PPure state ‘really’ informationally complete withrank-1 POVM
Yu Wang , , Yun Shang , Institute of Mathematics, AMSS, CAS, Beijing, 100190, China, University of Chinese Academy of Sciences, Beijing, 100049, China NCMIS, AMSS, CAS, Beijing, 100190, China. [email protected]
Abstract.
What is the minimal number of elements in a rank-1 positive-operator-valued measure (POVM) which can uniquely determine anypure state in d -dimensional Hilbert space H d ? The known result is thatthe number is no less than 3 d −
2. We show that this lower bound is nottight except for d = 2 or 4. Then we give an upper bound of 4 d −
3. For d =2, many rank-1 POVMs with four elements can determine any pure statesin H . For d = 3, we show eight is the minimal number by construction.For d = 4, the minimal number is in the set of { , , , } . We showthat if this number is greater than 10, an unsettled open problem can besolved that three orthonormal bases can not distinguish all pure statesin H . For any dimension d , we construct d + 2 k − H d , where 1 ≤ k ≤ d . Key words:
Quantum state tomography, Pure state, Quantum mea-surement, Rank-1 operators.
One of the central problems in quantum science and technology is the estimationof an unknown quantum state, via the measurements on a large number ofcopies of this state. Quantum state tomography is the process of determining anarbitrary unknown quantum state with appropriate measurement strategies.A quantum state ρ in d -dimensional Hilbert space H d is described by a den-sity matrix, namely by a positive semi-definite, unit-trace d × d matrix as S d . Ageneralized measurement can be described by a positive operator-valued mea-sure (POVM) [1]. The POVM elements, E k , satisfy the completeness condition: (cid:80) k E k = I . Performing this measurement on a system in state ρ , the probabilityof the k -th outcome is given from the Born rule, p k = tr( ρE k ). If the statisticsof the outcome probabilities are sufficient to uniquely determine the state, thePOVM is regarded as informationally complete (IC) [2].The IC-POVM can give a unique identification of an unknown state, whichshould distinguish any pair of different states from the statistics of probabilities.For example, we consider a POVM, { ( | (cid:105)±| (cid:105) )( (cid:104) |±(cid:104) | ) , ( | (cid:105)± i | (cid:105) )( (cid:104) |∓ i (cid:104) | ) } . a r X i v : . [ qu a n t - ph ] N ov It is not an IC-POVM, as the statistics of the outcome probabilities for states {| (cid:105) , | (cid:105)} under this measurement are the same. For any different quantum states ρ , ρ ∈ S d , an IC-POVM should distinguish them from the statistics of theoutcome probabilities. That is to say, we have tr( ρ E k ) (cid:54) = tr( ρ E k ) for someelements E k .We know that a quantum state ρ in H d is specified by d − ρ ) = 1. Caves et al. constructed anIC-POVM which contains the minimal d rank-1 elements [3], i.e., multiples ofprojectors onto pure states. If d + 1 mutually unbiased bases (MUBs) exist in H d , we can construct an IC-POVM with d ( d + 1) elements [4]. MUBs have theproperty that all inner products between projectors of different bases labeled by i and j are equal to 1 /d . Another related topic is the symmetric informationallycomplete positive operator-valued measure (SIC-POVM) [5]. It is comprised of d rank-1 operators. The inner products of all different operators are equal. ThisSIC-POVM appears to exist in many dimensions.For a state ρ in n -qubit system, d = 2 n . Thus the cost of measurement re-source with these measurement strategies grows exponentially with the increaseof number n . It is important to design schemes with lower outcomes to uniquelydetermine the state. This is possible when we consider a priori information aboutthe states to be characterized.Denote the rank of