Unextendible mutually unbiased bases in prime-squared dimensions
UUnextendible mutually unbiased bases in prime-squared dimensions
Vishakh Hegde and Prabha Mandayam
Department of Physics, IIT Madras, Chennai - 600036, India (Dated: August 25, 2015)A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist anotherbasis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets ofMUBs in prime-squared dimensions ( d = p ) that cannot be extended to a complete set using thegeneralized Pauli operators. We further observe an interesting connection between the existence ofunextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions.In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of p + 1 MUBs that saturate both a Shannon ( H ) and a collision ( H ) entropic lower bound. Such anidentification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applicationsas well as constructing optimal entanglement witnesses for higher dimensional systems. Two orthonormal bases A = {| a i (cid:105) , i = 1 , . . . , d } and B = {| b j (cid:105) , j = 1 , . . . , d } of a d -dimensional Hilbert space C d are said to be mutually unbiased if for all basisvectors | a i (cid:105) ∈ A and | b j (cid:105) ∈ B , |(cid:104) a i | b j (cid:105)| = 1 √ d , ∀ i, j = 1 , . . . , d. (1)In physical terms, if a system is prepared in an eigen-state of basis A and measured in basis B , all out-comes are equally probable. A set of orthonormal bases {B , B , . . . , B m } in C d is called a set mutually unbiasedbases (MUBs) if every pair of bases in the set is mutu-ally unbiased. MUBs form a minimal and optimal setof orthogonal measurements for quantum state tomog-raphy [1, 2]. Such bases play an important role in ourunderstanding of complementarity in quantum mechan-ics [3] and are central to quantum information tasks suchas entanglement detection [4], information locking [5],and quantum cryptography [6, 7].MUBs correspond to measurement bases that are most‘incompatible’, as quantified by uncertainty relations [8]and other incompatibility measures [9, 10], and, the secu-rity of quantum cryptographic tasks relies on this prop-erty of MUBs. In particular, protocols based on higher-dimensional quantum systems with larger numbers of un-biased basis sets can have certain advantages over thosebased on qubits [11, 12]. However, beyond the case oftwo measurements, being mutually unbiased is a neces-sary but not sufficient condition for satisfying a strongentropic lower bound [13]. It is therefore importantfor cryptographic applications to identify sets of MUBsin higher-dimensional systems that satisfy strong uncer-tainty relations.The maximum number of MUBs that can exist in a d -dimensional Hilbert space is d + 1 and explicit con-structions of such complete sets are known when d is aprime power [2, 14, 15]. However, in non-prime-powerdimensions, the question of whether a complete set ofMUBs exists remains unresolved. Related to the ques-tion of finding complete sets of MUBs is the importantconcept of unextendible sets of MUBs. A set of MUBs {B , B , . . . , B m } in C d is said to be unextendible ifthere does not exist another basis in C d that is unbiased with respect to all the bases B j , j = 1 , . . . , m . Examplesof such unextendible sets are known in the literature [16–20].More recently, a systematic construction of suchsmaller sets that are unextendible to a complete set wasobtained for two- and three-qubit systems [21]. In thecase of two-qubit systems, an interesting connection wasnoted between unextendible sets of Pauli classes andstate-independent proofs of the Kochen-Specker Theo-rem. It was also shown that the tightness of the an en-tropic uncertainty relation for any set of three MUBs in d = 4 follows as an important consequence of the ex-istence of weakly unextendible sets of MUBs [21]. Theexistence of similar unextendible sets was conjectured for d = 2 n ( n > n -qubit Pauli operators [23].Here, we provide a construction of weakly unextendiblesets of MUBs in prime-squared dimensions d = p , where p is prime. Each MUB is realized as the common eigen-basis of a maximal commuting class of tensor productsof the generalized Pauli operators. Our construction alsobrings to light an interesting connection between the ex-istence of unextendible sets and the tightness of entropiclower bounds in these dimensions. In particular, we iden-tify sets of p +1 MUBs that saturate both a Shannon anda collision EUR in d = p . This has important conse-quences for both cryptographic applications and for con-structing entanglement witnesses in higher dimensionalsystems.The rest of the paper is organized as follows. We beginwith a brief review of the standard construction of MUBsin Sec. I and formally define the notion of unextendibil-ity. We state our main result on the construction of un-extendible sets of MUBs in Sec. II and provide proofs inthe appendix (B). Finally, in Sec. III, we note the con-nection between the existence of unextendible MUBs andthe tightness of EURs in prime-squared dimensions. a r X i v : . [ qu a n t - ph ] A ug I. PRELIMINARIES
Our construction of unextendible MUBs is based onthe well known connection between mutually unbiasedbases and mutually disjoint maximal commuting opera-tor classes [14]. Consider a set S of d mutually orthog-onal unitary operators in a d -dimensional Hilbert space C d . Such a set constitutes a basis for M d ( C ), the space of d × d complex matrices. Since at most d such operatorscan mutually commute, we may consider a partitioningof the operator basis S into mutually disjoint maximalcommuting classes as follows. Definition 1 (Mutually Disjoint Maximal CommutingClasses) . A set of subsets C , C , . . . , C L |C j ⊂ S \ {I} ofsize |C j | = d − constitutes a (partial) partitioning of S \{I} into mutually disjoint maximal commuting classesif the subsets C j are such that • The elements of C j commute ∀ ≤ j ≤ L • C j ∩ C k = φ ∀ j (cid:54) = k The existence of such a partitioning of the operatorbasis S is directly related to the existence of mutuallyunbiased bases. We formally state this result in the fol-lowing Lemma, and refer to [14] for the proof. Lemma 2.
Let S be any unitary operator basis for M d ( C ) . There exist a set of L mutually unbiased basesin C d iff the S \ {I} can be partitioned into L mutuallydisjoint maximal commuting classes. Furthermore, theMUBs are simply realized as the common eigenvectors ofthe different maximal commuting operator classes. Since the maximum number of such classes that canbe formed in d -dimensions is d + 1, it follows that thenumber of MUBs in C d is at most d + 1. This bound issaturated for prime power dimensions [2].A simple example of such a unitary operator basis S isthe one comprising of products of the generalized Paulioperators acting on C d , which are defined as: X d | j (cid:105) = | ( j + 1) mod d (cid:105)Z d | j (cid:105) = ω j | j (cid:105) , (2)where ω = e πid . We will in fact make use of the uni-tary basis generated by the generalized Paulis in prime-dimensions for our construction of unextendible sets. A. Unextendible sets of MUBs and MaximalCommuting Operator Classes
We now proceed to formally define the notion of un-extendibility of MUBs, and the related notion of unex-tendible sets of operator classes.
