An Effective Iterative Method to Build the Naimark Extension of Rank-N

Home , Isospectral, POVM

arXiv:1703.01835v1 [quant-ph] 6 Mar 2017 etme 1 0831 SCISRCINFL itNEv3 FILE WSPC/INSTRUCTION 3:12 2018 11, September xesosadtetermas nue hta that ensures also theorem the and usually extensions is which space, Hilbert larger the a in measurement projective a as ssbett poetv)measurement (projective) a to subject is sculdt nidpnetypeae rb system probe prepared independently unde an system to the coupled where is measurement, indirect an as implementation n nqatmtcnlg.O h n ad tpoie concrete a provides it hand, measurement one the the On technology. quantum in one rbblt prtrvle esr PV)atn nteHlets Hilbert the theorem on Naimark is acting system. system (POVM) physical measure a operator-valued on probability performed measurement (generalised) Any Introduction 1. vlaeteps-esrmn tt n hst netgt th disturbance investigate measurement to and thus gain and information state post-measurement the evaluate rcdr ie tqatmcontrol quantum at aimed procedure ain ndffrn platforms different on tations unu ehooyLb iatmnod iia Universit Fisica, di Dipartimento Lab, Technology Quantum unu ehooyLb iatmnod iia Universit Fisica, di Dipartimento Lab, Technology Quantum h rbe ffidn h amr xesoso OMi nedacen a indeed is POVM a of extensions Naimark the finding of problem The e scnie e foperators of set a consider us Let amr extension Naimark hnoe ti lomr ffciei em fcmuainls computational of terms in Keywords effective el more containing also POVMs is extend It to also one. employed than be may it positivity as and ones, buil orthogonality of to requirements method sole iterative the an from suggest we mea particular, projective pr In a a space. as of implementation extension its Naimark i.e. the (POVM), finding measure of problem the revisit We NEFCIEIEAIEMTO OBIDTHE BUILD TO METHOD ITERATIVE EFFECTIVE AN amr xeso,Niakterm POVM theorem, Naimark extension, Naimark : AMR XESO FRN- POVMs RANK-N OF EXTENSION NAIMARK 10 , 11 n hst sesetnlmn cost entanglement assess to thus and , ftePV.A atro at hr r nnt Naimark infinite are there fact, of matter a As POVM. the of 1 , 2 eevdSpebr1,2018 11, September Received [email protected] matteo.paris@fisica.unimi.it , IOADLAPOZZA DALLA NICOLA 3 13 ATOG .PARIS A. G. MATTEO , 4 , , 14 5 nue htayPV a eimplemented be may POVM any that ensures , { 15 Π 26 , 16 m . 1 } 7 , 17 , htcnttt OMfrtephysical the for POVM a constitute that 8 , , 9 18 . nteohrhn,i emt to permits it hand, other the On . 19 aoia extension canonical , 20 u ehdipoe existing improves method Our . ad iao -03 iao Italy Milano, I-20133 Milano, à di a d iao -03 iao Italy Milano, I-20133 Milano, à di , h rjciemeasurement projective the d 21 ueeti agrHilbert larger a in surement 6 , 22 teps. bblt operator-valued obability n hnol h probe the only then and mnswt aklarger rank with ements , 23 12 , 24 rdo between tradeoff e n/rimplemen- and/or , 25 oe orealize to model swl sany as well as , ecie ya by described investigation r xss ..an i.e. exists, eerdt as to referred aeo the of pace tral September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

