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

TL;DR

This paper introduces an improved iterative method for constructing the Naimark extension of rank-n POVMs, enabling efficient implementation of POVMs as projective measurements in larger Hilbert spaces.

Contribution

The proposed method advances existing techniques by handling higher-rank POVMs and reducing computational complexity.

Findings

Method successfully extends rank-n POVMs.

Reduces computational steps compared to previous methods.

Applicable to a broader class of POVMs.

Abstract

We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.

Equations

$$\sum _{m=1}^M{\Pi }_m=I_S,{\Pi }_m={\Pi }_m^{†},{\Pi }_m\ge 0.$$

$$E_m{E}_n=E_m{\delta }_{m,n},E_m=E_m^{†},E_m\ge 0,$$

$$\hbox{Tr}_{A}\big{[}\Pi_{m}\,\rho\big{]}=\hbox{Tr}_{AS}\big{[}\left(|{e_{1}}\rangle\langle{e_{1}}|\otimes\rho\right)\,E_{m}\big{]}\,.$$

$${\Pi }_m={Tr}_A\left[\left(|e_1⟩⟨e_1|\otimes {\mathbb{I}}_S\right)E_m\right]$$

$$E_1\cdot E_1=E_1,E_1=E_1^{†},E_1\ge 0$$

$$E_1\cdot E_2=0$$

$$E_2\cdot E_2=E_2,E_2=E_2^{†},E_2\ge 0$$

$$E_m\cdot E_m=E_m.$$

$$E=\left(\begin{array}{cccc}\Pi & A& 0& \dots \\ A^{†}& B& 0& \dots \\ 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)$$

$$E\cdot E=\left(\begin{array}{cc}\Pi & A\\ A^{†}& B\end{array}\right)\cdot \left(\begin{array}{cc}\Pi & A\\ A^{†}& B\end{array}\right)=\left(\begin{array}{cc}\Pi & A\\ A^{†}& B\end{array}\right)=E$$

$${\Pi }^2+A{A}^{†}=\Pi$$

$$\Pi A+AB=A$$

$$A^{†}A+B^2=B$$

$$E=\left(\begin{array}{cccc}\Pi & A& 0& \dots \\ A^{†}& B& 0& \dots \\ 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)=\left(\begin{array}{cccc}V\Lambda V^{†}& V\sqrt{\Lambda (I-\Lambda )}& 0& \dots \\ \sqrt{\Lambda (I-\Lambda )}{V}^{†}& I-\Lambda & 0& \dots \\ 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)$$

$$A=Y\sqrt{I-Y^{†}Y},A^{†}=\sqrt{I-Y^{†}Y}{Y}^{†}.$$

$$E=$$

$$=\left(\begin{array}{c}Y\\ \sqrt{I-Y^{†}Y}\\ 0\\ ⋮\end{array}\right)\cdot \left(\begin{array}{cccc}{Y}^{†}& \sqrt{I-Y^{†}Y}& 0& \dots \end{array}\right)=Z{Z}^{†}$$

$$E_1\cdot E_2=\left(\begin{array}{cc}{Y}_1{Y}_1^{†}& Y_1\sqrt{I-Y_1^{†}{Y}_1}\\ \sqrt{I-Y_1^{†}{Y}_1}{Y}_1^{†}& I-Y_1^{†}{Y}_1\end{array}\right)\cdot \left(\begin{array}{cc}{\Pi }_2& A\\ A^{†}& B\end{array}\right)=0,$$

$$Y_1{Y}_1^{†}{\Pi }_2+Y_1\sqrt{I-Y_1^{†}{Y}_1}{A}^{†}=0$$

$$Y_1{Y}_1^{†}A+Y_1\sqrt{I-Y_1^{†}{Y}_1}B=0$$

$$\sqrt{I-Y_1^{†}{Y}_1}{Y}_1^{†}A+(I-Y_1^{†}{Y}_1)B=0.$$

$$E_2=\left(\begin{array}{cccc}{\Pi }_2& -{\Pi }_2{Y}_1\sqrt{I-Y_1^{†}{Y}_1}& \ast & \dots \\ -{\left(\sqrt{I-Y_1^{†}{Y}_1}\right)}^{-1}{Y}_1^{†}{\Pi }_2& {\left(\sqrt{I-Y_1^{†}{Y}_1}\right)}^{-1}{Y}_1^{†}{\Pi }_2{Y}_1\sqrt{I-Y_1^{†}{Y}_1}& \ast & \dots \\ \ast & \ast & \ast & \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right),$$