Definition 3 (Unextendible Sets of MUBs) . A set ofMUBs {B , B , . . . , B L } in C d is said to be unextendible ifthere does not exist another basis in C d which is unbiasedwith respect to all the bases in the set. For example, in dimension d = 6, the eigenbases of X , Z and X Z were shown to be an unextendible set ofMUBs [16]. This has the important consequence that theeigenbases of Weyl-Hiesenberg generators will not leadto a complete set of 7 MUBs in d = 6. In fact, severaldistinct families of unextendible triplets of MUBs havebeen constructed in d = 6 [17–19]. Moving away fromsix dimensions, the set of three MUBs obtained in d = 4using Mutually Orthogonal Latin Squares (MOLS) [24]is an example of an unextendible set of MUBs in prime-power dimensions [20].If there does not exist any vector v ∈ C d that is unbi-ased with respect to the MUBs {B , B , . . . , B L } in C d ,then the set of MUBs are said to be strongly unextendible .It has been shown that the eigenbases of X , Z and X Z are in fact strongly unextendible [16].A possible approach to constructing such unextendiblesets of MUBs is to start with maximal commuting classesof operators which are unextendible in the followingsense. Definition 4 (Unextendible Sets of Operator Classes) . A set of mutually disjoint maximal commuting classes C , C , . . . , C L of operators drawn from a unitary basis S is said to be unextendible if no other maximal class canbe formed out of the remaining operators in S \ ( { I } ∪ (cid:83) Li =1 C i )The eigenbases {B , B , . . . , B L } of the operator classes {C , C , . . . , C L ⊂ S} form a set of L weakly unextendibleMUBs in the following sense : There does not existanother basis unbiased with respect to {B , B , . . . , B L } that can be obtained as the common eigenbasis of a max-imal commuting class of operators in S .For example, consider the space C = C ⊗ C . ThePauli operators X , Z , Y = i X Z and their tensorproducts give rise to a set of 16 orthogonal two-qubitunitaries, including the identity operator I . It is knownthese can be partitioned into a set of five mutually dis-joint maximal commuting classes [14, 15] as for example, S = {Z ⊗ I , I ⊗ Z , Z ⊗ Z }S = {X ⊗ I , I ⊗ X , X ⊗ X }S = {X ⊗ Z , Z ⊗ Y , Y ⊗ X }S = {Y ⊗ I , I ⊗ Y , Y ⊗ Y }S = {Y ⊗ Z , Z ⊗ X , X ⊗ Y } , (3)thus giving rise to a set of five MUBS in d = 4.Suppose we consider the following set of operatorclasses instead: C = {Y ⊗ Y , I ⊗ Y , Y ⊗ I }C = {Y ⊗ Z , Z ⊗ X , X ⊗ Y }C = {X ⊗ I , I ⊗ Z , X ⊗ Z } . (4)The above partitioning makes use of just 9 of the 15 pos-sible two-qubit Pauli operators. It is easy to see that thispartitioning gives rise to an unextendible set of classes,since is not possible to form another maximal commutingclass from the remaining six operators: {I ⊗ X , X ⊗ X , Y ⊗ X Z ⊗ I , Z ⊗ Y , Z ⊗ Z } . The common eigenbases of C , C , C constitute a setof three weakly unextendible MUBs, as defined above.A systematic construction of such unextendible sets ofclasses in d = 2 , was obtained recently [21], and thecorresponding MUBs were shown to be strongly unex-tendible. II. UNEXTENDIBLE SETS OF CLASSES INPRIME-SQUARED DIMENSIONS
Here we examine whether it is possible to obtain a gen-eral construction of unextendible operator classes leadingto unextendible MUBs in prime-power dimensions. Theunitary basis of interest here is the one generated by ten-sor products of the generalized Paulis X p and Z p actingon a quantum systems of prime dimensions p as specifiedin Eq. (2): X p | j (cid:105) = | ( j + 1) mod p (cid:105) , Z p | j (cid:105) = ω j | j (cid:105) , ω = e πip . In particular, restricting our attention to prime-squareddimensions ( d = p , p is prime), we consider the unitaryoperator basis U ( p ) comprising operators of the form U = ( X p ) m ( Z p ) n ⊗ ( X p ) k ( Z p ) l , m, n, k, l ∈ F p , where F p is the prime field of order p . Every U ∈ U ( p ) satisfies ( U ) p = I p , where I p denotes the identity op-erator in the p -dimensional space. We show via explicitconstruction that using the operators in U ( p ) \{I p } it isindeed possible to construct unextendible sets of operatorclasses of cardinalities N ( p ) = p − p + 1 , p − p + 2in prime-squared dimensions. A. Structure of operator classes in d = p Our construction primarily relies on the properties ofmaximal commuting classes constructed out of operatorsin U ( p ) \ {I p } . We first list some of these propertiesand prove a few simple consequences of these properties,which are useful for our construction. Property 1.
Every maximal commuting class C of op-erators in U ( p ) is generated by a set of p + 1 independentoperators { U , U , U U , U U , . . . , U p − U } ∈ U ( p ) . Note that U and U are said to be independent if theredo not exist k, l ∈ F p such that U k = U l . To verify theabove property, we first observe that[ U , U ] = 0 ⇒ [ U k , U l ] = 0 , ∀ k, l ∈ F p . Therefore, if U , U ∈ C , then U k U l ∈ C ∀ k, l ∈ F p . Fur-thermore, since U p = U p = I p , this implies a cardinalityof p − C , as desired.It is easy to see that a maximal commuting class wedo not need more than two operators U , U to uniquelycharacterize a maximal commuting class. Suppose thereexists V ∈ C such that V (cid:54) = U k U l ∀ k, l ∈ F p . Sinceall integer powers modulo p of U , U , V would also com-mute, this would imply the class C is of cardinality p − d = p .We will often refer to such a set { U , U , U U , U U , . . . , U p − U } of p + 1 indepen-dent operators that give rise to a class C , as the generators of the class C . Furthermore, since theclass is completely determined once we pick a pair ofindependent, commuting operators { U , U } , we canrepresent C in terms of a pair of generators as follows: C ≡ (cid:104) U , U (cid:105) . Property 2.