2 Nicola Dalla Pozza and Matteo G. A. Paris

system S described by the Hilbert space , i.e. HS M Π = I , Π = Π† , Π 0 . (1) m S m m m ≥ m=1 X The elements of the set are not necessarily projectors, Π Π = Π δ . The n m 6 n n,m Naimark theorem states that it is possible to extend each POVM elements to a larger (product) Hilbert space (see Appendix A) such that the extended HA ⊗ HS measurement operators are projectors in the product space. In particular, it is possible to define the auxiliary Hilbert space such that the system Hilbert space HA is isomorphic to a subspace in , where the density operator ρ defined on HS HA ⊗HS corresponds to the density operator e e ρ defined on . The state HS | 1ih 1|⊗ HA ⊗ HS e may be chosen as the state corresponding to the first vector of the canonical | 1i basis of . Naimark theorem states that we can find projectors E HA { m} E E = E δ , E = E† , E 0, (2) m n m m,n m m m ≥ each of them corresponding to a POVM element Πm in the following sense. The distributions of the m-th outcome, as obtained from E and Π on the states { m} { m} e e ρ and ρ respectively, are the same, i.e. | 1ih 1|⊗ Tr Π ρ = Tr ( e e ρ) E . (3) A m AS | 1ih 1|⊗ m At the operatorial level, this is expressed by the following the set of relations Π = Tr [( e e I ) E ] (4) m A | 1ih 1|⊗ S m which, solved for the Em given the Πm, provide the desired Naimark extension of the POVM. As it was originally suggested by Helstrom4 the projectors E may be built { m} by placing a copy of Πm in the upper-left block position of the matrix representation of E (corresponding to the element 1 in the matrix e eT ). At the same time, { m} 1 · 1 no explicit recipes had been provided on how to find the remaining blocks. The aim of this paper is to describe an iterative method for effectively building those blocks upon exploiting the sole requirements of orthogonality and positivity. The problem has been addressed before 27,28, and constructive methods to find the projective measurement have been suggested. In short, these methods amount to set up and solve a linear problem which gives the coefficients of the projectors in the canonical basis of the enlarged Hilbert space . However, the focus HA ⊗ HS has been on solving the problem for rank-1 POVM elements. Our iterative method, also based on solving a linear problem, shows two main advantages compared to existing techniques. On the one hand, it is more efficient in terms of computational steps and, on the other hand, it may applied also to POVMs containing elements with rank greater than one. The paper is structured as follows. In the next Section we introduce the iterative method, first illustrating the basic idea and then, in Sections 2.1 and 2.2, describing in details its two building blocks, i.e. the constrained building of an idempotent September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 3

matrix and the constrained building of a matrix orthogonal to a given one. In Section 2.3 we put everything together and illustrate the overall algorithm to build the Naimark extension of a generic rank-n POVM. In Section 3, we illustrate few examples of application, whereas Section 4 closes the paper with some concluding remarks.

2. An iterative method to build the Naimark extension of rank-n POVMs In the following, we will write projectors as matrices of suitable sizes composed by blocks. The first step in building the projectors Em is analogue to the original Helstrom recipe, i.e. we define the upper-left block in the matrix of Em equal to Πm. The algorithm then builds the projectors one at a time, upon defining their blocks iteratively. As we will see soon, initially the blocks of the first projector are mostly zero, and the building of the following projectors populates other blocks. In this sense, the amount of non-zero rows and columns grows during the building of the projectors, and the size of the necessary auxiliary Hilbert space is obtained HA only at the end of the procedure. The algorithm initially builds the blocks of E1 in order to satisfy the constraints (2) on itself, i.e.

E E = E , E = E†, E 0 (5) 1 · 1 1 1 1 1 ≥ Then, we build some blocks of E2 in order to satisfy the orthogonality with E1, E E = 0 (6) 1 · 2 and then imposing the other constraints

E E = E , E = E†, E 0 (7) 2 · 2 2 2 2 2 ≥ we define the remaining blocks. As we will see, this second step do not modify the previously defined blocks of E2. Analogously, the algorithm builds E3 (if any) imposing its orthogonality with E , E , and then imposing that E E = E . The generalisation is straightfor- 1 2 3 · 3 3 ward, the element Em is built in order to satisfy at first the ortogonality with the previously built projectors, and then imposing the condition E E = E . The m · m m algorithm is thus an iterative one, since it employs the projectors already found, until all the elements are built. The algorithm requires basically two steps repeated several times: building a matrix with some assigned blocks such that it is orthogonal to another matrix, and the completion of the matrix in order to make it idempotent, that is, satisfying E E = E . m · m m These steps are analysed in some details in the following two Sections, whereas the overall algorithm is summarised in Section 2.3. September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