a density matrix for state ρ as k , 1 ≤ k ≤ d . And makea decomposition that S d = ⊕ dk =1 S d,k , where S d,k is the set of all the densitymatrices with rank k . When k = 1, the state in S d, is pure. A pure state isspecified by d complex numbers, which correspond to 2 d real numbers. For thereason of normalization condition and freedom of a global phase, there are 2 d − d elements can not distinguish all pair of different states ρ , ρ in S d, , noteven in a subset ˜ S d, , where S d, \ ˜ S d, is a set of measure zero. They gave adefinition of pure-state informationally complete (PSI-complete) POVM, whoseoutcome probabilities are sufficient to determine any pure states (up to a globalphase), except for a set of pure states that is dense only on a set of measure zero.That is to say, if a pure state was selected at random, then with probability 1 itwould be located in ˜ S d, and be uniquely identified. A PSI-complete POVM with2 d elements is constructed, but not all the elements in this POVM are rank-1.They constructed another PSI-complete POVM with 3 d − d elements.Finkelstein proved this by a precise construction [7]. Moreover, he gave astrengthened definition of PSIR-completeness, which indicates that all purestates are uniquely determined. For any pair of different pure states ρ , ρ ∈ S d, ,a PSIR-complete POVM should distinguish them. He showed that a rank-1PSIR-complete POVM must have at least 3 d − d − ρ , ρ ∈ S d, , Heinosaari, Mazzarella, and Wolf gave the minimal number of POVM elements to identify them [8]. The number is 4 d − − c ( d ) α ( d ), where c ( d ) ∈ [1 ,
2] and α ( d ) is the number of ones appearing in the binary expansion of d −
1; the results in papers [20, 21, 11] showed that four orthonormal bases, cor-responding to four projective measurements, can distinguish all pure states. Forany pair of different states ρ ∈ S d, , ρ ∈ S d , Chen et al. showed that a POVMmust contain at least 5 d − et al. gave five orthonormal bases that are enough to distinguish them[13]. For a statein S d,k , it can be reconstructed with a high probability with rd log ( d ) outcomesvia compressed sensing techniques [14]. Goyeneche et al. [15] constructed fiveorthonormal bases to determine all the coefficients of any unknown input purestates. The first basis is fixed and used to determine a subset s d, ⊂ S d, , wherethe pure state belongs to. The other four bases are used to uniquely determineall the states in s d, .In this paper, we consider the pure-state version of informational complete-ness with rank-1 POVM. Firstly, we show that the lower bound of 3 d − d = 2 and possibly be reachedwhen d = 4. Then we show a result that there exist a large number of rank-1PSIR-complete POVMs with 4 d − d = 2 , ,
4. For dimension d = 2and d = 3, we construct the rank-1 PSIR-complete POVMs with the minimalnumber of elements, which are 4 and 8 correspondingly. All the coefficients ofan unknown pure state in H and H can be calculated by these POVMs. Fordimension d = 4, the minimal number is in the range of { , , , } . If it isbigger than 10, an answer can be given to a related unsolved problem, i.e., threeorthonormal bases can not distinguish all pure states in H . Lastly, we construct d + 2 k − H d , here 1 ≤ k ≤ d . This is an adaptive strategy. For any inputpure state, we use d operators to determine a subset s d, ⊂ S d, , where the purestate belongs to. Together with the other 2 k − s d, . Thus using this adaptive method, any inputpure states can be determined with at most 3 d − In this section, we will give the upper and lower bounds of the minimal numberof elements in a rank-1 PSIR-complete POVM. Denote this minimal number as m ( d ). It is in the range of [4 d − − c ( d ) α ( d ) , d − In this part, we show that a rank-1 PSIR-complete POVM with 3 d − d = 2 or 