Every operator in a class commutes withexactly p − operators from another class.Proof. Consider a pair of mutually disjoint maximalcommuting classes C , C with generators { U , U } and { V , V } respectively: C ≡ (cid:104) U , U (cid:105) , C ≡ (cid:104) V , V (cid:105) . We first show that any operator in C must commutewith atleast one operator in C . Suppose U ∈ C doesnot commute with any operator in C . Then the followingcommutation relations hold.[ U , V ] (cid:54) = 0 ⇒ U V = αV U , where α is a p th root of unity. Similarly,[ U , V ] (cid:54) = 0 ⇒ U V = α j V U , where j ∈ F p , and j (cid:54) = 0. These relations imply that, U ( V k V ) = α k + j ( V k V ) U ∀ k ∈ F p . If U does not commute with any element of C , we re-quire k + j (cid:54) = 0 mod p ∀ k ∈ F p . This is not possible if j (cid:54) = 0. Hence U ∈ C must commute with at least oneoperator in C .We may therefore assume without loss of generalitythat [ U , V ] = 0. This in turn implies that [ U , V k ] =0 ∀ k ∈ F p . Hence, U ∈ C commutes with p − C .Finally, we show that an operators in C cannot com-mute with more than p − U were to commute with another operator, say V in C . This would imply [ U , V k V l ] = 0 ∀ k, l ∈ F p . Inother words we would have an operator U ∈ C whichcommutes with all operators in C ! But this would giverise to a set of commuting operators with cardinalitygreater p −
1, which is not possible for operators on a p -dimensional space. Therefore, no operator can commutewith more than p − U in a given class C commutes with exactly one independent operator V inanother class C (cid:48) – the remaining ( p −
2) commuting op-erators in C (cid:48) are just powers of V . We also note twoadditional properties, which follow directly from Prop-erty 2. Property 3. If U ∈ C , and V ∈ C such that [ U , V ] = 0 , then the operators U k V and U l V with k (cid:54) = l ∈ F p must necessarily belong to different classes.Proof. Note that [ U , V ] = 0 implies [ U , ( U k V ) j ] =0 , ∀ k, j ∈ F p . Therefore, if U k V and U l V with k (cid:54) = l were to belong to the same class C , the operator U ∈ C would commute with 2( p −
1) operators in the class C ,in violation of Property 2. Hence, U k V and U l V mustnecessarily belong to two different classes. Property 4.
Given two classes C ≡ (cid:104) U , U (cid:105) , C ≡ (cid:104) V , V (cid:105) , such that, [ U , V ] = [ U , V ] = 0 , [ U , V ] (cid:54) = 0 , [ V , U ] (cid:54) = 0 . (5) Then, for a given l ∈ F p , there exists a unique m ∈ F p such that [ U l V , U m V ] = 0 .Proof. Let ω = e πi/p . Since [ V , U ] (cid:54) = 0, we may assumewithout loss of generality, that U V = αV U , α ∈ { ω, ω , . . . , ω p − } ,V U = α k U V , k ∈ F p . Then we have,( U l V )( U m V ) = α ( km + l ) ( U m V )( U l V ) . Thus, for the operators U l V and U m V to commute, werequire, km + l = 0 mod p .Finally, for a given pair k, l ∈ F p , we need to show thatthis expression holds true for a unique m ∈ F p . Let usassume that for a given l , m is not unique. Thereforewe have, km + l = km + l = 0 mod p . Hence m = m mod p , a contradiction, since m , m ∈ F p . Therefore m = m = m ∈ F p . B. Existence and Construction of UnextendibleMUBs
Having noted a few basic properties of operator classesin prime-squared dimensions, we now proceed to discussour construction of unextendible sets of classes in thesedimensions. Our starting point will be a a partitioningof the unitary basis U ( p ) into a complete set of p + 1mutually disjoint maximal commuting classes in d = p .We know that such a complete set always exists in prime-power dimensions from Lemma 2 and the earlier resultsWootters and Fields [2]. Starting with a complete set, weseek to identify subsets of classes whose elements mightbe used to construct newer classes. We first note that itsuffices to restrict our attention to subsets of classes ofcardinality p + 1. Lemma 5.
Given a partitioning of the unitary basis U ( p ) into a complete set of p + 1 classes, to form a newmaximal commuting class of operators from U ( p ) , we re-quire operators from exactly p + 1 of the p + 1 classes.Proof. Consider a complete set Σ ≡ {C , C , . . . , C p +1 } ofmaximal commuting classes in dimension d = p . Nowconsider any new maximal commuting class C I = (cid:104) U , V (cid:105) not belonging to Σ. We know from 1 that there are p + 1independent operators characterizing the new class C I .These p + 1 operators must surely come from the classesbelonging to Σ since it is a complete partitioning of theunitary operators in U p . We also know from 2 that everyoperators in a class commutes with only one independent operators in a different class, and so each of the p + 1generators of C I must come from different maximal com-muting classes belonging to Σ. Thus the new class C I isnecessarily formed by picking ( p −
1) operators each from p + 1 classes.More specifically, using 4 and the commutation re-lations in Eq. (5), we can pick the p − C I as follows: U ∈ C , V ∈ C , U V ∈ C , . . . , U p − V ∈ C p +1 , where C , C , . . . , C p +1 ∈ Σ.Suppose we do identify a set of p + 1 classes {C , C , . . . , C p +1 } that belong to a complete set of classes,such that a new class C I can be formed using the opera-tors in ∪ p +1 i =1 C i , is it possible to form more classes usingthe same set of p + 1 classes? This is answered in thefollowing lemma. We merely state the result here andrefer to the appendix B for the proof. Lemma 6.
No more than two new operator classes canbe constructed from a set of p + 1 classes belonging to acomplete set of classes. Once we have such a bound on the number of newclasses that can be formed from a subset of the completeset if classes, we are naturally led to the following state-ment on the existence of unextendible sets of classes.
Theorem 7.
In dimensions d = p , there exist unex-tendible sets of classes of cardinality N ( p ) = p − p + 1 or N ( p ) = p − p + 2 , the common eigenbases of whichform weakly unextendible sets of N ( p ) = p − p + 1 or N ( p ) = p − p + 2 MUBs.Proof.