4 Nicola Dalla Pozza and Matteo G. A. Paris

2.1. Building an idempotent matrix At first, let us consider the problem of building an idempotent matrix when some of its blocks are assigned. This is the case of the evaluation of E1, which has the block Π1 in the upper-left position. If Π1 is already idempotent, we can just put Π1 in the corner and set the remaining blocks to zero. If this is not the case, we can define the blocks around Π such that E E = E , possibly employing the 1 1 · 1 1 minimum amounts of blocks, and setting the others to zero. In what follows, we ignore the subscripts that refers to the m-th element. The general problem becomes to find the adjacent blocks of the upper-left corner in order to make the matrix E idempotent. As we will see in the following, it is enough to assume the following matrix structure for E Π A 0 ... A† B 0 ...   E = 0 00 ... (8)  . . . .   . . . ..  . . .    with Π a given block, while A, B are blocks to find (A† and B 0 have been used 2 ≥ so that E = E†). In the case Π = Π we can omit A, B since the matrix is already idempotent. Otherwise, we have to add the blocks A, A†, B and the matrix E grows in sizes. The constraint Π A Π A Π A E E = = = E (9) · A B · A B A B † † † gives the following equations:

2 Π + AA† = Π (10) ΠA + AB = A (11) 2 A†A + B = B (12) Equation (10) can be solved exploiting the singular value decomposition (SVD) for Π = V ΛV † and A = USW †. Setting U = V, W = I, S = Λ(I Λ) leads to − A = V Λ(I Λ). Assuming for the moment a full rank matrix Π, with eigenvalues − p strictly included in the range (0, 1), equation (11) allows us to ﬁnd B = I Λ. p − Finally, equation (12) is veriﬁed with the above solutions, and the blocks of E can be built as

Π A 0 ... V ΛV † V Λ(I Λ) 0 ... − A† B 0 ... Λ(I Λ)V † I Λ 0 ...    − p −  E = 0 00 ... = 0 00 ... (13) p  . . . .   . . . .   . . . ..  . . . ..    . . .      A diﬀerent route may be also employed upon exploiting positivity of the elements of the POVM. Indeed, for positive semi-deﬁnite Π we have the decomposition September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 5

Π = Y Y † (with Y having no particular properties), which may be used instead of SVD, which is generally demanding in terms of computational time. Notice that if Π is not full rank, the decomposition is still available, with Y being a rectangular matrix with the same rank. With this decomposition, equation (10) is solved by

A = Y I Y Y , A† = I Y Y Y †. (14) − † − † p p and B = I Y †Y follows. Finally, equation (12) is veriﬁed by re-writing A†A = − √B(I B)√B. − 1 For rectangular Y , equation (14) still holds upon deﬁning Y − as the Penrose 1 inverse of Y, that is, the rectangular matrix satisfying Y − Y = I on the support of Y . In addition, the decomposition E = ZZ† is also readily available from Y ,

Y Y † Y √I Y Y 0 ... − † √I Y Y Y † I Y †Y 0 ...  − † −  E = 0 00 ... (15)  . . . .   . . . ..  . . .    Y

√I Y †Y =  −  Y † √I Y Y 0 ... = ZZ† (16) 0 · − †  .   .   .   

2.2. Building a matrix orthogonal to a given one In this section we consider the problem of building a matrix (with some assigned blocks) such that it is orthogonal to a given one. This occurs in building, e.g., E2, which has the upper-left block equal to Π and must verify E E = 0. If we have 2 1 · 2 Π Π = 0, it is enough to set the blocks adjacent to Π equal to zero. In the most 1 · 2 2 general case, this does not hold, and to satisfy the orthogonality condition we have to explicitly determine the blocks around Π2. The expression

Y1Y † Y1 I Y †Y1 Π A E E = 1 − 1 2 =0 , (17) 1 2  q  A B · I Y †Y Y † I Y †Y · † − 1 1 1 − 1 1 q  where Π1 = Y1Y1†, provides the constraints