$$\left(\begin{array}{cc}X{X}^{†}& -X{X}^{†}{Y}_1\sqrt{I-Y_1^{†}{Y}_1}\\ -{\left(\sqrt{I-Y_1^{†}{Y}_1}\right)}^{-1}{Y}_1^{†}X{X}^{†}& {\left(\sqrt{I-Y_1^{†}{Y}_1}\right)}^{-1}{Y}_1^{†}X{X}^{†}{Y}_1\sqrt{I-Y_1^{†}{Y}_1}\end{array}\right)$$

$$=\left(\begin{array}{c}X\\ -{\left(\sqrt{I-Y_1^{†}{Y}_1}\right)}^{-1}{Y}_1^{†}X\end{array}\right)\cdot \left(\begin{array}{cc}{X}^{†}& -X^{†}{Y}_1\sqrt{I-Y_1^{†}{Y}_1}\end{array}\right)=Y_2\cdot Y_2^{†}$$

$$E_1$$

$$E_2$$

$$\displaystyle E_{3}=\left(\begin{array}[]{c|c}\overbrace{\begin{array}[]{c|c}\overbrace{\begin{matrix}\Pi_{3}&-\Pi_{3}\ Y_{1}R_{1}^{-1}\\ \ast&R_{1}^{-1}Y_{1}^{\dagger}\ \Pi_{3}\ Y_{1}R_{1}^{-1}\end{matrix}}^{X_{3}^{(2)}X_{3}^{(2)\dagger}}&\qquad-X_{3}^{(2)}X_{3}^{(2)\dagger}\ Y_{2}R_{2}^{-1}\\ \hline\cr\ast&\displaystyle R_{2}^{-1}Y_{2}^{\dagger}\ X_{3}^{(2)}X_{3}^{(2)\dagger}\ Y_{2}R_{2}^{-1}\end{array}}^{Y_{3}Y_{3}^{\dagger}}&Y_{3}R_{3}\\ \hline\cr\ast&I-Y_{3}^{\dagger}Y_{3}\end{array}\right)$$

$$|{\psi_{1}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\ 1\end{array}\right)\,,\quad|{\psi_{2}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}e^{-i\pi/3}\\ e^{i\pi/3}\end{array}\right)\,,\quad|{\psi_{3}}\rangle=-\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}e^{i\pi/3}\\ e^{-i\pi/3}\end{array}\right)\,.$$

$${\Pi }_1=$$

$$|{\xi }_1⟩=\sqrt{2/3}|v_1⟩|{\psi }_1⟩+\sqrt{1/3}|v_2⟩|{\psi }_3⟩,$$

$$|{\xi }_2⟩=\sqrt{2/3}|v_1⟩|{\psi }_2⟩-\sqrt{1/3}|v_2⟩|{\psi }_3⟩,$$

$$|{\xi }_3⟩=\sqrt{2/3}|v_1⟩|{\psi }_3⟩+\sqrt{1/3}|v_2⟩|{\psi }_3⟩,$$

$$|{\xi }_4⟩=|v_2⟩|{\psi }_3^{\prime }⟩,$$

$$\displaystyle|{\psi_{3}^{\prime}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-e^{i\pi/3}\\ e^{-i\pi/3}\end{array}\right)\,.$$

$$\displaystyle E_{1}=\frac{1}{3}\left(\begin{array}[]{cccc}1&1&1&0\\ 1&1&1&0\\ 1&1&1&0\\ 0&0&0&0\end{array}\right),\quad E_{2}=\frac{1}{3}\left(\begin{array}[]{cccc}1&e^{-\frac{2i\pi}{3}}&e^{\frac{2i\pi}{3}}&0\\ e^{\frac{2i\pi}{3}}&1&e^{-\frac{2i\pi}{3}}&0\\ e^{-\frac{2i\pi}{3}}&e^{\frac{2i\pi}{3}}&1&0\\ 0&0&0&0\end{array}\right),$$

Full text

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1 Introduction

Any (generalised) measurement performed on a physical system is described by a probability operator-valued measure (POVM) acting on the Hilbert space of the system. Naimark theorem [1, 2, 3, 4, 5] ensures that any POVM may be implemented as a projective measurement in a larger Hilbert space, which is usually referred to as the Naimark extension of the POVM. As a matter of fact, there are infinite Naimark extensions and the theorem also ensures that a canonical extension exists, i.e. an implementation as an indirect measurement, where the system under investigation is coupled to an independently prepared probe system [6] and then only the probe is subject to a (projective) measurement [7, 8, 9].

