina-tion (B3), and then we acquire inequality (B5) by em-ploying (B1) and (B2). The next inequality (B6) is at-tributed to the concavity of a real-valued function f p p, q q : “ a p ´ p qp ´ q q (B8)defined on r , s ˆ r , s , and the last equality is againbecause of the combination (B3). In conclusion, thecombined-probability space ω is a convex set in R d . Be-side, to recognize that f p p, q q is a concave function, wepresent the Hessian matrix ˜ B f B p B f B p B q B f B q B p B f B q ¸ “ ¨˝ ´p ´ q q { p ´ p q { p ´ p q { p ´ q q { p ´ p q { p ´ q q { ´p ´ p q { p ´ q q { ˛‚ (B9)that is a negative semidefinite matrix for every p and q inthe interval r , q . For p “ or q “ or both, f p p, q q “ ,and the Hessian matrix is the ˆ zero matrix. Appendix C: Preliminary calculations for the nextappendix
With (3), (4), (9), and (10), let us again acknowledgethat probability “ cos (angle) , and the quantum anglesbelong to the interval r , π s . Now we consider j ‰ l and q j ` q l “ cos β j ` cos β l “ ` cos p β j ` β l q cos p β j ´ β l q . (C1)Since the difference between angles β j ´ β l P r´ π , π s , wehave ď cos p β j ´ β l q . Hence, with (C1), one can estab-lish q j ` q l ď ô cos p β j ` β l q ď , (C2)and then q j ` q l ď ô π ď β j ` β l p j ‰ l q (C3)due to the arccos function; note that arccos p cos ς q “ ς for ς P r , π s . One can also perceive π ď β j ` β l as a TI.Next we are going to validate a result that is appliedin Appendix D.If j ‰ l , ď θ ij ´ β j , and ď θ kl ´ β l ,then ď cos p θ ij ´ β j q ` cos p θ kl ´ β l q . (C4)Let us designate θ ij ´ β j and θ kl ´ β l by ϕ ij and ϕ kl ,respectively, and write cos ϕ ij ` cos ϕ kl “ ` cos p ϕ ij ` ϕ kl q cos p ϕ ij ´ ϕ kl q (C5)just like (C1). One can show that the sum ϕ ij ` ϕ kl “ p θ ij ` θ kl q ´ p β j ` β l q ď π (C6)due to θ ij ` θ kl ď π and (C3). Clearly ϕ ij , ϕ kl ď π because θ, β P r , π s , and if ď ϕ ij , ϕ kl [see the re-quirements in (C4)] then we have ď ϕ ij ` ϕ kl and4 ϕ ij ´ ϕ kl P r´ π , π s . As a net result, ď cos p ϕ ij ˘ ϕ kl q ,the last term in (C5) turns out to be a nonnegative func-tion, and thus we achieve ď cos ϕ ij ` cos ϕ kl . It com-pletes a proof of (C4).In addition to the requirements in(C4), if and only if θ ij “ π “ θ kl and β j ` β l “ π , then we acquire the equal-ity “ cos p θ ij ´ β j q ` cos p θ kl ´ β l q in (C4). (C7)If θ ij “ π “ θ kl and β j ` β l “ π then evidently wehave the equality of (C7). Now let us prove theconverse under the requirements ď ϕ ij , ϕ kl of (C4).If cos ϕ ij ` cos ϕ kl “ then the last term in (C5)must vanish, which occurs—provided ď ϕ ij , ϕ kl —when the sum in (C6) attains its upper bound π or ϕ ij ´ ϕ kl “ ˘ π . The case ϕ ij ´ ϕ kl “ π arises when ϕ ij “ π and ϕ kl “ , and ϕ ij ´ ϕ kl “ ´ π happens when ϕ ij “ and ϕ kl “ π . Both these cases come under ϕ ij ` ϕ kl “ π —that is when the sum in (C6) reachesits upper bound—which materialize if and only if θ ij “ π “ θ kl and β j ` β l “ π ; it validates (C7).Similar to (C3) we have p i ` p k ď ô π ď α i ` α k p i ‰ k q , (C8)and to (C4) plus (C7) we haveif i ‰ k , ď θ ij ´ α i , and ď θ kl ´ α k ,then ď cos p θ ij ´ α i q ` cos p θ kl ´ α k q .In addition, if and only if θ ij “ π “ θ kl and α i ` α k “ π , then we own the equality “ cos p θ ij ´ α i q ` cos p θ kl ´ α k q . (C9) Appendix D: Extreme points of ω In Appendix B, we demonstrate that the combined-probability space ω is a compact convex set in R d . Ac-cording to the Krein-Milman theorem (see Theorem . . and Appendix A.3 in [38]), every point of such a set canbe decomposed into a convex combination of its extremepoints. In this appendix, starting from an arbitrary in-terior point of ω , we move toward its extreme points.