Y Y †Π + Y I Y †Y A† = 0 (18) 1 1 2 1 − 1 1 q Y Y †A + Y I Y †Y B = 0 (19) 1 1 1 − 1 1 q I Y †Y Y †A + (I Y †Y )B =0 . (20) − 1 1 1 − 1 1 q September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

6 Nicola Dalla Pozza and Matteo G. A. Paris

Equation (18) allows us to ﬁnd A = Π2Y1 I Y1†Y1, whereas equation (19) pro- − 1 − − q vides the expression B = I Y †Y Y †Π Y I Y †Y . The third equation, − 1 1 1 2 1 − 1 1 (20), is indeed veriﬁed by theseq solutions. q At this stage, upon imposing the orthogonality with E1, we found that E2 has the structure

Π2 Π2Y1 I Y1†Y1 ... 1 − 1 − ∗  − −q  I Y1†Y1 Y1†Π2 I Y1†Y1 Y1†Π2Y1 I Y1†Y1 ... E2 = − − − − ∗ ,    q q q ...    ∗. ∗. ∗. .   . . . ..  . . .   (21) where the blocks indicated by are left unused and may be exploited to impose ∗ other conditions on E2. If a decomposition Π2 = XX† is available, the big block just deﬁned in (21) has a simple decomposition,

XX† XX†Y1 I Y1†Y1 1 − 1 −  − − q  I Y †Y Y †XX† I Y †Y Y †XX†Y I Y †Y − − 1 1 1 − 1 1 1 1 − 1 1  q q q   X  1 = − = Y Y † (22) X† X†Y1 I Y1†Y1 2 2  I Y1†Y1 Y1†X · − − · − − q  q  Notice that the blocks just deﬁned depend on Π1 (via its decomposition) and upon Π2. If we have to impose the orthogonality of matrix Em with E1, only the non-zero blocks in E1 would be involved. Thus, the solution would be the same

substituting Πm = XmXm† in place of Π2 = X2X2†.

2.3. The algorithm The algorithm builds the projectors one at a time, using the previously built projectors. For each projector Em two steps are performed: first the orthogonal construction of Section 2.2, which defines some blocks of Em such that the projector is orthogonal to all the projectors previously evaluated. In the second step, leveraging the idempotent construction illustrated in Section 2.1, some other blocks are defined 2 so that Em = Em. Before applying the orthogonal or idempotent construction, it is checked whether Em is already ortogonal to the other projectors or idempotent. If this is the case, the step is simply skipped. The algorithm starts building E1 with the idempotent construction, as the orthogonal one is not necessary. Π1 is copied in the upper-left block of E1 and

the solution (15) is evaluated with Y = Y1, Y1Y1† = Π1, where Y1 has been obtained for instance from the singular value decomposition of Π1 = V1Λ1V1†, giving September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 7

Y = V √Λ . If Π is full rank, the projector E has a nonzero 2 2 blocks in the 1 1 1 1 1 × upper-left corner. The remaining blocks are zero.

Y1Y † Y1 I Y †Y1 1 − 1 0  Iq Y †Y1  E1 = ∗ − 1  0 0        The projector E2 is then built using the two steps. The block Π2 is copied in the (1) (1) upper-left corner, a decomposition Π2 = X2 X2 † is evaluated (by SVD if needed) and the three blocks around are deﬁned as in (21) leveraging on the decompositon

Π1 = Y1Y1† previously evaluated. At this point, the big block just deﬁned has a decomposition Y2Y2† as in (22), and if not idempotent, the adjacent blocks need to be evaluated accordingly employing the idempotent construction of equation (15).