4. For the other dimensions, any rank-1POVM with 3 d − Definition 1 : (PSI really-completeness [7]). A pure-state informationally re-ally complete POVM on a d -dimensional quantum system H d is a POVM whoseoutcome probabilities are sufficient to uniquely determine any pure state (up toa global phase).As we introduced above, the PSIR-complete POVM can distinguish any pairof different states ρ , ρ ∈ S d, . Neglecting the restriction of rank-1, we denote m ( d ) to be the minimal number of elements in a PSIR-complete POVM. Cer-tainly, a rank-1 PSIR-complete POVM is PSIR-complete. Thus m ( d ) ≥ m ( d ).From the result in [8], m ( d ) = 4 d − − c ( d ) α ( d ), where c ( d ) ∈ [1 ,
2] and α ( d )is the number of ones appearing in the binary expansion of d −
1. From the con-clusion by Finkelstein, m ( d ) ≥ d −
2. But it is not clear when they are equalor whether a greater number than 3 d − m ( d ) and 3 d − f ( d ) = 4 d − − c ( d ) α ( d ) − (3 d − α ( d ), we havelog d ≥ α ( d ). So f ( d ) > d − − d . Define g ( d ) ≡ d − − d . Then g (cid:48) ( d ) = 1 − /d . If d >
2, it holds that g (cid:48) ( d ) >
0. And when d = 8, g (8) = 1 > d ∈ [8 , + ∞ ), m ( d ) > d −
2. When d ∈ [2 , m ( d )is given in [8]. We have m (2 , , , , ,
7) = (4 , , , , , d −
2: (4 , , , , , d = 2 or 4, m ( d ) can be 3 d −
2. For the other dimensions, m ( d ) ≥ m ( d ) > d − d − In this section, we show that 4 d − m ( d ). This up-per bound is given by constructing rank-1 POVMs from the minimal sets oforthonormal bases which can determine all pure states in H d . Definition 2 : Let B = {| φ k (cid:105)} , · · · , B m − = {| φ km − (cid:105)} be m orthonormal basesof H d , k = 0 , · · · , d −
1. For different pure states ρ , ρ ∈ H d , they are distin-guishable if tr( ρ | φ kj (cid:105)(cid:104) φ kj | ) (cid:54) = tr( ρ | φ kj (cid:105)(cid:104) φ kj | ) (1)for some | φ kj (cid:105) . If any pair of different pure states is distinguishable by B , · · · , B m − ,the bases {B j } can distinguish all pure states [11].Obviously m bases correspond to m · d rank-1 projections. E kj = | φ kj (cid:105)(cid:104) φ kj | , j = 0 , · · · , m − k = 0 , · · · , d −
1. Since (cid:80) d − k =0 E kj = I , we have tr( ρI ) = 1 forall pure state ρ . One projection for each basis can be left out as the probabilitycan be expressed by others. Thus m ( d −
1) rank-1 self-adjoint operators candistinguish all pure states. Can these operators be transformed to a rank-1 PSIR-complete POVM?From the proposition 3 in paper [8], we know that m ( d −
1) self adjointoperators can be used to construct a POVM with m ( d −
1) + 1 elements. A kj ≡ ( I + (cid:107) E kj (cid:107) − E kj ) / [ m ( d − j = 0 , · · · , m − k = 0 , · · · , d −
2. Then O ≤ A kj ≤ I/m ( d −
1) and by setting the new element A ≡ I − (cid:80) j,k A kj we geta new POVM. This POVM have the same power with the self-adjoint operators { E k } , as there exists a bijection between the outcome probabilities of both sides. But not all of the elements are rank-1. The following conversion can keep theelements of transformed POVM to be rank-1.
Rank-1 conversion:
Given n rank-1 positive self-adjoint operators { E k : k = 1 , · · · , n } , G = (cid:80) dk =1 E k >
0, a rank-1 POVM denoted by { F k : k =1 , · · · , n } can be constructed. F k = G − / E k G − / and (cid:80) nk =1 F k = I .From the discussion in [6, 7] , if positive operators { E k } are information-ally complete with respect to generic pure states (a set of measure zero canbe neglected), and they can determine all (normalized and unnormalized) purestates in this set, { F k } is a PSI-complete POVM. Furthermore, if positive oper-ators { E k } are informationally complete with respect to all pure states, can theconverted POVM { F k } be PSIR-complete? Here we give a sufficient condition. Theorem 1.