1. Interior of ω A point ` (cid:126)p, (cid:126)q ˘ P ω that obeys each of the constraints(6), (8), and (13) with strict inequality, ă p i , ă q j , θ ij ă α i ` β j for all ď i, j ď d , (D1)is called an interior point of ω . In certain cases, suchas d “ and θ P t , π u , there exist—no interior point—only extreme points, then the following analysis is notneeded. However, for d ą , there is always an interior point: with θ ij ď π ă ? d , one can show that thecenter—specified by p i “ d “ q j for all i, j —of ω is aninterior point when d ą .We begin our journey from a general but fixed interiorpoint ` (cid:126)p, (cid:126)q ˘ along a straight line, which is the locus ofpoints (cid:126)P “ ` p , p , (cid:126)p rest , (cid:126)q ˘ P R d , where p , p obey thelinear equation p ` p “ ´ ř di “ p i “ p ` p ď (D2)and (cid:126)p rest “ p p , ¨ ¨ ¨ , p d q . One can acknowledge that twopoints on this line differ from each other only in the firsttwo coordinates, hence p , p are the only variables here.In (D2), the inequality saturates for d “ and becomesstrict due to (D1) when d ą .Since we never want to move outside of the combinedspace, we only consider those points on the line that liein ω . From Sec. II recall that a point of R d lies in Ω if and only if it meets all the requirements (5)–(8), andif it also satisfies all the TIs (13) only then it belongs to ω . So a point (cid:126)P “ ` p , p , (cid:126)p rest , (cid:126)q ˘ on the line, definedby (D2), is contained in Ω if and only if ď p and ď p . (D3)With (D2) and (D3), one can derive ď p , p ď p ` p . (D4)As per (3) and (4), we can attach angles α and α with p and p , correspondingly. If these angles comply with θ j ´ β j ď α , θ k ´ β k ď α for all ď j, k ď d , (D5)only then (cid:126)P P ω . Observe that the other demands for (cid:126)P to be in ω —(D1) for ď i ď d and (7)—are automati-cally met, because (cid:126)p rest and (cid:126)q are also parts of the interiorpoint ` (cid:126)p, (cid:126)q ˘ P ω .Considering the suprema θ J ´ β J “ max ď j ď d (cid:32) θ j ´ β j ( and (D6) θ K ´ β K “ max ď k ď d (cid:32) θ k ´ β k ( , (D7)we can convert all the conditions in (D5) into two θ J ´ β J ď α and θ K ´ β K ď α . (D8)Throughout the paper, in the subscripts of angles, cap-ital letters are used to highlight a supremum. Asupremum, say θ J ´ β J , cannot be a negative number: θ J ´ β J ă implies θ j ă β j for every j by the defini-tion (D6). Which leads to r j ą q j for each j by the rela-tions (3), (4), (9), and (10), and then to the contradiction “ ř dj “ r j ą ř dj “ q j “ . Furthermore, θ J ´ β J “ if and only if θ j “ β j for every j . So, both suprema (D6)and (D7) lie in r , π s .5Since the cosine function is monotonically decreasingand nonnegative on r , π s , we can translate the con-straints (D8) as cos α ď cos p θ J ´ β J q , cos α ď cos p θ K ´ β K q (D9)and then as p “ cos α ď cos p θ J ´ β J q , (D10) p “ cos α ď cos p θ K ´ β K q . (D11)By the way, inequalities (A17) and (A13) impose strongerrestrictions than (D5), (D10), and (D11). Since p fol-lows p with Eq. (D2), all the restrictions (D4), (D10),and (D11) can be put together as ď max ! , p ` p ´ cos p θ K ´ β K q ) ď p ď min ! cos p θ J ´ β J q , p ` p ) ď . (D12)One can witness that these bounds on p depend on thechosen interior point ` (cid:126)p, (cid:126)q ˘ . In short, only those (cid:126)P thatfulfill the requirements (D2) and (D12) belong to thecombined space ω .From the interior point ` (cid:126)p, (cid:126)q ˘ , we can travel on theline in two directions: where p increases and where p decreases. While moving we pass four points (cid:126)P , ¨ ¨ ¨ , (cid:126)P of R d that are presented in Table I. When we proceed inthe direction where p increases, then we reach first either (cid:126)P or (cid:126)P . It all depends on the minimum value in (D12).The point that we reach first belongs to ω . Whereas theother point, then, fails to satisfy (D12), and thus it liesoutside of ω . While moving in the other direction, where p decreases, we encounter first either (cid:126)P or (cid:126)P . Depend-ing on the maximum value in (D12) one of t (cid:126)P , (cid:126)P u willbe in, other will be out of, ω (unless both these pointsare the same).All the above possibilities are communicated throughTable II. For any ` (cid:126)p, (cid:126)q ˘ , only two of these possibilitiescan and will materialize, thus ω contains only a duo of(distinct) points from Table I. In Table III, we presentevery such duo. In fact, the interior point ` (cid:126)p, (cid:126)q ˘ can beexpressed as a convex combination λ ` p , p , (cid:126)p rest , (cid:126)q ˘loooooooomoooooooon (cid:126)P `p ´ λ q ` p , p , (cid:126)p rest , (cid:126)q ˘loooooooomoooooooon (cid:126)P (D13)of points of the one duo (cid:126)P , (cid:126)P that lies in ω . For eachduo, λ P p , q is presented in Table III.By varying λ from 0 to 1 in the combination (D13),one can generate the line segment from (cid:126)P to (cid:126)P . Recallthat the line