† Y2Y2 1  −  Π2 Π2Y1 I Y1†Y1 E =  z − }| − {  2  1q 1 Y2 I Y2†Y2   − − −   I Y1†Y1 Y1†Π2Y1 I Y1†Y1 q   ∗ − −   q q   I Y †Y2   ∗ − 2    Notice that in this case in the original matrix E the 4 4 blocks in the upper left 2 × corner are defined, for a total of 4D rows and 4D columns (if Π1, Π2 are full rank), with D being the dimension of the system Hilbert space. In the evaluation of E3 the same orthogonal and idempotent construction are repeated, with the difference that the first must be repeated twice to get the orthogonality with E1 ed E2. As (1) (1) usual, first the block Π3 = X3 X3 † is copied in the upper left corner. The first blocks around Π3 are evaluated with (21). (2) (2) The newly defined big block has decomposition X3 X3 † obtained from (22) (1) (2) where X = X3 , Y2 = X3 . The orthogonal construction (21) is repeated to get the (2) (2) orthogonality with E2, employing the block X3 X3 † and the term Y2 previously defined in the idempotent contruction of E2. A new bigger block is obtained with

decomposition Y3Y3† as in (22). The idempotent construction is then used employing Y = Y3 as in equation (15). Finally, we get the matrix (note that not all the blocks have the same size)

† Y3Y3

(2) (2)†  X3 X3  z 1 }| { E =  Π3 Π3 Y1R1− (2) (2) 1  3  X X † Y R− Y3R3   −1 1 3 3 2 2   z R1− Y1†}|Π3 Y1R1−{ −   ∗ 1 (2) (2) 1   R− Y † X X † Y2R−   ∗ 2 2 3 3 2   I Y †Y   ∗ − 3 3    September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

8 Nicola Dalla Pozza and Matteo G. A. Paris

with R = I Y †Y , R = I Y †Y , R = I Y †Y . Notice that the 1 − 1 1 2 − 2 2 3 − 3 3 expression ofqE3 depends upon theq decompositions Yq1Y1†, Y2Y2† of the upper-left blocks of the preceding projectors. This holds for each projector Em. The method used to evaluate E3 may be iterated for any subsequent projector. (1) (1) First, the block Πm = Xm Xm † is copied in the upper left corner. The adjacent blocks are deﬁned imposing the orthogonality with E1, following the orthogonal (2) (2) construction. The just deﬁned block has decomposition Xm Xm †, and the orthogonal construction is repeated using Yn, which is the term used in the idempotent construction of En, n

3. Examples Here we apply our procedure to obtain the Naimark extension of POVMs already presented in the literature. In this way, we are able to show the main features of the algorithm, and its advantages compared to existing ones.

3.1. Three elements POVM Helstrom considered the example a three-elements POVM Π , Π , Π , Π + Π + { 1 2 3} 1 2 Π = I , deﬁned by Π = 2 ψ ψ , k =1, 2, 3, where 4 3 S k 3 | kih k| 1 1 1 e iπ/3 1 eiπ/3 ψ = , ψ = − , ψ = . (23) | 1i √ 1 | 2i √ eiπ/3 | 3i −√ e iπ/3 2 2 2 − i.e. 1 11 1 1 e 2iπ/3 1 1 e2iπ/3 Π = , Π = − , Π = (24) 1 3 11 2 3 e2iπ/3 1 3 3 e 2iπ/3 1 − The extension originally obtained by Helstrom was based on a two-dimensional auxiliary Hilbert space with basis v = (1, 0)T , v = (0, 1)T , and it is given by | 1i | 2i September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 9