Let { E k } be a set of rank-1 positive self-adjoint operators, whoseoutcome probabilities are sufficient to uniquely determine all pure states (up toa global phase). Some of the elements satisfy the following condition: (cid:88) k ∈ B E k = I. (2) After the rank-1 conversion, we will get a rank-1 PSIR-complete POVM { F k } .Proof. Here we prove that any pair of different pure states is distinguishable bythis POVM.Let ρ and ρ be an arbitrary pair of different pure states. Define q i =tr( G − ρ i ) for i = 1 ,
2. As G = I + (cid:80) k / ∈ B E k , we have det( G ) (cid:54) = 0. So G , G − / and G − are of full rank. And q i = tr( G − ρ i ) (cid:54) = 0.Then define another pair of pure states σ i = G − / ρ i G − / /q i , i = 1 ,
2. Forany k , tr( F k ρ i ) = q i tr( E k σ i ) . (3)When pure states σ and σ are the same, as ρ (cid:54) = ρ , the number q shouldnot be equal to q . Thus tr( F k ρ ) (cid:54) = tr( F k ρ ) for any k .When pure states σ and σ are different, by the assumption of { E k } , thereexists some E k satisfying tr( E k σ ) (cid:54) = tr( E k σ ). If q = q , then tr( F k ρ ) (cid:54) =tr( F k ρ ). If not, we have (cid:80) k ∈ B tr( E k σ ) = (cid:80) k ∈ B tr( E k σ ) = 1. As we have theassumption (cid:80) k ∈ B tr( E k ) = I . Then (cid:80) k ∈ B tr( F k σ ) (cid:54) = (cid:80) k ∈ B tr( F k σ ). Thus itcan also be deduced that tr( F k ρ ) (cid:54) = tr( F k ρ ) for some k ∈ B .So the POVM { F k } can distinguish the different pure states ρ , ρ ∈ S d, .This indicates that given a set of outcome probabilities { p k } , there is a uniquepure state ρ such that p k = tr( ρF k ) for all k . For any other different pure state σ , we can always get tr( σF k ) (cid:54) = p k for some k . Thus { F k } is enough to uniquelydetermine any pure states from the other different pure states. With the priorknowledge that the state is pure, it can be uniquely determined. Remark:
In this proof, we consider the case where { E k } can distinguish allpure states. We can make a extension to this theorem. If { E k } can distinguishall different states ρ , ρ ∈ H d , and the equation 2 still holds, then the conversedPOVM { F k } can is informationally complete with respect to all quantum states,pure or mixed. Theorem 2.
Assume that m orthonormal bases can distinguish all pure states in H d , a large number of PSIR-complete POVMs with m ( d −
1) + 1 rank-1 elementscan be constructed.Proof.
Denote these orthonormal bases as {B j } , j = 0 , · · · , m −
1. The elementsin basis B j are {| φ kj (cid:105)} , k = 0 , · · · , d −
1. Now we pick up m ( d −
1) + 1 elementsfrom these bases. We can randomly choose one basis B j and keep all the elementsin it. The corresponding projectors satisfy (cid:80) d − k =0 | φ kj (cid:105)(cid:104) φ kj | = I . Then we select d − m ( d −
1) + 1 elements. There are m · d m − collections totally.Each collection will correspond to m ( d −
1) + 1 rank-1 projectors, which candistinguish all pure states. They satisfy the condition in Theorem 1. After therank-1 conversion, we will get a rank-1 PSIR-complete POVM with m ( d −
1) + 1elements. Moreover, we can construct a large number of PSIR-complete POVMsfor each collection. Denote the projectors to be { E , · · · , E d , E d +1 , · · · , E m ( d − } ,where (cid:80) dk =1 E k = I . We can multiply E j by an arbitrary non-negative number e j , where j = d + 1 , · · · , m ( d −
1) + 1. So a new set of operators is constructed, { E , · · · , E d , e d +1 · E d +1 , · · · , e m ( d − · E m ( d − } . They also satisfy the con-dition in Theorem 1. The proof is complete.Various researches focus on the minimal number of orthonormal bases thatcan distinguish all pure states [17, 18, 19, 20, 21]. This problem is almost solved.The minimal number of orthonormal bases is summarized in [11]. Moreover, fourbases are constructed from a sequence of orthogonal polynomials. For dimension d = 2, at least three orthonormal bases are needed to distinguish all pure quan-tum states. For d = 3 and d ≥
5, the number is four. For d = 4, four bases areenough but it is not clear whether three bases can also distinguish.So we can give the upper bound of m ( d ). When d = 2, m (2) = 4. When d ≥ m ( d ) = 4 d − H , H and H In this section, we will present some results about the rank-1 PSIR-completePOVMs for lower dimensions d . In Figure 1, we show the relations betweendifferent kinds of informationally complete POVM. An IC-POVM is a PSIR-complete POVM. For dimension d = 2, four is the minimal number of elements in a rank-1 PSIR-complete POVM. One example showed in [6] is the following: E c = a c I + b c n c · σ , c = 1 , , , . (4)The parameter: a c = b c = 1 / n = (0 , , n = (2 √ / , , − / n =( −√ / , √ / , − / n = ( −√ / , −√ / − / σ = ( σ x , σ y , σ z ). This is Fig. 1.