is described by (D2). If (cid:126)P , (cid:126)P belong to thecombined space, then obviously the whole segment willbe in ω thanks to its convexity. The line segments con-necting (cid:126)P with (cid:126)P (provided (cid:126)P ‰ (cid:126)P ) and connecting (cid:126)P with (cid:126)P p (cid:126)P ‰ (cid:126)P q remain outside of ω . Therefore,these two duos are not listed in Table III. TABLE I. A list of four points (cid:126)P “ ` p , p , (cid:126)p rest , (cid:126)q ˘ P R d thatlie on the line characterized by (D2). From the interior point ` (cid:126)p, (cid:126)q ˘ , (cid:126)P , (cid:126)P are in the direction where p increases, and (cid:126)P , (cid:126)P are in the direction where p decreases. So, the valueof p for a point here is one of the four bounds [stated in(D12)]. Once we have p —in the center column—then p isretrieved with (D2) and placed in the right column. (cid:126)P p p (cid:126)P cos p θ J ´ β J q p ` p ´ cos p θ J ´ β J q (cid:126)P p ` p (cid:126)P p ` p (cid:126)P p ` p ´ cos p θ K ´ β K q cos p θ K ´ β K q TABLE II. The conditions that—rely on the minimum andthe maximum values in (D12)—determine whether a pointfrom Table I will be in or out of ω . If a condition from the leftcolumn holds, only then the related case in the right columnoccurs, and vice versa. One can realize that at most twoconditions can hold at a time.If and only if Then cos p θ J ´ β J q ă p ` p (cid:126)P P ω and (cid:126)P R ω cos p θ J ´ β J q ą p ` p (cid:126)P R ω and (cid:126)P P ω cos p θ J ´ β J q “ p ` p (cid:126)P “ (cid:126)P P ω cos p θ K ´ β K q ă p ` p (cid:126)P R ω and (cid:126)P P ω cos p θ K ´ β K q ą p ` p (cid:126)P P ω and (cid:126)P R ω cos p θ K ´ β K q “ p ` p (cid:126)P “ (cid:126)P P ω TABLE III. Duos (cid:126)P , (cid:126)P of points from Table I. Only one outof these duos—unless two or more duos are the same—lies in ω and expresses the interior point ` (cid:126)p, (cid:126)q ˘ through the convexcombination (D13) with a real number λ . Corresponding toeach duo, λ is registered in the right column. One can con-firm that ă λ ă by realizing ă p ă cos p θ J ´ β J q and ă p ă cos p θ K ´ β K q . (cid:126)P , (cid:126)P λ(cid:126)P , (cid:126)P p cos p θ J ´ β J q (cid:126)P , (cid:126)P cos p θ K ´ β K q ´ p cos p θ J ´ β J q ` cos p θ K ´ β K q ´ p ´ p (cid:126)P , (cid:126)P ´ p p ` p (cid:126)P , (cid:126)P ´ p cos p θ K ´ β K q ` (cid:126)p, (cid:126)q ˘ in ω can be decomposed as a convex combination of boundary points of ω , which are decomposed in the nextpart. Note that the subsequent analysis is for d ą . Inthe case of d “ , p ` p “ , and Table I already carriesthe extreme points of ω . In fact, for d “ , we only need (cid:126)P and (cid:126)P , because ω contains (cid:126)P and (cid:126)P if and only if (cid:126)P “ (cid:126)P and (cid:126)P “ (cid:126)P , respectively.
2. Boundary of ω The boundary of ω is made of d ` d regions, wherea region is characterized by equality in one of the con-straints (6), (8), and (13): P i : “ (cid:32) p (cid:126)p, (cid:126)q q P ω ˇˇ p i “ ( , (D14) Q j : “ (cid:32) p (cid:126)p, (cid:126)q q P ω ˇˇ q j “ ( , and (D15) R ij : “ (cid:32) p (cid:126)p, (cid:126)q q P ω ˇˇ α i ` β j “ θ ij ( (D16)for ď i, j ď d . A point from Table I, provided it is in ω , called a boundary point because it belongs to one ofthe regions (D14)–(D16). To reveal that the boundarypoints of ω can be decomposed into certain convex com-binations, let us suppose that the duo (cid:126)P , (cid:126)P belongs to ω and analyze first (cid:126)P P P and then (cid:126)P P R J . Of course,an identical treatment can be delivered in the case ofother duos from Table III. TABLE IV. A list of four points (cid:126)P “ ` , p , p , (cid:126)p rest , (cid:126)q ˘ sim-ilar to Table I. The upper bounds on p [see (D18)] specifythe points (cid:126)P and (cid:126)P , while the lower bounds determine (cid:126)P and (cid:126)P . These bounds are stated in the middle column for p , and then the corresponding p are obtained by (D17) [seethe right column]. (cid:126)P p p (cid:126)P cos p θ K ´ β K q ř i “ p i ´ cos p θ K ´ β K q (cid:126)P ř i “ p i (cid:126)P ř i “ p i (cid:126)P ř i “ p i ´ cos p θ L ´ β L q cos p θ L ´ β L q TABLE V. The necessary and sufficient conditions—that arisefrom the restraint (D18)—for a point of Table IV to be in orout of the region P Ă ω . The