The problem of finding the Naimark extensions of a POVM is indeed a central one in quantum technology. On the one hand, it provides a concrete model to realize the measurement[10, 11], and thus to assess entanglement cost[12] and/or implementations on different platforms [13, 14, 15, 16, 17, 18]. On the other hand, it permits to evaluate the post-measurement state and thus to investigate the tradeoff between information gain and measurement disturbance [19, 20, 21, 22, 23, 24, 25], as well as any procedure aimed at quantum control [26].

Let us consider a set of operators $\{\Pi_{m}\}$ that constitute a POVM for the physical system $S$ described by the Hilbert space $\mathcal{H}_{S}$, i.e.

[TABLE]

The elements of the set are not necessarily projectors, $\Pi_{n}\Pi_{m}\neq\Pi_{n}\delta_{n,m}$. The Naimark theorem states that it is possible to extend each POVM elements to a larger (product) Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{S}$ (see A) such that the extended measurement operators are projectors in the product space. In particular, it is possible to define the auxiliary Hilbert space $\mathcal{H}_{A}$ such that the system Hilbert space $\mathcal{H}_{S}$ is isomorphic to a subspace in $\mathcal{H}_{A}\otimes\mathcal{H}_{S}$, where the density operator $\rho$ defined on $\mathcal{H}_{S}$ corresponds to the density operator $|{e_{1}}\rangle\langle{e_{1}}|\otimes\rho$ defined on $\mathcal{H}_{A}\otimes\mathcal{H}_{S}$. The state $|{e_{1}}\rangle$ may be chosen as the state corresponding to the first vector of the canonical basis of $\mathcal{H}_{A}$. Naimark theorem states that we can find projectors $\{E_{m}\}$

[TABLE]

each of them corresponding to a POVM element $\Pi_{m}$ in the following sense. The distributions of the $m$-th outcome, as obtained from $\{E_{m}\}$ and $\{\Pi_{m}\}$ on the states $|{e_{1}}\rangle\langle{e_{1}}|\otimes\rho$ and $\rho$ respectively, are the same, i.e.

[TABLE]

At the operatorial level, this is expressed by the following the set of relations

[TABLE]

which, solved for the $E_{m}$ given the $\Pi_{m}$, provide the desired Naimark extension of the POVM.

As it was originally suggested by Helstrom[4] the projectors $\{E_{m}\}$ may be built by placing a copy of $\Pi_{m}$ in the upper-left block position of the matrix representation of $\{E_{m}\}$ (corresponding to the element 1 in the matrix $e_{1}\cdot e_{1}^{T}$). At the same time, 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 $\mathcal{H}_{A}\otimes\mathcal{H}_{S}$. However, the focus 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 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 $E_{m}$ is analogue to the original Helstrom recipe, i.e. we define the upper-left block in the matrix of $E_{m}$ equal to $\Pi_{m}$. 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 $\mathcal{H}_{A}$ is obtained only at the end of the procedure.

The algorithm initially builds the blocks of $E_{1}$ in order to satisfy the constraints (2) on itself, i.e.

[TABLE]

Then, we build some blocks of $E_{2}$ in order to satisfy the orthogonality with $E_{1}$,

[TABLE]

and then imposing the other constraints

[TABLE]

we define the remaining blocks. As we will see, this second step do not modify the previously defined blocks of $E_{2}$.

Analogously, the algorithm builds $E_{3}$ (if any) imposing its orthogonality with $E_{1},\ E_{2}$, and then imposing that $E_{3}\cdot E_{3}=E_{3}$. The generalisation is straightforward, the element $E_{m}$ is built in order to satisfy at first the ortogonality with the previously built projectors, and then imposing the condition $E_{m}\cdot E_{m}=E_{m}$. The 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

[TABLE]

These steps are analysed in some details in the following two Sections, whereas the overall algorithm is summarised in Section 2.3.