EH = ξ ξ , k =1, .., 4, where k | kih k| ξ = 2/3 v ψ + 1/3 v ψ , (25) | 1i | 1i| 1i | 2i| 3i ξ = p2/3 v ψ p1/3 v ψ , (26) | 2i | 1i| 2i− | 2i| 3i ξ = p2/3 v ψ + p1/3 v ψ , (27) | 3i | 1i| 3i | 2i| 3i ξ4 = pv2 ψ3′ , p (28) | i | i| i 1 eiπ/3 ψ3′ = − iπ/3 . (29) | i √2 e− The iterative algorithm in this case is particularly efficient since the orthogonality construction gives also idempotent matrices. Overall, a two-dimensional auxiliary Hilbert space is still required, but only the upper left 3-by-3 corner has non-zero coefficients. 2iπ 2iπ 1110 1 e− 3 e 3 0 2iπ 2iπ 1 1110 1 e 3 1 e− 3 0 E1 =   , E2 =  2iπ 2iπ  , (30) 3 1110 3 e− 3 e 3 1 0    0000   0 0 00       2iπ 2iπ   1 e 3 e− 3 0 2iπ 2iπ 1 e− 3 1 e 3 0 E3 =  2iπ 2iπ  . (31) 3 e 3 e− 3 1 0    0 0 00    The correctness of both solutions is verified by checking the properties of orthogonality, idempotence, and the upper left corner equal to the original POVM. The extension proposed by Helstrom gives 4-by-4 matrices with no zero coefficients, and therefore differs for the block adjacent the left upper corner. Here we H report the matrix expression of E1 for comparison with E1 in Eq. (30) iπ iπ 2 − 2 1 1 e 3 e 3 √2 √2 iπ iπ 2 − 2  1 1 e 3 e 3  H 1 √2 √2 iπ iπ E1 = − 2 − 2 iπ . (32) e 3 e 3 1 1 3  e 3   √2 √2 2 2 −  iπ iπ  2 2 iπ −   e 3 e 3 1 1   e 3   √2 √2 − 2 2    3.2. Four elements POVM Helstrom also considered a four-elements POVM Π , Π , Π , Π , with4 { 1 2 3 4} i(k 1) π 1 1 e− − 4 Πk = ψk ψk , ψk = i(k 1) π , k =1, 2, 3, 4 , 2| ih | | i √2 e − 4 i.e. 1 11 1 1 i 1 1 1 1 1 i Π1 = , Π2 = − , Π3 = − , Π4 = . (33) 4 11 4 i 1 4 1 1 4 i 1 − − September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

10 Nicola Dalla Pozza and Matteo G. A. Paris

Again, the iterative algorithm easily ﬁnds the extension since the orthogonal construction directly gives idempotent matrices, without the need of the idempotent construction. iπ iπ 1 1 √20 1 i e− 4 e− 4 − − iπ iπ 1 1 1 √20 1 i 1 e 4 e 4 E1 =   E2 =  iπ iπ −  (34) 4 √2 √2 2 0 4 e 4 e− 4 1 1  − iπ − iπ −   0 0 00   e 4 e− 4 1 1     −     iπ iπ  1 1 0 i√2 1 i e 4 e 4 − − iπ −3iπ 1 1 10 i√2 1 i 1 e− 4 e 4 E3 =  − −  E4 =  − iπ iπ −  (35) 4 0 000 4 e− 4 e 4 1 1  − 3iπ − iπ   i√2 i√20 2   e 4 e 4 1 1   −   −      3.3. Rank-2 POVMs In a more recent paper, rank-2 POVM elements have been introduced to describe generalized measurements involving sets of Pauli quantum observables chosen at random, the so-called quantum roulettes 29. More precisely, quantum roulettes are generalized measurements obtained by selecting the observable σk with a probability z in the set of nondegenerate and isospectral observables σ . The POVM { k} { k} elements are deﬁned as linear combination of the projectors associated with the observables outcomes. In Ref. 29, the canonical Naimark extension is sought, i.e. the implementation of the generalized measurement in a larger Hilbert space using a projective indirect measurement on the ancillary system after its coupling with the system. In this scenario, Eq. (3) is rewritten as

Tr Π ρ = Tr ( ω ω ρ) U †(P I )U , A m AS | Aih A|⊗ m ⊗ S where ωA is the ancillary state,U describes the coupled evolution between the | i systems, and Pm is the projective measurement in the ancillary system. A ﬁrst example of POVM is that of a roulette obtained from the Pauli operators σ , σ { 1 3} with probabilities z, 1 z , z (0, 1), giving the elements { − } ∈ 1 2 z z 1 z z Π1 = − , Π 1 = − . 2 z z − 2 z 2 z − − The solution proposed uses the ancillary state ω = 1 0 + eiφ 1 , the projec- | Ai √2 | i | i tors 1 2 z z(2 z) P1 = − − , P 1 = I P1, 2 z(2 z) z − − − p and the unitary p f 0 00 0 0 if ∗ 0 2 2z z U =   , f = − + i . 0 if 0 0 2 z 2 z ∗ sr r   − −  00 0 f    September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 11