table is like Table II.If and only if Then cos p θ K ´ β K q ă ř i “ p i (cid:126)P P P and (cid:126)P R P cos p θ K ´ β K q ą ř i “ p i (cid:126)P R P and (cid:126)P P P cos p θ K ´ β K q “ ř i “ p i (cid:126)P “ (cid:126)P P P cos p θ L ´ β L q ă ř i “ p i (cid:126)P R P and (cid:126)P P P cos p θ L ´ β L q ą ř i “ p i (cid:126)P P P and (cid:126)P R P cos p θ L ´ β L q “ ř i “ p i (cid:126)P “ (cid:126)P P P TABLE VI. Depending on (cid:126)P and the conditions in Table V,at most two separate points of Table IV can belong to P .Here, the left column carries all such couples of points. Tothe right side of each couple (cid:126)P , (cid:126)P , the value of λ is written,which associates the couple (provided it is in P ) back to (cid:126)P “ λ (cid:126)P ` p ´ λ q (cid:126)P . Taking ă p ă cos p θ L ´ β L q and ă p ` p ď cos p θ K ´ β K q —that determines (cid:126)P P P [seeTable II]—one can check that each λ lies in the interval p , s . (cid:126)P , (cid:126)P λ(cid:126)P , (cid:126)P p ` p cos p θ K ´ β K q (cid:126)P , (cid:126)P cos p θ L ´ β L q ´ p cos p θ K ´ β K q ` cos p θ L ´ β L q ´ ř i “ p i (cid:126)P , (cid:126)P ´ p ř i “ p i (cid:126)P , (cid:126)P ´ p cos p θ L ´ β L q Now we start from (cid:126)P and travel within the region P along a new set of points (cid:126)P “ ` , p , p , (cid:126)p rest , (cid:126)q ˘ bychanging p , p according to p ` p “ ´ ř di “ p i “ ř i “ p i ď , (D17)where (cid:126)p rest “ p p , ¨ ¨ ¨ , p d q . Repeating the procedure sim-ilar to Appendix D 1, here we have ď max ! , ř i “ p i ´ cos p θ L ´ β L q ) ď p ď min ! cos p θ K ´ β K q , ř i “ p i ) ď , (D18)which is like (D12). The supremum θ K ´ β K is definedby (D7) and θ L ´ β L “ max ď l ď d (cid:32) θ l ´ β l ( . (D19)If and only if p respects (D18) and p follows p with(D17), then a new (cid:126)P P P Ă ω .Analogous to Tables I–III, here we compose Tables IV–VI, in that order. Table IV holds a collection of fourpoints. Table V has the conditions that decide whether apoint of Table IV is in or out of P . Table VI supplies allpossible couples—of points from Table IV—out of whichone belongs to P , that one is determined by (cid:126)P . Theline segment—connecting the one couple—carries (cid:126)P andcompletely occupies in the region P .Now we are going to focus on (cid:126)P P R J . Let us proceedfrom (cid:126)P by altering only p , p of another new vector (cid:126)P “ ` cos p θ J ´ β J q , p , p , (cid:126)p rest , (cid:126)q ˘ with respect to p ` p “ ´ ř di “ p i ´ cos p θ J ´ β J q “ ř i “ p i ´ cos p θ J ´ β J q . (D20)7 TABLE VII. A set of four points (cid:126)P “ ` cos p θ J ´ β J q , p , p , (cid:126)p rest , (cid:126)q ˘ like Tables I and IV. Here t (cid:126)P , (cid:126)P u and t (cid:126)P , (cid:126)P u areobtained with the upper and lower bounds in (D21), correspondingly. These bounds are arranged in the center column, and p is drawn from p with (D20). (cid:126)P p p (cid:126)P cos p θ K ´ β K q ř i “ p i ´ cos p θ J ´ β J q ´ cos p θ K ´ β K q (cid:126)P ř i “ p i ´ cos p θ J ´ β J q (cid:126)P ř i “ p i ´ cos p θ J ´ β J q (cid:126)P ř i “ p i ´ cos p θ J ´ β J q ´ cos p θ L ´ β L q cos p θ L ´ β L q TABLE VIII. If there is a case from the left column, then we have the corresponding consequence in the right column. Allthese cases are implications of (D21)–(D23). The table is built in the same way as Table II and V.If Then K ‰ J (cid:126)P R R J and (cid:126)P P R J K “ J and cos p θ J ´ β J q ` cos p θ K ´ β K q ă ř i “ p i (cid:126)P P R J and (cid:126)P R R J cos p θ J ´ β J q ` cos p θ K ´ β K q ą ř i “ p i (cid:126)P R R J and (cid:126)P P R J cos p θ J ´ β J q ` cos p θ K ´ β K q “ ř i “ p i (cid:126)P “ (cid:126)P P R J L ‰ J (cid:126)P P R J and (cid:126)P R R J L “ J and cos p θ J ´ β J q ` cos p θ L ´ β L q ă ř i “ p i (cid:126)P R R J and (cid:126)P P R J cos p θ J ´ β J q ` cos p θ L ´ β L q ą ř i “ p i (cid:126)P P R J and (cid:126)P R R J cos p θ J ´ β J q ` cos p θ L ´ β L q “ ř i “ p i (cid:126)P “ (cid:126)P P R J TABLE IX. Taking the case K “ J and L “ J , we havefour duos of points, and the table is arranged in the samemanner as Table III and VI. Right side to each duo, weplace λ that relates the duo (when it is in R J ) to thepoint (cid:126)P “ λ (cid:126)P ` p ´ λ q (cid:126)P . Having ă p i ă cos p θ iJ ´ β J q for i “ , , and the condition cos