Properties 5 and 6 imply that we can form ei-ther 0, 1 or 2 new classes using a set of p + 1 classesbelonging to the complete set. Let {C , C , . . . , C p +1 } be a set of p + 1 classes such that exactly one newclass can be formed using the operators in ∪ p +1 i =1 . Thisnew classes, together with the remaining set of p − p classes ( {C p +2 , . . . , C p +1 } ) is an unextendible set of p − p + 1 classes. Suppose {C , C , . . . , C p +1 } were aset of p + 1 classes such that exactly 2 new classes canbe formed using the operators in ∪ p +1 i =1 . Then, thesenew classes, together with the remaining set of p − p classes ( {C p +2 , . . . , C p +1 } ) form an unextendible set of N ( p ) = p − p + 2 classes.For example, consider the case of p = 3, d = 3 . In3 -dimensions, we can construct an unextendible set ofeight classes as illustrated in Fig. 1. Starting with a pairof classes C , C , we pick the remaining set of 7 classesrequired to form a complete set, in such a way that thereexist two new classes C I , C II in ∪ i =1 C i . Then, C I , C II along with {C , . . . , C } is an unextendible set of eightclasses. C. The case of p = 3 We can further restrict the cardinality of the unex-tendible sets in dimension d = 3 , using certain addi-tional properties which hold in this case. We state andprove these additional properties in Appendix C, leadingto the following result. Theorem 8. In d = 3 , consider a set of four classes C , C , C and C that belongs to a complete set of classes.If one new class can be constructed using the operatorsin ∪ i =1 C i , then, it is possible to construct one more classusing the same set of four classes. Therefore, the car-dinality of an unextendible set of classes N (3) (cid:54) = 7 in d = 3 . In other words, either (a) it is possible to find exactlytwo more classes C I , C II using the operators in ∪ i =1 C i giv-ing rise to an unextendible set of eight classes (as shownin Fig. 1), or, (b) no new classes can be formed using theoperators in ∪ i =1 C i .We further show that if a set of four classes C , C , C , C belongs to an unextendible set of 8 classes in d = 3 , then,it is possible to form exactly one more class C III usingthe operators in ∪ i =1 C i . This implies the existence of anunextendible set of 5 classes in d = 3 , as shown in Fig. 2. III. TIGHTNESS OF EURS INPRIME-SQUARED DIMENSIONS C I C II C C C C C C C C C C bbbb bbbb b bb bb bb bbbbbbb bbbbbb bbbbbb bbbbbb (a) Complete set of classes C II C C C C C C b b b bbbbbbbb bbbbbbb bb bbb bb bb bb bbb C I (b) Unextendible set ofclasses FIG. 1: Construction of unextendible set of 8 classes in d = 3 . Dots represent operators and horizontal linesrepresent classes. Vertical lines represent new classesconstructed using existing classes. C I and C II togetherwith C − C forms an unextendible set of 8 classes. C I C II C C C bbbbb C A C C C b bb b bbbbb bb b bbbbb b b bbbbbbbbb (a) Unextendible set of 8classes C I C A C C C bbbbbb bb b bbbbb bb b bbbb (b) Unextendible set of 5classes FIG. 2: Construction of unextendible set of 5 classes in d = 3 using operators from the unextendible set of 8classes. Dots represent operators and horizontal linesrepresent classes. Vertical lines represent new classesconstructed using existing classes. C I and C A togetherwith C − C forms an unextendible set of 5 classes.In this section, we describe an interesting connectionbetween the existence of unextendible sets of classes inquantum systems of prime-squared dimensions and thetightness of two well known EURs in these dimensions.Indeed, our approach of constructing unextendible setsof classes leads to a systematic way of identifying MUBsthat saturate both a Shannon ( H ) entropic and a H -entropic lower bound in prime-squared dimensions.An entropic uncertainty relation is a lower bound onthe sum of the entropies associated with a set of mea-surement bases, measured independently on identicallyprepared copies of a quantum state. Recall that a mea-surement of basis B ( i ) ≡ {| b ( i ) j (cid:105)} in state | ψ (cid:105) induces aprobability distribution p ( i ) ( j ) | ψ (cid:105) = |(cid:104) b ( i ) j | ψ (cid:105)| . Since theentropy is a measure of the spread of the distribution, wemay use any valid entropic function H ( { p ( i ) ( j ) | ψ (cid:105) ) } toquantify the uncertainty in the outcome of the measure-ment B ( i ) in state | ψ (cid:105) .Here, we focus on two of the R´enyi class of en-tropies [25], namely the Shannon entropy ( H , the R´enyientropy of order 1) and the collision entropy ( H , theR´enyi entropy of order 2), defined as: H ( { p ( i ) } ) := − (cid:88) i p ( i ) log p ( i ) ,H ( { p ( i ) } ) := − log (cid:88) i p ( i ) . (6)We denote the entropies associated with a measurementof basis B ( i ) on state | ψ (cid:105) as H α ( B ( i ) || ψ (cid:105) ) , α = 1 ,
2. It isa well known that a pair of measurement bases B (1) , B (2) in a d -dimensional system satisfy the following Shannonentropic uncertainty bound [26]:12 (cid:104) H ( B (1) ; | ψ (cid:105) ) + H ( B (2) ; | ψ (cid:105) ) (cid:105) ≥
12 log d, ∀| ψ (cid:105) . (7)Furthermore, this bound is saturated iff the bases are mu-tually unbiased. As a trivial consequence of the aboverelation, we have a bound on the average Shannon en-tropy of any set of L MUBs in d -dimensions (previouslynoted in [13, 27]):1 L L (cid:88) i =1 H ( B ( i ) ; | ψ (cid:105) ) ≥
12 log d, ∀| ψ (cid:105) . (8)It is known that there exist incomplete set of MUBsthat saturate this weak entropic bound. In particular, itis known [13] that a set of s + 1 MUBs in square dimen-sions d = s which are realized by the action of productunitary operators on the computational basis saturatethe bound in Eq. (8).Our construction of unextendible classes provides analternate way of identifying sets of p + 1 MUBs that sat-urate the EUR in Eq. (8), in prime-square dimensions( d = p ). Furthermore, we also show that the same setof p + 1 MUBs in d = p also saturates the following H entropic entropic relation, which was shown to hold forany set of L MUBs in d -dimensions [28, 29],1 L L (cid:88) i =1 H ( B ( i ) ; | ψ (cid:105) ) ≥ − log (cid:18) L + d − Ld (cid:19) . (9) Indeed, the two lower bounds in Eqs. (8) and (9) coincidefor the case of L = p + 1 MUBs in d = p . Theorem 9.