The relations of different kinds of informationally complete POVM. The la-bels { , , , } stand for SIC-POVM, IC-POVM, PSIR-complete POVM, PSI-completePOVM respectively. For example, an IC-POVM is a PSIR-complete POVM. also a SIC-POVM. It can distinguish all quantum states in H , pure or mixed.There are two SIC-POVMs for d = 2 introduced in paper [5]. The other SIC-POVM is used [22], which shows the efficiency of qubit tomography.Now we can construct 12 rank-1 IC-complete POVMs with four elements.There are three mutually unbiased bases for d = 2. B = {| (cid:105) , | (cid:105)} , B = { ( | (cid:105) ± | (cid:105) ) / √ } , B = { ( | (cid:105) ± i | (cid:105) ) / √ } . (5)These three mutually unbiased bases can distinguish all quantum states in H .We can select four elements as introduced in Theorem 2. There are 12 col-lections totally. For example, the elements for one collection are | (cid:105) , | (cid:105) , ( | (cid:105) + | (cid:105) ) / √ , ( | (cid:105) + i | (cid:105) ) / √
2. The corresponding rank-1 projectors are | (cid:105)(cid:104) | , | (cid:105)(cid:104) | ,( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ) / | (cid:105) + i | (cid:105) )( (cid:104) | − i (cid:104) | ) /
2. After the rank-1 conversion,we will get a rank-1 POVM with 4 elements. Interestingly, this POVM is thespecial case when d = 2 constructed by Caves et al. [3]. For dimension d = 3, there are four mutually unbiased bases. By Theorem 2, wehave 4 × collections with 9 elements. We can construct rank-1 IC-completePOVMs with 9 elements from each selection. By a reference to Heinosaari et al. [8], m (3) = 8. So the minimal number of elements is either 8 or 9 for a rank-1PSIR-complete POVM. Now we show that this number is 8 by constructing 8rank-1 operators satisfying Theorem 1. After the rank-1 conversion, we will geta PSIR-complete POVM with 8 elements. The operators are as follows: E = | (cid:105)(cid:104) | , E = | (cid:105)(cid:104) | , E = | (cid:105)(cid:104) | , E = ( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ), E =( | (cid:105) + i | (cid:105) )( (cid:104) |− i (cid:104) | ), E = ( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ), E = ( | (cid:105) + | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | + (cid:104) | ), E = ( | (cid:105) + | (cid:105) + i | (cid:105) )( (cid:104) | + (cid:104) | − i (cid:104) | ).Let an arbitrary unknown pure state in H be | φ (cid:105) = (cid:80) k =0 a k e iθ k | k (cid:105) . Let a k be non-negative real numbers for k = 0 , ,
2. As e iπ = −
1, we can modify thevalue of θ k to guarantee a k ≥
0. Let θ k be in the range of [0 , π ), as e iθ k = e i ( θ k +2 tπ ) for integer t . For the freedom choice of global phase, we let θ = 0.The outcome probabilities can be calculated as follows:tr( E k | φ (cid:105)(cid:104) φ | )= a k , for k = 0 , , E | φ (cid:105)(cid:104) φ | )= a + a + 2 a a cos θ ,tr( E | φ (cid:105)(cid:104) φ | )= a + a + 2 a a sin θ ,tr( E | φ (cid:105)(cid:104) φ | )= a + a + 2 a a cos θ ,tr( E | φ (cid:105)(cid:104) φ | )= a + a + a + 2 a a cos θ + 2 a a cos θ + 2 a a cos θ cos θ +2 a a sin θ sin θ ,tr( E | φ (cid:105)(cid:104) φ | )= a + a + a + 2 a a cos θ + 2 a a sin θ + 2 a a cos θ sin θ − a a sin θ cos θ .The coefficients of a k can be calculated by E k , where k = 0 , ,