p θ J ´ β J q ď p ` p thatcertifies (cid:126)P P R J [see Table II], one can show that ď λ ă in every case. (cid:126)P , (cid:126)P λ(cid:126)P , (cid:126)P p ` p ´ cos p θ J ´ β J q cos p θ J ´ β J q (cid:126)P , (cid:126)P cos p θ J ´ β J q ´ p ř i “ ´ cos p θ iJ ´ β J q ´ p i ¯ (cid:126)P , (cid:126)P ´ p ř i “ p i ´ cos p θ J ´ β J q (cid:126)P , (cid:126)P ´ p cos p θ J ´ β J q Note that (cid:126)p rest “ p p , ¨ ¨ ¨ , p d q , and (D20) identifies astraight line, a segment of which is contained in the re- gion R J . In addition to (D20), if p agrees to ď max ! , ř i “ p i ´ cos p θ J ´ β J q ´ cos p θ L ´ β L q ) ď p ď min ! cos p θ K ´ β K q , ř i “ p i ´ cos p θ J ´ β J q ) ď (D21)only then the new vector (cid:126)P P R J . Like Tables I and IV,here we assemble Table VII of four points using the fourbounds in (D21).Due to (C4) and (C7) from Appendix C, we haveif K ‰ J then ă cos p θ J ´ β J q ` cos p θ K ´ β K q , and (D22)if L ‰ J then ă cos p θ J ´ β J q ` cos p θ L ´ β L q . (D23)These inequalities are strict because a requirements in(C7), β J ` β K “ π , cannot be met since q J ` q K ă is caused by (D1). Now taking (D21)–(D23) with ř i “ p i ď , one can deduce that the vectors (cid:126)P and (cid:126)P of Table VII can not belong to R J unless K “ J and L “ J , respectively. This fact is recorded in Ta-ble VIII with some other conditions, together they tell8when a point of Table VII will be in or out of the region R J .A duo, out of the four listed in Table IX, resides in R J and expresses (cid:126)P through a convex combination. AsTables I–III are linked with the interior point ` (cid:126)p, (cid:126)q ˘ P ω and Tables IV–VI are attached to (cid:126)P P P , Tables VII–IXare associated with (cid:126)P P R J . Tables I, IV, and VII carrythe boundary points of ω , P , and R J , respectively.
3. Extreme of ω In the above parts, it is demonstrated that every in-terior point ` (cid:126)p, (cid:126)q ˘ P ω can be decomposed into a con-vex combination of the boundary points of ω , which canfurther be decomposed into convex combinations of theboundary points of regions (D14)–(D16). Continuing thisdecomposition process, we reach at a point ` ˚ (cid:126)p , (cid:126)q ˘ , where ˚ (cid:126)p “ ` cos ˚ α , ¨ ¨ ¨ , cos ˚ α m , , ˚ p s , ˘ , (D24) ˚ α i “ θ iJ ´ β J p for all i “ , ¨ ¨ ¨ , m q , (D25) ˚ p s “ ´ ř mi “ cos ˚ α i p m ` ď s ď d q , (D26) ” , ¨ ¨ ¨ , , and (D27) ď m ď d ´ . (D28)Since every ˚ α i of (D25) is a supremum, ď ˚ α i [see theexplanation below (D8)] and ˚ α i ă α i ă π due to (D1),we deduce that ď ˚ α i ă π p for all i “ , ¨ ¨ ¨ , m q . (D29)The point ` ˚ (cid:126)p , (cid:126)q ˘ , designated by (D24)–(D28), satis-fies m and d ´ p m ` q number of equality constraints oftype (13) and (6), respectively. If ˚ p s of (D26) follows ď ˚ p s ď cos p θ sZ ´ β Z q (D30)then ` ˚ (cid:126)p , (cid:126)q ˘ P ω , where θ sZ ´ β Z “ max ď z ď d (cid:32) θ sz ´ β z ( (D31)is a supremum like (D6), (D7), (D19), and (D25). Onecan check that points in Table I for d “ and in Tables IVas well as VII—provided K “ J and L “ J —for d “ are like ` ˚ (cid:126)p , (cid:126)q ˘ ; remember that ř di “ p i “ due to (5).Furthermore, one can easily recognize ˚ p s in each of thesepoints. Then, one can see through Table II, V, and VIIIthat one of the two inequalities in (D30) is required for apoint to be in ω . The other inequality is automaticallyobeyed due to (D1) and the conditions appeared in theearlier decompositions.If we start our journey from a point ` (cid:126)p , (cid:126)q ˘ , where (cid:126)q “ p r , ¨ ¨ ¨ , r d q , (D32)then we will arrive at the point ` ˚ (cid:126)p , (cid:126)q ˘ , where ˚ (cid:126)p “ ` , ˘ (D33) TABLE X. Four points (cid:126)Q “ ` ˚ (cid:126)p , q , q , q , (cid:126)q rest ˘ P R d thatrest on the line specified by (D35). From the point ` ˚ (cid:126)p , (cid:126)q ˘ , thecoordinate q increases towards t (cid:126)Q , (cid:126)Q u , while it decreasestowards t (cid:126)Q , (cid:126)Q u . The middle column carries the four boundsgiven in (D36), and then q is obtained with (D35). The tableis prepared in the same fashion as Tables I, IV, and VII. (cid:126)Q q q (cid:126)Q cos p θ K ´ ˚ α K q q ` q ´ cos p θ K ´ ˚ α K q (cid:126)Q q ` q (cid:126)Q q ` q (cid:126)Q q ` q ´ cos p θ L ´ ˚ α L q cos p θ L ´ ˚ α L q [for , see (D27)]. This point represents an