In dimension d = p , consider a set of ( p +1) classes {C , C , . . . , C p +1 } , such that at least one moremaximal commuting class C I can be constructed by pick-ing ( p − operators (of the form U, U , . . . , U p − ) fromeach of the classes. Then, the MUBs {B , B , . . . , B p +1 } corresponding to the classes {C , C , . . . , C p +1 } saturate the following entropic uncertainty relations: p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) ≥ log p, p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) ≥ log p, (10) with the lower bound attained by the common eigenstatesof the newly constructed class C I . We provide the proof for this theorem in the appendix(D). We may note that the states saturating the bound inEq. (10) are indeed states that look alike [30] with respectto each of the bases in the set {B , B , . . . , B p +1 } . IV. CONCLUDING REMARKS
We show by explicit construction the existence ofweakly unextendible sets of MUBs of cardinalities N ( p ) = p − p + 1 , p − p + 2 in prime squared ( d = p )dimensions. Our construction is based on grouping thegeneralized Pauli operators in these dimensions into setsof mutually disjoint, maximal commuting classes that areunextendible to a complete set of ( d + 1) classes. We fur-ther demonstrate a general connection between the exis-tence of unextendible sets and the tightness of entropicuncertainty relations for the H and H R´enyi entropies.Numerical evidence suggests that the MUBs obtainedin our construction are in fact strongly unextendible.This is also borne out by a recent construction of un-extendible sets of MUBs by exploring the connection be-tween MUBs and complementary decompositions of max-imal abelian subalgebras [31]. Finally, it remains an in-teresting question to determine the cardinality of unex-tendible sets in prime-power dimensions ( d = p n ). 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Appendix A: Unextendible set of operator classesin d = 3 Here is an example of a complete set of classes in d = 3 dimensions: C = (cid:8) I ⊗ X Z , I ⊗ X Z , X Z ⊗ I, X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ I, X Z ⊗ X Z , X Z ⊗ X Z (cid:9) C = (cid:8) Z ⊗ X Z , Z ⊗ X Z , X ⊗ X , X Z ⊗ X Z , X Z ⊗ Z , X ⊗ X , X Z ⊗ Z , X Z ⊗ X Z (cid:9) C = (cid:8) Z ⊗ Z , Z ⊗ Z , X ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z (cid:9) C = (cid:8) I ⊗ Z , I ⊗ Z , X ⊗ I, X ⊗ Z , X ⊗ Z , X ⊗ I, X ⊗ Z , X ⊗ Z (cid:9) C = (cid:8) Z ⊗ X Z , Z ⊗ X Z , X ⊗ X , X Z ⊗ X Z , X Z ⊗ Z , X ⊗ X , X Z ⊗ Z , X Z ⊗ X Z (cid:9) C = (cid:8) I ⊗ X , I ⊗ X , X Z ⊗ I, X Z ⊗ X , X Z ⊗ X , X Z ⊗ I, X Z ⊗ X , X Z ⊗ X (cid:9) C = (cid:8) I ⊗ X Z , I ⊗ X Z , Z ⊗ I, Z ⊗ X Z , Z ⊗ X Z , Z ⊗ I, Z ⊗ X Z , Z ⊗ X Z (cid:9) C = (cid:8) Z ⊗ Z , Z ⊗ Z , X ⊗ X Z , X Z ⊗ X , X Z ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X (cid:9) C = (cid:8) Z ⊗ X , Z ⊗ X , X ⊗ X Z , X Z ⊗ Z , X Z ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ Z (cid:9) C = (cid:8) Z ⊗ X , Z ⊗ X , X ⊗ X Z , X Z ⊗ Z , X Z ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ Z (cid:9) We see that from C , C , C , C , we can form two new classes: C I = (cid:8) I ⊗ X Z , I ⊗ X Z , Z ⊗ X Z , Z ⊗ X Z , Z ⊗ X Z , Z ⊗ X Z , Z ⊗ I, Z ⊗ I (cid:9) C II = (cid:8) X Z ⊗ I, X Z ⊗ I, X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , I ⊗ X Z , I ⊗ X Z (cid:9) Therefore C I , C II , C , C , C , C , C , C form a set of 8 unextendible maximal commuting classes.We make use operators from 4 of the above 8 classes (Where either C I or C II are necessarily used) to form a newclass. This new class along with the remaining set of 4 classes form a set of 5 classes which are unextendible to acomplete set of classes. In particular, we make use of the classes C I , C , C and C to form a new class C A given by: C A = (cid:8) I ⊗ Z , Z ⊗ I, Z ⊗ Z , Z ⊗ Z , I ⊗ Z , Z ⊗ I, Z ⊗ Z , Z ⊗ Z (cid:9) The classes C II , C , C , C and C A form a set of 5 classes which are unextendible to a complete set of classes as givenbelow: C II = (cid:8) X Z ⊗ I, X Z ⊗ I, X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , X Z ⊗ X Z , I ⊗ X Z , I ⊗ X Z (cid:9) C = (cid:8) I ⊗ X , I ⊗ X , X Z ⊗ I, X Z ⊗ X , X Z ⊗ X , X Z ⊗ I, X Z ⊗ X , X Z ⊗ X (cid:9) C = (cid:8) Z ⊗ X , Z ⊗ X , X ⊗ X Z , X Z ⊗ Z , X Z ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ Z (cid:9) C = (cid:8) Z ⊗ X , Z ⊗ X , X ⊗ X Z , X Z ⊗ Z , X Z ⊗ X Z , X ⊗ X Z , X Z ⊗ X Z , X Z ⊗ Z (cid:9) C A = (cid:8) I ⊗ Z , Z ⊗ I, Z ⊗ Z , Z ⊗ Z , I ⊗ Z , Z ⊗ I, Z ⊗ Z , Z ⊗ Z (cid:9) Appendix B: Construction of unextendible sets ofoperator classes in d = p Lemma 10 (6) . No more than classes can be con-structed using a set of p + 1 classes belonging to a com-plete set of classes.Proof. Consider a set of p +1 classes C , C , . . . , C p +1 usingthe elements of which, two more classes C I , C II can beformed. Without loss of generality, we may assume the p + 1 classes are of the following form: the classes C and C are first defined as C ≡ (cid:104) U , U (cid:105) and C ≡ (cid:104) V , V (cid:105) ,with generators that satisfy the commutation relations inEq. (5); the remaining p − C , C , as: C j ≡ (cid:104) U j − V , U j − V (cid:105) , j = 3 , . . . , p + 1 . The two new classes C I and C II are then given by C I ≡ (cid:104) U , V (cid:105) , C II ≡ (cid:104) U , V (cid:105) Let us now assume that we can form a third class C III using the operators in ∪ p +1 i =1 C i . We can assume withoutloss of generality that the generators of C III are of theform, U l U and V m V . Since they commute, we have km + l = 0 mod p . We now need ( U l U ) r V m V ∈ C j forsome j ∈ { , , . . . , p + 1 } and l, r, m ∈ { , , . . . , p − } .As per our assumptions, the generators of the class C j are U j − V and U s V , where s of course depends on thevalue of j and s ∈ { , , . . . , p − } .We only need to match the exponents of the individualcomponents, that is, we need: U lr U r V m V = ( U t V ) m U s V = U tm U s V m V , where we have defined t = j −
2, so that t ∈ { , , . . . , p − } . This in turn implies that lr = tm mod p and r = s mod p , which holds only if, • Case I: r = t = s and l = m , or • Case II: r = m = s and l = t Consider Case I: We know that U t V and U s V commute.This immediately means that ks + t = 0 mod p . Since k (cid:54) = 0, t = s iff s = t = 0, which is a contradiction because we know that s, t ∈ { , , . . . , p − } . Hence CaseI is not possible.Consider Case II: This would require [ U l V , U m V ] = 0and [ U l U , V m V ] = 0. The latter commutation relationtranslates to ( p − k ) m + l = 0 mod p , while the for-mer translates to km + l = 0 mod p , which is possibleif and only if (a) p is an even number, which leads acontradiction because we only consider primes p >
2, or,(b) if m = l = 0, which is again not possible because m, l ∈ { , , . . . , p − } . Hence Case II is not possible.This proves that we cannot form a third class fromthe set of p + 1 classes mentioned before, for all prime p > p = 2, Case II is possible and hence an-other class can be formed, as already noted in [21]. Appendix C: The case of d = 3 We first observe two properties of a complete set of p + 1 classes in d = p , which are not specific to p = 3. Property 5.
Let C , C , . . . , C p +1 be a set of p + 1 classesout of a complete set of classes in d = p , such that a newclass C I ≡ (cid:104) U , V (cid:105) can be formed using U ∈ C , V ∈ C , U V ∈ C , etc. Then, there exist ˜ U ∈ C and ˜ V ∈ C such that ˜ U ˜ V ∈ C .Proof. Let C ≡ (cid:104) U , U (cid:105) and C ≡ (cid:104) V , V (cid:105) , with the gen-erators satisfying the commutation relations in Eq. (5).We are given that U V ∈ C . We need to find anotherindependent operator in C so as to completely generate C .Starting with any operator of the form W = U a V b U c V d , we have the following constraints on a, b, c, d ∈ F p so that W is a valid operator in C , giventhe classes C and C .(i) [ U , U a V b U c V d ] = 0. We impose this conditionsince we need to have an operator in C commutingwith U . This implies that b = 0 mod p . Theoperator W is thus of the form U a U c V d .(ii) [ U V , U a U c V d ] = 0. Therefore ( p − c + d =0 mod p . Hence c = d mod p . The operator W isthus of the form U a U c V c .If W = U a U c V c is indeed a generator of C all productsof U V and U a U c V c must belong in C . Hence( U a U c V c ) ( U V ) p − a = U a V p − a U c V c ∈ C . The last operator is obtained as a product of U a U c ∈ C and V p − a V c ∈ C . Therefore, if we let,˜ U := U a U c ˜ V := V p − a V c we have ˜ U ˜ V ∈ C as desired. Property 6.
Consider C , C , . . . , C p +1 to be part of acomplete set of classes in d = p system such that at leastone new class can be formed by picking p − operatorsfrom each of these p + 1 classes. Let the generators of C ≡ (cid:104) U , U (cid:105) and C ≡ (cid:104) V , V (cid:105) , with U V = αV U .Then, we can always redefine U ∈ C such that U V = αV U , where α is a p th root of unity.Proof. Let us suppose that U V = α j V U for some j ∈ F p \ { } . We know that U k ∈ C ∀ k ∈ F p . Therefore U j − ∈ F p . Letting U := U j − , we see that U V = αV U . Theorem (8) . In d = 3 , given C , C , C and C belong-ing to a complete set of classes and using which one newclass can be constructed, it is always possible to constructanother class using the set of classes. In other words,it is not possible to construct N (3) = 3 − unextendible set of classes.Proof. As before, we assume that the classes C , C are ofthe form C ≡ (cid:104) U , U (cid:105) , C ≡ (cid:104) V , V (cid:105) . Further, we assume that U V ∈ C and U V ∈ C , im-plying the existence of one more class C I ≡ (cid:104) U , V (cid:105) .We also know from properties 5 and 6 that the gener-ators of C , C can be chosen to satisfy U V = αV U , U V = αV U , such that U V ∈ C .Let us now assume that it is not possible to find onemore class using operators in the set {C , C , C , C } . Thiswould imply that U V / ∈ C , and therefore, U V U V / ∈C . Since we are given that the set {C , C , C , C } is parta complete set of classes, the operator U V U V mustbelong some other class, say, C n , ≤ n ≤ p (