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 $E_{1}$, which has the block $\Pi_{1}$ in the upper-left position. If $\Pi_{1}$ is already idempotent, we can just put $\Pi_{1}$ in the corner and set the remaining blocks to zero. If this is not the case, we can define the blocks around $\Pi_{1}$ such that $E_{1}\cdot E_{1}=E_{1}$, possibly employing the 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 $E$

[TABLE]

with $\Pi$ a given block, while $A,\ B$ are blocks to find ($A^{\dagger}$ and $B\geq 0$ have been used so that $E=E^{\dagger}$). In the case $\Pi^{2}=\Pi$ we can omit $A,\ B$ since the matrix is already idempotent. Otherwise, we have to add the blocks $A,\ A^{\dagger},\ B$ and the matrix $E$ grows in sizes. The constraint

[TABLE]

gives the following equations:

[TABLE]

Equation (10) can be solved exploiting the singular value decomposition (SVD) for $\Pi=V\Lambda V^{\dagger}$ and $A=USW^{\dagger}$. Setting $U=V,\ W=I,\ S=\sqrt{\Lambda(I-\Lambda)}$ leads to $A=V\sqrt{\Lambda(I-\Lambda)}$. Assuming for the moment a full rank matrix $\Pi$, with eigenvalues strictly included in the range $(0,1)$, equation (11) allows us to find $B=I-\Lambda$.

Finally, equation (12) is verified with the above solutions, and the blocks of $E$ can be built as

[TABLE]

A different route may be also employed upon exploiting positivity of the elements of the POVM. Indeed, for positive semi-definite $\Pi$ we have the decomposition $\Pi\leavevmode\nobreak\ =\leavevmode\nobreak\ YY^{\dagger}$ (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 $\Pi$ 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

[TABLE]

and $B=I-Y^{\dagger}Y$ follows. Finally, equation (12) is verified by re-writing $A^{\dagger}A=\sqrt{B}(I-B)\sqrt{B}$.

For rectangular $Y$, equation (14) still holds upon defining $Y^{-1}$ as the Penrose inverse of Y, that is, the rectangular matrix satisfying $Y^{-1}Y=I$ on the support of $Y$. In addition, the decomposition $E=ZZ^{\dagger}$ is also readily available from $Y$,

[TABLE]

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., $E_{2}$, which has the upper-left block equal to $\Pi_{2}$ and must verify $E_{1}\cdot E_{2}=0$. If we have $\Pi_{1}\cdot\Pi_{2}=0$, it is enough to set the blocks adjacent to $\Pi_{2}$ equal to zero. In the most general case, this does not hold, and to satisfy the orthogonality condition we have to explicitly determine the blocks around $\Pi_{2}$. The expression

[TABLE]

where $\Pi_{1}=Y_{1}Y_{1}^{\dagger}$, provides the constraints

[TABLE]

Equation (18) allows us to find $A=-\Pi_{2}Y_{1}\sqrt{I-Y_{1}^{\dagger}Y_{1}}$, whereas equation (19) provides the expression $B=\left(\sqrt{I-Y_{1}^{\dagger}Y_{1}}\right)^{-1}Y_{1}^{\dagger}\Pi_{2}Y_{1}\sqrt{I-Y_{1}^{\dagger}Y_{1}}$. The third equation, (20), is indeed verified by these solutions.

At this stage, upon imposing the orthogonality with $E_{1}$, we found that $E_{2}$ has the structure

[TABLE]

where the blocks indicated by $\ast$ are left unused and may be exploited to impose other conditions on $E_{2}$. If a decomposition $\Pi_{2}=XX^{\dagger}$ is available, the big block just defined in (21) has a simple decomposition,

[TABLE]

Notice that the blocks just defined depend on $\Pi_{1}$ (via its decomposition) and upon $\Pi_{2}$. If we have to impose the orthogonality of matrix $E_{m}$ with $E_{1}$, only the non-zero blocks in $E_{1}$ would be involved. Thus, the solution would be the same substituting $\Pi_{m}=X_{m}X_{m}^{\dagger}$ in place of $\Pi_{2}=X_{2}X_{2}^{\dagger}$.

2.3 The algorithm

The algorithm builds the projectors one at a time, using the previously built projectors. For each projector $E_{m}$ two steps are performed: first the orthogonal construction of Section 2.2, which defines some blocks of $E_{m}$ 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 so that $E_{m}^{2}=E_{m}$. Before applying the orthogonal or idempotent construction, it is checked whether $E_{m}$ is already ortogonal to the other projectors or idempotent. If this is the case, the step is simply skipped.