2. As thecoefficient a k is non-negative, we have a k = (cid:112) tr(E k | φ (cid:105)(cid:104) φ | ). The remaining taskis to determine θ k .When only one element in { a , a , a } is nonzero, it is the trivial case. Thestate can be | (cid:105) , | (cid:105) or | (cid:105) .When two elements in { a , a , a } are nonzero, the state can also be de-termined. For example, a = 0 and a , a (cid:54) = 0. We can write the state as | φ (cid:105) = a | (cid:105) + a e iθ | (cid:105) . The global phase of θ is extracted. So θ = 0. Theremaining unknown coefficient θ can be calculated by the effect of E and E .If a = 0 and a , a (cid:54) = 0, the state is | φ (cid:105) = a | (cid:105) + a e iθ | (cid:105) . The coefficient θ can be calculated by the effect of E and E . If a = 0 and a , a (cid:54) = 0, coefficient θ can be calculated by the effect of E and E .When all elements in { a , a , a } are nonzero, we let θ = 0. Now we de-termine the remaining coefficients θ and θ . From the effect of E and E ,cos θ and sin θ can be calculated correspondingly, thus the coefficient θ canbe uniquely determined. After we know the values of a k and θ , we can calculatecos θ by the effect of E . At the same time, sin θ can be calculated by theeffect of E or E , as cos θ and sin θ can not be both zero. Then θ is uniquelydetermined.Thus any pure state in H can be uniquely determined by the eight rank-1positive self-adjoint operators. These operators satisfies the condition in Theo-rem 1. After the rank-1 conversion, we will get a PSIR-complete POVM witheight elements. By a reference to Heinosaari et al. [8], such POVM is one of theminimal possible resource. For dimension d = 4, the known result is that m (4) = 10 [8]. There are fivemutually unbiased bases. Thus we can construct many rank-1 IC-POVMs with16 elements. Four orthonormal bases can distinguish all pure states in H [11].By theorem 1 and theorem 2, we can construct many PSIR-complete POVMswith 13 elements. So the true value of m (4) is in the range of { , , , } .It is still not clear whether three bases can distinguish all pure states in H . No results show three bases would fail and no results give the potentialsupport. There are some partial answers to this question. Three orthonormal bases consisting solely of product vectors are not enough. In fact, even fourproduct bases are not enough [11]. Eleven is the minimum number of Paulioperators needed to uniquely determine any two-qubit pure state [23].We can conclude that there is no gap between m ( d ) and m ( d ) when d =2 ,
3. If a gap exists when d = 4, three orthonormal bases are not enough todistinguish all pure states. Consider the contrapositive form. If three orthonormalbases can distinguish all pure states in H , we can construct a PSIR-completePOVM with 10 elements by Theorem 2. d + 2 k − Goyeneche et al. took an adaptive method to demonstrate that any input purestate in H d is unambiguously reconstructed by measuring five observables, i.e.,projective measurements onto the states of five orthonormal bases [15]. Thus ∼ d rank-1 operators are needed. The adaptive method is that the choice of somemeasurements is dependent on the result of former ones. The fixed measurementbasis is the standard, B = {| (cid:105) , · · · , | d − (cid:105)} . We measure the pure state withthis basis first. The results of this basis will determine a subset s d, ⊂ S d, , wherethe input pure state belongs to. They construct four bases {B , B , B , B } todetermine all pure states in s d, .Let an arbitrary unknown input pure state in H d be | φ (cid:105) = (cid:80) d − s =0 a s e iθ s | s (cid:105) ,where a s is a non-negative real number and θ s ∈ [0 , π ) for s = 0 , · · · , d −