2) + 1.We then show that it is not possible to complete theclass C n , in such a way that it is mutually disjoint with {C , C , C , C } . We first note that a general operator G = U a V b U c V d ∈ C n must satisfy the following conditions: (cid:2) U V U V , U a V b U c V d (cid:3) = 0 , (C1) (cid:2) U V , U a V b U c V d (cid:3) (cid:54) = 0 (C2) (cid:2) U V , U a V b U c V d (cid:3) (cid:54) = 0 (C3) (cid:2) U U , U a V b U c V d (cid:3) (cid:54) = 0 (C4) (cid:2) V V , U a V b U c V d (cid:3) (cid:54) = 0 (C5) Eq.(1) along with inequalities (2) and (3) has two pos-sible solutions: (i) b = 2 a + 1 and d = 2 c + 2, or, (ii) b = 2 a + 2 and d = 2 c + 1.Option (i) along with inequalities (4),(5) above implythat c = ( a + 2)mod3. This implies that the secondgenerator G of the class C n must be of the following form: U a V a +11 U a +22 V a . Substituting different values of a ∈ F , we get, G ∈ { U V , V U , U V U V } . We We note that option (ii), along with inequalities (4),(5) do not give any further solutions: they simply yieldthe squares of these operators.It is easy to see that the set of solutions for G , alongwith the operator U V U V is closed under multiplica-tion. Therefore, all three possible solutions for the secondgenerator lead to the same class: C n ≡ (cid:104) U V U V , U V U V (cid:105) . However, we have chosen the operators U , V that U V U V ∈ C . Thus the class C n containing the opera-tor U V U V cannot be mutually disjoint with the initialset of three classes. Therefore, either U V U V ∈ C ,leading to a second class C II ≡ (cid:104) U , V (cid:105) , or, C , C , C , C cannot be extended to a complete set of p +1 classes. Appendix D: Proof of tightness of EURs
A key ingredient of the proof pf Theorem 9 is a pa-rameterization of the basis-vectors of the bases {B i } cor-responding to classes {C i } , in terms of vectors in a p + 1-dimensional vector space over the field F p . We first indexthe operators in class C i as follows: starting with a pairof generators σ (1) i , σ (2) i , we may obtain a set of p + 1 in-dependent operators as products of these two generators: σ ( k ) i = ( σ (1) i ) k − ( σ (2) i ) ∀ k ∈ [3 , p + 1] . The remaining operators in C i are simply powers of σ ( k ) i , k ∈ [1 , p + 1]. A general operator in C i is thus de-noted as ( σ ( k ) i ) j , j ∈ F p .Consider an operator in the span of the operators con-stituting the class C i , of the following form: ρ x i = Ip + p +1 (cid:88) l =1 (cid:104) ω x l σ ( l ) i + ω x l ( σ ( l ) i ) (cid:105) + . . . + p +1 (cid:88) l =1 (cid:104) ω ( p − x l ( σ ( l ) i ) p − (cid:105) , (D1)where, x ≡ ( x , x , . . . , x p +1 ) ∈ F p +1 p , with x k = ( k − x + x mod p, k ∈ [3 , p + 1] . Clearly,
T r ( ρ x i ) = 1, and, ( ρ x i ) † = ρ x i . The latter followsfrom the fact that,( σ ( l ) i ) † = ( σ ( l ) i ) p +1 . T r (( ρ x i ) ) = 1: T r (( ρ x i ) ) = 1 p T r ( I + p +1 (cid:88) l =1 p − (cid:88) k =1 I ) . = 1 p T r ( I + ( p − I ) = 1 p T r ( I ) = 1 . Finally, the following lemma proves that ρ x i is in facta rank-1 projector. Lemma 11.
The operator defined in Eq. (D1) satisfies ( ρ x i ) = ρ x i .Proof. ( ρ x i ) = 1 p ( ρ x i + ( ω x σ (1) i ρ x i + ω x σ (2) i ρ x i + . . . + ω x p +1 σ ( p +1) i ρ x i )+ ( ω x ( σ i ) (2) ρ x i + ω x ( σ (2) i ) ρ x i + . . . + ω x p +1 ( σ ( p +1) i ) ρ x i )+ . . . + ( ω ( p − x ( σ (1) i ) p − ρ x i + ω ( p − x ( σ (2) i ) p − ρ x i + . . . + ω ( p − x p +1 ( σ ( p +1) i ) p − ρ x i )) . (D2)We require that ( ρ x i ) = 1 p ( p ρ x i ) = ρ x i . This would imply that we need each term of the summa-tion in Eq. (D2) be ρ x i . We show that this is indeed thecase, since ω ax b ( σ ( b ) i ) a ρ x i = ρ x i ∀ a ∈ F p , ∀ b ∈ [1 , p + 1] . To see this, consider a generic term in the operator sum ω ax b ( σ bi ) a ρ x i of the form, ω ax b ( σ ( b ) i ) a × ω cx d ( σ ( d ) i ) c = ω ax b + cx d ( σ ( b ) i ) a ( σ ( d ) i ) c . We have the following: ax b + cx d = a (( b − − δ b, )) x + (1 − δ b, ) x ) + c (( d − − δ d, )) x + (1 − δ d, ) x ) . = ( ab + cd − a (1 − δ b, ) + c (1 − δ d, ))) x + ( a (1 − δ b, ) + c (1 − δ d, )) x . ( σ ( b ) i ) a ( σ ( d ) i ) c = (( σ i ) ( b − ( σ (2) i ) − δ b, ) a (( σ (1) i ) d − ( σ (2) i ) − δ d, ) c . = ( σ (1) i ) ab + cd − a (1 − δ b, )+ c (1 − δ d, )) ( σ (2) i ) a (1 − δ b, )+ c (1 − δ d, ) . We now have two possibilities:( A) a (1 − δ b, ) + c (1 − δ d, ) = 0 mod p . Therefore ax b + cx d = ( ab + cd ) x = γx , ( σ ( b ) i ) a ( σ ( d ) i ) c = ( σ (1) i ) ab + cd = ( σ (1) i ) γ , where, γ = ( ab + cd ) mod p .