extreme pointof ω and a special case m “ with “ ˚ p s (D34)of (D28) and (D26). In the case (D34), the supremum ˚ α “ θ J ´ β J “ that is possible if and only if θ j “ β j ,means r j “ q j , for every j . Indeed, it is so [see (D32)].In all other cases, ă ˚ α i for every ď i ď m [see the lim-its (D29) on ˚ α i of (D25)], and ` ˚ (cid:126)p , (cid:126)q ˘ can be decomposedfurther by adopting the same procedure as before.Without loss of generality, let us suppose J “ for the subsequent analysis. Here we begin with (cid:126)Q “ ` ˚ (cid:126)p , q , q , q , (cid:126)q rest ˘ , where q ` q “ ´ ř di “ q j ´ cos β loomoon q “ q ` q (D35)and (cid:126)q rest “ p q , ¨ ¨ ¨ , q d q . One can acknowledge that (cid:126)Q represents all those points, including ` ˚ (cid:126)p , (cid:126)q ˘ , that fall onthe straight line characterized by (D35).If q stays on the line with q , which follows ď max ! , q ` q ´ cos p θ L ´ ˚ α L q ) ď q ď min ! cos p θ K ´ ˚ α K q , q ` q ) ď , (D36)then (cid:126)Q P ω . Here θ K ´ ˚ α K “ max ď k ď d (cid:32) θ k ´ ˚ α k ( and (D37) θ L ´ ˚ α L “ max ď l ď d (cid:32) θ l ´ ˚ α l ( (D38)are suprema, and the angles ˚ α are related to the compo-nents of ˚ (cid:126)p through (3) and (4) [see also (D24) and (D25)].The constraints (D36) look alike (D12) and (D18). Iden-tical to Tables I, IV, and VII, we enter a list of fourpoints in Table X, where the points are drawn from thefour bounds on q given in (D36).Now, to establish criteria for a point of Table X to bein or out of ω , we are going to address the two cases m “ with ă ˚ p s and (D39) m ą with ă ˚ p s (D40)9individually [see Eq. (D26) for ˚ p s and the range (D28)of m ]. Let us first take the case (D40): whatever thesuprema (D37) and (D38) are, we have ă cos β ` cos p θ K ´ ˚ α K q and (D41) ă cos β ` cos p θ L ´ ˚ α L q . (D42)To demonstrate this, we consider m “ , the cases with m ą can be handled likewise. For m “ , we have β “ θ i ´ ˚ α i (where i “ , ) due to (D25). If K as-sociated with the supremum (D37) is 1, then by taking β “ θ ´ ˚ α we can validate the strict inequality (D41)thanks to (C9). If K ‰ , we can do the same by nowconsidering β “ θ ´ ˚ α . In a similar fashion, we canestablish the other inequality (D42).We draw the following inferences from inequalities(D41) and (D42). cos p θ K ´ ˚ α K q ą ´ q “ ř dj “ q j ě q ` q , (D43) cos p θ L ´ ˚ α L q ą ´ q “ ř dj “ q j ě q ` q (D44)implies that the maximum and the minimum values in(D36) are 0 and q ` q , respectively. Consequently,the points (cid:126)Q and (cid:126)Q of Table X never, whereas (cid:126)Q and (cid:126)Q always, belong to ω in the case (D40). More-over, ` ˚ (cid:126)p , (cid:126)q ˘ can be broken into the convex combination λ (cid:126)Q ` p ´ λ q (cid:126)Q , where λ “ q q ` q [see Table XII].Next, it is not difficult to realize that both (cid:126)Q and (cid:126)Q can be decomposed further and further until we arrive ata point ` ˚ (cid:126)p , ˚ (cid:126)q ˘ , where ˚ (cid:126)q “ ` q , , ˚ q t , ˘ with ˚ q t “ ´ q p ď t ď d q . (D45)In the decomposition process one will encounter inequali-ties, such as (D41) and (D42), that can be tacked like theabove. For m ą , a point ` ˚ (cid:126)p , ˚ (cid:126)q ˘ defined by (D24)–(D27)and (D45) is an extreme point of ω , because it cannotbe written into a convex combination of other points of ω . Furthermore, ` ˚ (cid:126)p , ˚ (cid:126)q ˘ is a vector-valued function of β since θ -angles are fixed by (10) once the measurementsettings are selected in (1).Let us now turn to the case (D39), where β “ θ ´ ˚ α according to (D25), (cid:126)Q “ ` ˚ (cid:126)p , cos p θ ´ ˚ α q , q , q , (cid:126)q rest ˘ , and (D46) ˚ (cid:126)p “ ` ˚ p , , ˚ p s , ˘ with ´ ˚ p s “ ˚ p “ cos ˚ α . (D47)Since supremum (D37) is a nonnegative number, K caneither be s or 1 here. It is due to θ i ´ ˚ α i ď when i ‰ s and i ‰ , because then ˚ α i “ π and every θ ď π .Similarly, L related to the supremum (D38) can eitherbe s or 1 here.When K “ s or L “ s or both, we encounter situa-tion similar to the case (D40): When K “ s then—dueto (C9)—we have cos p θ K ´ ˚ α K q ` cos p θ ´ ˚ α q ě and thus (D48) cos p θ K ´ ˚ α K q ě ´ q “ ř dj “ q j ě q ` q . (D49) TABLE XI. Group of conditions for the case (D39), where ˚ α “ θ ´ β . A condition from the left column delivers whatis on its right side. These conditions originate