1. Wecan extract the global phase to let one phase θ s be 0.Now we construct d + 2 k − ≤ k ≤ d . Thus at most 3 d − d operators to be measured with are E s = | s (cid:105)(cid:104) s | , s = 0 , · · · , d − . (6)We can calculate the amplitudes a s by the effect of E s , a s = (cid:112) tr( E s | φ (cid:105)(cid:104) φ | ).Then we keep track of the sites { s } of nonzero amplitudes { a s } to determine asubset s d, . Let k be the number of nonzero amplitudes, 1 ≤ k ≤ d .For example, the sites of nonzero amplitudes are { , · · · , d − } . Then k = d .The subset s d, is { (cid:80) d − k =0 a k e iθ k | k (cid:105) : a k (cid:54) = 0 } . The remaining 2 d − F s = ( | (cid:105) + | s (cid:105) )( (cid:104) | + (cid:104) s | ) , G s = ( | (cid:105) + i | s (cid:105) )( (cid:104) | − i (cid:104) s | ); (7)where s = 1 , · · · , d − θ = 0. We have the equations: (cid:40) tr( F s | φ (cid:105)(cid:104) φ | ) = a + a s + 2 a a s cos θ s , tr( G s | φ (cid:105)(cid:104) φ | ) = a + a s + 2 a a s sin θ s . (8) From the assumption and measurement results of E s , all the amplitudes a s are nonzero and known. Then cos θ s and sin θ s can be calculated by theeffect of F s and G s . All the coefficients θ s can be uniquely determined. Thus allcoefficients of the unknown pure state in H d are calculated.The operators E s and G s appear in the construction of PSI-complete POVMgiven by Finkelstein [7]. And operators E s , F s / G s / d rank-1 elements in the IC-POVM constructed by Caves et al. [3]. In fact, F s and G s can be the other types to calculate θ s . For example, F s = ( | (cid:105) + | s (cid:105) )( (cid:104) | + (cid:104) s | ), G s = ( | (cid:105) + i | s (cid:105) )( (cid:104) | − i (cid:104) s | ), s = 0 , , · · · , d − { n , · · · , n k − } .The subset s d, is { (cid:80) k − j =0 a n j e iθ nj | n j (cid:105) : a n j (cid:54) = 0 } . The remaining 2 k − F s = ( | n (cid:105) + | n s (cid:105) )( (cid:104) n | + (cid:104) n s | ) , G s = ( | n (cid:105) + i | n s (cid:105) )( (cid:104) n | − i (cid:104) n s | ); (9)where s = 1 , · · · , d − k −
1. Let the phase θ n = 0. With similar analysis, we canuniquely calculate cos θ j and sin θ j by the effect of F j and G j . All the phases θ s and amplitudes a s of | φ (cid:105) can be uniquely determined. We analyse the minimal number of elements in rank-1 PSIR-complete POVM.The bound is in [4 d − − c ( d ) α ( d ) , d − d − d = 2, 4. For d = 2, we construct many rank-1 POVMs with fourelements which can distinguish all quantum states. For d = 3, we show that eightis the minimal number in a PSIR-complete POVM by construction. For d = 4,if m (4) >
10, we can give a answer to an unsolved problem. Three orthonormalbases can not distinguish all pure states in H . Finally, we construct d + 2 k − H d , where 1 ≤ k ≤ d . Thus we can determine an arbitrary unknown pure statein H d with at most 3 d − Acknowledgements
This work was partially supported by National Key Research and DevelopmentProgram of China under grant 2016YFB1000902, National Research Foundationof China (Grant No.61472412), and Program for Creative Research Group ofNational Natural Science Foundation of China (Grant No. 61621003).
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