( B) a (1 − δ b, ) + c (1 − δ d, ) (cid:54) = 0 mod p , in which case,( a (1 − δ b, ) + c (1 − δ d, )) − exists. Therefore wehave, ax b + cx d = ( a (1 − δ b, ) + c (1 − δ d, ))((( ab + cd )( a (1 − δ b, ) + c (1 − δ d, )) − − x + x ) . = γx β . ( σ ( b ) i ) a ( σ ( d ) i ) c = (( σ (1) i ) β − σ (2) i ) γ = ( σ ( βi )) γ where γ = ( a (1 − δ b, ) + c (1 − δ d, )) mod p and β = ( ab + cd )( a (1 − δ b, )+ c (1 − δ d, )) − mod ( p +1).To summarize, for a, c ∈ F p and b, d ∈ [1 , p + 1] we haveshown, ω ax b ( σ ( b ) i ) a × ω cx d ( σ ( d ) i ) c = ω γx β ( σ ( β ) i ) γ , for some γ ∈ F p and β ∈ [1 , p + 1]. Thus, ω γx β ( σ ( β ) i ) γ isone of the terms that occurs when ρ x i is expanded in the { σ ( i ) } operator basis. Every term in the operator expan-sion for the product of ρ x i with ω ax b ( σ ( b ) i ) a gives a uniqueterm in the operator expansion of ρ x i . We therefore con-clude that ω ax b ( σ ( b ) i ) a ρ x i = ρ x i ∀ a ∈ F p , ∀ b ∈ [1 , p +1]. ρ x i is therefore a valid pure state in the span of theoperators belonging to the i th class. We may therefore1rewrite the state as ρ x i = | b x i (cid:105)(cid:104) b x i | , where, | b x i (cid:105) is a com-mon eigenvector of the operators belonging to the class C i . Recall that the vector x ∈ F p +1 p is determined whenwe pick a pair of elements x , x ∈ F p . Since x and x can each take on p values, we can find p such pure statesin the space of the operators belonging to a given class C i .Furthermore, we show that these p states are indeed or-thogonal to each other, thus constituting an orthonormalbasis for the class C i .Consider states | b x i (cid:105) , | b y i (cid:105) , with x (cid:54) = y . Then,tr[ | b x i (cid:105)(cid:104) b x i | b y i (cid:105)(cid:104) b y i | ]= 1 p + 1 p p +1 (cid:88) l =1 [ ω x l +( p − y l + ω x l +( p − y l ]+ 1 p [ . . . + p +1 (cid:88) l =1 ω ( p − x l + y l ] . (D3)For the RHS to vanish, we require that x l = y l for oneand only one l ∈ { , , . . . , p + 1 } . In other words, thestrings given by x = x x . . . x p +1 and y = y y . . . y p +1 must agree at exactly one position, in order for the cor-responding states | b x i (cid:105) , | b y i (cid:105) to be mutually orthogonal.Given a vector x , there exist exactly p vectors that coin-cide with x at exactly one location. Thus, correspondingto each class C i , we have an orthonormal basis for theentire Hilbert space C p , with basis vectors | b x i (cid:105)(cid:104) b x i | = Ip + 1 p p +1 (cid:88) l =1 (cid:2) ω x l σ li + ω x l ( σ li ) (cid:3) + . . . + 1 p p +1 (cid:88) l =1 ω ( p − x l ( σ li ) p − . (D4)We are now ready to prove Theorem 9, the statement ofwhich we recall here. Theorem (9) . In dimension d = p , consider aset of ( p + 1) classes {C , C , . . . , C p +1 } , such that atleast one more maximal commuting class C I can beconstructed by picking ( p − operators (of the form U, U , . . . , U p − ) from each of the classes. Then, theMUBs {B , B , . . . , B p +1 } corresponding to the classes {C , C , . . . , C p +1 } saturate the following entropic uncer-tainty relations: p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) ≥ log p, (D5)1 p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) ≥ log p, (D6) with the lower bound attained by the common eigenstatesof the newly constructed class C I .Proof. Let C , C , . . . , C p +1 be a set of p + 1 classes suchthat at least one maximal commuting class can be con-structed by picking p − {| b a m i (cid:105)} the states that constitute the basis B i associated with the class C i , using a set of vectors a m ∈ ( F p ) p +1 , m ∈ [1 , p ]. Thus | b a m i (cid:105) is the m th basisvector of the i th basis.Now consider a class constructed from {C , C , . . . , C p +1 } by picking ( p −
1) operators fromeach class. Without loss of generality, we can indexthese operators as { σ ( i ) i , ( σ ( i ) i ) , . . . , ( σ ( i ) i ) p − } , where σ ( i ) i ∈ C i . Further, we parameterize a common eigenstateof this new class C I as follows: | ψ (cid:105)(cid:104) ψ | = Ip + p +1 (cid:88) l =1 (cid:2) ω e l σ ll + ω e l ( σ ll ) (cid:3) + . . . + 1 p ω ( p − e l ( σ ll ) p − . The H entropy of any state is given by: H ( B i || ψ (cid:105) ) = − log (cid:88) x = a , a ,..., a p2 ( T r [ | b x i (cid:105)(cid:104) b x i | ψ (cid:105)(cid:104) ψ | ]) = − log (cid:88) x = a , a ,..., a p2 ( 1 p T r { I + I p +1 (cid:88) l =1 ω x l +( p − e l + ω x l +( p − e l + . . . + ω ( p − x l + e l } ) For a given e and a given l , there are only p vectors x with x l = e l . Therefore we have: H ( B i || ψ (cid:105) ) = − log p × p = log p We see that this is true for any basis B i ∀ i ∈ [1 , , . . . , p +1] corresponding to any of the p + 1 classes which wereused to construct the new class C I . Therefore we have,1 p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) = log p This implies that the H -entropic uncertainty relation inEq. (D5) is tight for these p + 1 measurement bases.We now show that the Shannon entropic uncertaintyrelation in Eq. (D6) is also tight for any prime squareddimensions. The Shannon entropy associated with a mea-surement of B i on state | ψ (cid:105) is given by, H ( B i || ψ (cid:105) ) = − (cid:88) x = a ,a ,...,a p p xi, | ψ (cid:105) log p xi, | ψ (cid:105) , where, the probability p xi, | ψ (cid:105) of obtaining outcome x whenmeasuring basis B i on state | ψ (cid:105) is given by p x i, | ψ (cid:105) = tr[ | b xi (cid:105)(cid:104) b xi | ψ (cid:105)(cid:104) ψ | ]= − p + 1 p tr p +1 (cid:88) l =1 (cid:104) ω x l +( p − e l + ω x l +( p − e l (cid:105) + . . . + 1 p p +1 (cid:88) l =1 ω ( p − x l + e l . l , x l = e l only for p x ∈{ a , a , . . . , a p } . Hence, we have: H ( B j || ψ (cid:105) ) = − p × ( pp log 1 p ) = log p Therefore we have:1 p + 1 p +1 (cid:88) i =1 H ( B i || ψ (cid:105) ) = log p, thus proving that the Shannon uncertainty relation inEq. (D6)is tight for these pp