On the other hand, upon applying the iterative algorithm gives these solutions straightaway,

√(1 z)z 1 z z − 0 − 2 2 √2 √(1 z)z  z z 0 −  2 2 √2 E1 = , E 1 = IAS E1,  √(1 z)z z z  − −  − 0   √2 2 − 2   √(1 z)z   0 − z 1 z   √2 − 2 − 2    which is equivalent to the canonical one up to a rotation in the ancillary state. The paper presents also another example with rank-2 diagonal POVM elements, 1 2 + f 0 I Π1 = 1 , Π 1 = Π1. 0 f − − 2 − The proposed extension employs the ancillary state ω = e , the projectors of | Ai | 1i the observable σ3, i.e. P1 = e1 e1 , P 1 = e2 e2 , and the unitary | ih | − | ih | 1 + f 0 0 i 1 f 2 2 −  q 0 1 f i 1 + f q 0  U = 2 − 2 ,  q 1 q1   0 i 2 + f 2 f 0   −   1 q q 1   i f 0 0 + f   2 − 2   q q  which gives

1 1 2 2 + f 0 0 2 i 1 4f 1 1 2 − 0 2 f 2 i 1 4f 0 U † (P1 IS)U =  − − p  . (36) ⊗ 0 1 i 1 4f 2 1 + f 0  − 2 − p2   1 i 1 4f 2 0 0 1 f   − 2 − p 2 −    In this case the iterativep algorithm is particularly easy to apply since we have diagonal POVM elements, and it gives the solution

1 1 2 2 + f 0 2 1 4f 0 1 − 1 2 0 2 f 0 2 1 4f E1 =  − p −  1 1 4f 2 0 1 f 0  2 − 2 − p   0 1 1 4f 2 0 1 + f   p 2 − 2    which is equivalent to (36) since inp both cases we can see Π1 in the upper left bock.

4. Conclusions In this paper we have addressed the problem of ﬁnding the Naimark extension of a probability operator-valued measure, i.e. its implementation as a projective measurement in a larger Hilbert space. As a matter of fact, the extension of a POVM is not unique and we have exploited this degree of freedom to introduce an iterative September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

12 Nicola Dalla Pozza and Matteo G. A. Paris

method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it is more effective in terms of computational steps needed to determine the POVM extension. Even more importantly, our method may be employed also to extend POVMs containing elements with rank larger than one. Since a Naimark extension provides a concrete model to realize the generalized measurement, we foresee applications of our method to assess technological solutions on different platforms and to investigate the tradeoff between information gain and measurement disturbance in generalized measurements.

Acknowledgments This work has been supported by EU through the Collaborative Project QuProCS (Grant Agreement 641277) and by UniMI through the H2020 Transition Grant 15-6-3008000-625.

Appendix A. Kronecker product convention The product space is usually defined as , with the system Hilbert space HS ⊗ HA on the left. However, given the definition of Kronecker product HS a B a B 11 ··· 1n A B = . .. . , ⊗  . . .  a B a B  m1 mn   ···  the opposite convention, i.e. describing the composite system by the Hilbert space , makes it easier to graphically visualize the product matrix. For instance, HA ⊗ HS for a matrix given by the product of the first element of the canonical basis only one block is non-zero B 0 0 ··· 0 0 0 T  ···  (e1 e1 ) B = . . . . , · ⊗ . . .. .    0 0 0    ···  The standard convention would make the notation more cumbersome.