from (D36) andthe discussion around (D49). At most two conditions can holdsimultaneously, thus more than two distinct points of Table Xcannot be a part of ω . The table looks like Table VIII.If Then K “ s (cid:126)Q P ω K “ , cos p θ K ´ ˚ α K q ă q ` q (cid:126)Q P ω and (cid:126)Q R ω cos p θ K ´ ˚ α K q ą q ` q (cid:126)Q R ω and (cid:126)Q P ω cos p θ K ´ ˚ α K q “ q ` q (cid:126)Q “ (cid:126)Q P ω L “ s (cid:126)Q P ω L “ , cos p θ L ´ ˚ α L q ă q ` q (cid:126)Q R ω and (cid:126)Q P ω cos p θ L ´ ˚ α L q ą q ` q (cid:126)Q P ω and (cid:126)Q R ω cos p θ L ´ ˚ α L q “ q ` q (cid:126)Q “ (cid:126)Q P ω TABLE XII. Collection of duplets (cid:126)Q , (cid:126)Q of points from Ta-ble X. Only one of these duplets—except if two or more arethe same—belongs to ω and represents the point ` ˚ (cid:126)p , (cid:126)q ˘ withthe convex combination λ (cid:126)Q ` p ´ λ q (cid:126)Q . Here we assume K “ and L “ , otherwise (cid:126)Q and (cid:126)Q can not belong to ω without being equal to (cid:126)Q and (cid:126)Q , respectively [see Ta-ble XI]. The right column has the values of λ for each duplet,provided the duplet lies in ω . One can check that λ P r , s with ă q ď cos p θ K ´ ˚ α K q and ă q ď cos p θ L ´ ˚ α L q [see (D36)]. (cid:126)Q , (cid:126)Q λ(cid:126)Q , (cid:126)Q q cos p θ ´ ˚ α q (cid:126)Q , (cid:126)Q cos p θ ´ ˚ α q ´ q cos p θ ´ ˚ α q ` cos p θ ´ ˚ α q ´ q ´ q (cid:126)Q , (cid:126)Q ´ q q ` q (cid:126)Q , (cid:126)Q ´ q cos p θ ´ ˚ α q One can perceive that (D48) and (D49) are analogues to(D41) and (D43), respectively. The inequalities in (D49)suggest that q ` q is the minimum value in (D36).Therefore, without exception (cid:126)Q lies in ω , if (cid:126)Q “ (cid:126)Q then (cid:126)Q P ω . Identically, for L “ s , always (cid:126)Q P ω , and (cid:126)Q belongs to ω only when it is (cid:126)Q .When K “ and L “ only then (cid:126)Q and (cid:126)Q can bein ω without being equal to (cid:126)Q and (cid:126)Q , respectively [seeTable XI]. With Table XI, for the case (D39), one canfind out whether or not a duplet of points from Table Xlies in ω . All such duplets are gathered in Table XII,which reveals that the point ` ˚ (cid:126)p , (cid:126)q ˘ can be split into a0convex combination. As before, we can break the pointsof Table X further and further until we reach extremepoints of ω .In the case (D39), the decomposition process leads to ˚ (cid:126)q “ ` cos ˚ β , ¨ ¨ ¨ , cos ˚ β n , , ˚ q t , ˘ , where (D50) ˚ β j “ θ j ´ ˚ α p for all j “ , ¨ ¨ ¨ , n q , (D51) ˚ q t “ ´ ř nj “ cos ˚ β j p n ` ď t ď d q , and (D52) ď n ď d ´ . (D53)If ˚ q t of (D52) obeys ď ˚ q t ď cos p θ Zt ´ ˚ α Z q , where (D54) θ Zt ´ ˚ α Z “ max ď z ď d (cid:32) θ zt ´ ˚ α z ( , (D55)then the point ` ˚ (cid:126)p , ˚ (cid:126)q ˘ stated by (D47) and (D50) belongsto ω . It is an extreme point of ω in the case (D39).One can also realize that both there ˚ (cid:126)p and ˚ (cid:126)q are func-tions of β by noticing ˚ β j “ θ j ´ θ ` β in (D51) with ˚ α “ θ ´ β . In fact, the extreme point identified by(D33) and (D32) in the case (D34) can also be repre-sented with these ˚ (cid:126)p and ˚ (cid:126)q of (D47) and (D50) by taking ˚ α “ , which make it as an endpoint of the parametriccurve ` ˚ (cid:126)p p ˚ α q , ˚ (cid:126)q p ˚ α q ˘ . In conclusion, we realize the struc-ture of extreme points of ω :The point ` ˚ (cid:126)p , ˚ (cid:126)q ˘ , where “ ˚ (cid:126)p is specified by (D24)–(D27) and ˚ (cid:126)q is given by (D45)" when m ą and “ ˚ (cid:126)p is describe by (D47) and ˚ (cid:126)q is presentedby (D50)–(D53)" when m “ , represents an ex-treme point of ω provided β is within suitablelimits presented in the next part. For every ď m ď p d ´ q , ` ˚ (cid:126)p p β q , ˚ (cid:126)q p β q ˘ is a vector-valuedfunction of a real parameter β , thus it charac-terizes an m -parametric curve in ω . Such curvesare presented in Sec. II. (D56)