Appendix B. Building the decomposition of Em The procedure explained in Section 2.3 suggests a recursive construction to di-

rectly obtain the decomposition Em = ZmZm† . In order to evaluate Zm, we ini- (1) (1) tially need a decomposition Πm = Xm Xm †, obtained for instance from its singular value decomposition. Then, the orthogonal construction (22) is applied with (i) (i+1) X = Xm , Y1 = Yi to evaluate Y2 = Xm . This step is repeated for i = 1 to (m) m. The last block calculated, Xm , is deﬁned as Ym and used in the idempotent construction (16) employing Y = Ym to get Z = Zm. September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

An eﬀective iterative method to build the Naimark extension of rank-n POVMs 13

This construction can be summarized by the following matrix (in general rectangular)

(1) Xm 1 X(2)  − (1) m     I Y1†Y1 Y1†Xm − −   (i)   Xm    q           1    (i+1)   −  Xm   (2)    I Y2†Y2 Y2†Xm     − −    (m)   q   Xm = Ym      1     Zm =      − (i)    I Yi†Yi Yi†Xm    − −     q .     .     .     1    − (m 1)    I Ym† 1Ym 1 Ym† 1Xm −    −    − − − −    q          I Ym† Y   − m   q (B.1) Notice that to obtain the term Zm the decomposition Xm of Πm is used, as well as all the terms Y1, Y2, ...Ym 1 used in the preceding idempotent constructions. − 1 − This is an eﬃcient procedure, since the terms such as I Y †Y Y †, i

Bibliography 1. M. A. Naimark, Iza. Akad. Nauk USSR, Ser. Mat. 4 277 (1940); C.R. Acad. Sci. URSS 41, 359, (1943). 2. N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1963), Vol. 2. 3. C. W. Helstrom, Int. J. Theor. Phys. 8, 361, (1973). 4. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976) 5. A. S. Holevo, Statistical Structure of Quantum Theory, Lect. Not. Phys 61, (Springer, Berlin,2001). 6. A. Peres, Found. Phys. 20, 1441 (1990). 7. B. He, J. A. Bergou, Z. Wang, Phys. Rev. A 76, 042326 (2007). 8. J. Bergou, J. Mod. Opt. 57, 160 (2010). 9. M. G. A. Paris, Eur. Phys. J. ST 203, 61 (2012). 10. M. Ban, Int. J. Theor. Phys. 36, 2583 (1997). 11. M. G. A. Paris, G. Landolﬁ, G. Soliani, J. Phys. A 40, F531 (2007). September 11, 2018 3:12 WSPC/INSTRUCTION FILE itNEv3

14 Nicola Dalla Pozza and Matteo G. A. Paris

12. R. Jozsa, M. Koashi, N. Linden, S. Popescu, S. Presnell, D. Shepherd, A. Winter, Quantum Inf. Comp. 3, 405 (2003). 13. D. De Falco, D. Tamascelli, RAIRO Th. Inf. Appl. 40, 93 (2006). 14. R. Beneduci, J. Math. Phys. 48, 022102 (2007). 15. R. Beneduci, Int. J. Th. Phys. 49, 3030 (2010). 16. R. Y. Levine, R. R. Tucci, Found. Phys. 19, 175 (1989). 17. C. Sparaciari, M. G. A. Paris, Int. J. Quantum Inf. 12, 1461012 (2014). 18. D. Tamascelli, S. Olivares, C. Benedetti, M. G. A. Paris, Phys. Rev. A 94, 042129 (2016). 19. K. Banaszek, Phys. Rev. Lett. 86, 1366 (2001). 20. K. Banaszek, Open Sys. Information Dyn. 13, 1 (2006) 21. M. G. Genoni, M. G. A. Paris, Phys. Rev. A 71, 052307 (2005). 22. L. Miˇsta, R. Filip, Phys. Rev. A 72, 034307 (2005). 23. M. G. Genoni, M. G. A. Paris, Phys. Rev. A 74, 012301 (2006). 24. M. G. Genoni, M. G. A. Paris, J. Phys. CP 67, 012029 (2007). 25. M. Wilde, Proc. Roy. Soc. A 469, 20130259 (2013). 26. A. Mandilara, J. W. Clark, Phys. Rev. A 71, 013406 (2005). 27. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, New York,1995). 28. J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, California Institute of Technology, 1998. 29. C. Sparaciari, M. G. A. Paris, Phys. Rev. A 87, 012106 (2013).