4. Limits on β We start with the m -parametric curve ` (cid:126)p p β q , (cid:126)q p β q ˘ identified by (16)–(21). According to (D56), a part of thecurve that lies in ω represents its extreme points. Thispart is specified by the upper and lower limits of β . Tocompute these limits, here, we only need to consider ď p s , (D57) θ it ď α i ` β t p for i “ , ¨ ¨ ¨ , m , s q , and (D58) θ sj ď α s ` β j p for j “ , t q . (D59)When i ą m and i ‰ s then α i “ π , and when j ‰ and j ‰ t , then β j “ π . So one can easily perceive that thepoints ` (cid:126)p p β q , (cid:126)q p β q ˘ fulfill rest of the requirements (13)as well as (5)–(8) to be in ω . For i “ s in (D58) or j “ t in (D59), the TI is alwaysobeyed: due to π ď α s ` α (D60) “ α s ` θ ´ β (D61) “ α s ` θ ´ π ` β t , we have (D62) π ď π ´ θ ď α s ` β t . (D63)With (C8), (16), and (C3) one can sequentially gothrough the steps (D60)–(D62), and the left-hand sideinequality in (D63) is a consequence of θ ď π . Since α s and β t obey π ď α s ` β t , they certainly follow the TI θ st ď α s ` β t as every θ ď π .If we decrease β then α s ` β decreases, and β reaches its lower limit β when the inequality (D59), for j “ , gets saturated. It means that β is a solution ofthe equation θ s ´ β “ α s and thus of cos p θ s ´ β q “ p s “ ´ ř mi “ cos p θ i ´ β q (D64)[by (16) and (19)]. If we increase β then p s and α i ` β t p i “ , ¨ ¨ ¨ , m q decrease, and β attains its upper limit β as soon as one of the inequalities (D57) and (D58) getssaturated. Using (16), (19), and β t “ π ´ β [owing to(C3)], these inequalities can be expressed as ď ´ ř mi “ cos p θ i ´ β q and (D65) β ď θ i ´ θ it ` π p for i “ , ¨ ¨ ¨ , m q . (D66)Now we need to investigate the two cases, m “ and ă m ď p d ´ q listed in (D56), separately for β .In the case m “ , (D65) clearly holds, and the upperlimit β “ θ ´ θ t ` π (D67)is obtained when (D66) is saturated. Corresponding to β of (D67), we have α “ θ ´ β “ θ ` θ t ´ π (D68)which is a root of the equation cos p θ ´ α q ` cos p θ t ´ α q “ . (D69)In the case ă m ď p d ´ q , when we increase β thenthe inequality (D65), rather than (D66), gets saturatedfirst. Hence, β is now a solution of ř mi “ cos p θ i ´ β q “ . (D70)One can justify these statements by proving β ď θ i ` θ i ´ π loooooomoooooon r β ď θ i ´ θ it ` π , (D71)where ď i, i ď m . As β is a root of Eq. (D70), r β is aroot of cos ` θ i ´ r β ˘ ` cos ` θ i ´ r β ˘ “ . (D72)1Equations (D64), (D70), and (D72) are of the form ř m i “ cos p θ i ´ β q “ , (D73)where m angles—the m -set t θ , ¨ ¨ ¨ , θ m u —are takenfrom the first column of Θ matrix [given in (11)]. Al-ways, we must choose the root of Eq. (D73) that respects ď β ď θ i for every i “ , ¨ ¨ ¨ , m . Furthermore, as weadd more angles from the first column to the m -set, thenumber of nonnegative terms increases on the left-handside of Eq. (D73). Then β of smaller value will satisfyEq. (D73). So, by comparing Eqs. (D70) and (D72) inthis way, we can certify the left-hand side inequality in(D71). Whereas, after a simplification, the right-handside inequality turns into θ i ` θ it ď π , which is true asevery θ ď π .In conclusion, the lower limit β is the root ofEq. (D64) for every ď m ď p d ´ q . The upperlimit β , for m “ , is given by (D67) and canbe derived from Eq. (D69). For ă m , β is thesolution of Eq. (D70). (D74)In fact, Eq. (D69)—where two angles are taken fromthe first column of Θ —is also like Eq. (D73). Basi-cally, one needs to solve equation such as (D73)—where ď m ď d angles are picked from a row or a columnof Θ —to get a limit and then an endpoint of an m -parametric curve. When m “ then m can only be 2[see (D64) and (D69)]. And, when ă m ď p d ´ q then m can either be m or m ` [see (D70) and (D64)].To solve Eq. (D73) for β , we transform it into x cos β ` y sin β cos β ` z “ , where (D75) x : “ ř m i “ cos 2 θ i “ ř m i “ r i ´ m , (D76) y : “ ř m i “ sin 2 θ i “ ř m i “ a r i p ´ r i q , and (D77) z : “ ř m i “ sin θ i ´ “ m ´ ř m i “ r i ´ . (D78) Calling cos β “ q by the relations (3) and (4), we canwrite Eq. (D75) as x q ` y a q p ´ q q ` z “ . (D79)The two roots of Eq. (D79) are cos β “ q “ p y ´ x z q ˘ y a y ´ z p x ` z q p x ` y q , (D80)which only depend on the m -set t θ , ¨ ¨ ¨ , θ m u associatedwith Eq. (D73).We pick the root (D80) with + sign due to the fol-lowing reasons. First, for m “ , we have equation suchas (D72), and its root r β —given in (D71)—correspondsto the + sign solution [see also (D68) with (D69)]. Sec-ond, for m “ d , β “ is the only permissible solution ofEq. (D73). It is because angles θ i are not random realnumbers, they follow ř di “ cos θ i “ . When m “ d , z “ d ´ “ ´ x [see (D76) and (D78)], and always the so-lution (D80) with + sign offers β “ . 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