The algorithm starts building $E_{1}$ with the idempotent construction, as the orthogonal one is not necessary. $\Pi_{1}$ is copied in the upper-left block of $E_{1}$ and the solution (15) is evaluated with $Y=Y_{1},\ Y_{1}Y_{1}^{\dagger}=\Pi_{1}$, where $Y_{1}$ has been obtained for instance from the singular value decomposition of $\Pi_{1}=V_{1}\Lambda_{1}V_{1}^{\dagger}$, giving $Y_{1}=V_{1}\sqrt{\Lambda_{1}}$. If $\Pi_{1}$ is full rank, the projector $E_{1}$ has a nonzero $2\times 2$ blocks in the upper-left corner. The remaining blocks are zero.

[TABLE]

The projector $E_{2}$ is then built using the two steps. The block $\Pi_{2}$ is copied in the upper-left corner, a decomposition $\Pi_{2}=X_{2}^{(1)}X_{2}^{(1)\dagger}$ is evaluated (by SVD if needed) and the three blocks around are defined as in (21) leveraging on the decompositon $\Pi_{1}=Y_{1}Y_{1}^{\dagger}$ previously evaluated. At this point, the big block just defined has a decomposition $Y_{2}Y_{2}^{\dagger}$ as in (22), and if not idempotent, the adjacent blocks need to be evaluated accordingly employing the idempotent construction of equation (15).

[TABLE]

Notice that in this case in the original matrix $E_{2}$ the $4\times 4$ blocks in the upper left corner are defined, for a total of $4D$ rows and $4D$ columns (if $\Pi_{1},\ \Pi_{2}$ are full rank), with $D$ being the dimension of the system Hilbert space. In the evaluation of $E_{3}$ the same orthogonal and idempotent construction are repeated, with the difference that the first must be repeated twice to get the orthogonality with $E_{1}$ ed $E_{2}$. As usual, first the block $\Pi_{3}=X_{3}^{(1)}X_{3}^{(1)\dagger}$ is copied in the upper left corner. The first blocks around $\Pi_{3}$ are evaluated with (21).

The newly defined big block has decomposition $X_{3}^{(2)}X_{3}^{(2)\dagger}$ obtained from (22) where $X=X_{3}^{(1)},\ Y_{2}=X_{3}^{(2)}$. The orthogonal construction (21) is repeated to get the orthogonality with $E_{2}$, employing the block $X_{3}^{(2)}X_{3}^{(2)\dagger}$ and the term $Y_{2}$ previously defined in the idempotent contruction of $E_{2}$. A new bigger block is obtained with decomposition $Y_{3}Y_{3}^{\dagger}$ as in (22).

The idempotent construction is then used employing $Y=Y_{3}$ as in equation (15). Finally, we get the matrix (note that not all the blocks have the same size)

[TABLE]

with $R_{1}=\sqrt{I-Y_{1}^{\dagger}Y_{1}},\ R_{2}=\sqrt{I-Y_{2}^{\dagger}Y_{2}},\ R_{3}=\sqrt{I-Y_{3}^{\dagger}Y_{3}}$. Notice that the expression of $E_{3}$ depends upon the decompositions $Y_{1}Y_{1}^{\dagger},\ Y_{2}Y_{2}^{\dagger}$ of the upper-left blocks of the preceding projectors. This holds for each projector $E_{m}$.

The method used to evaluate $E_{3}$ may be iterated for any subsequent projector. First, the block $\Pi_{m}=X_{m}^{(1)}X_{m}^{(1)\dagger}$ is copied in the upper left corner. The adjacent blocks are defined imposing the orthogonality with $E_{1}$, following the orthogonal construction. The just defined block has decomposition $X_{m}^{(2)}X_{m}^{(2)\dagger}$, and the orthogonal construction is repeated using $Y_{n}$, which is the term used in the idempotent construction of $E_{n},\ n<m$. At the end of each orthogonal constructions, the newly defined big block is decomposed as $X_{m}^{(i)}X_{m}^{(i)\dagger},\ i<m$, and the orthogonal construction is repeated until $i$ reach $m$. The term $X_{m}^{(m)}=Y_{m}$ is then used in the idempotent construction to get the final block structure of $E_{m}$.

At this stage, upon following the procedure leading to $E_{m}$, a recursive construction may be also obtained for its decomposition $E_{m}=Z_{m}Z_{m}^{\dagger}$. For further details, see B. If all the $\Pi_{m}$ are full rank and with eigenvalues in the range $[0,1]$, then the size of the projectors grows exponentially. In fact, the projector $E_{1}$ has in this case $2\times 2$ non-zero blocks, for a total of $2D$ rows and $2D$ columns; the projector $E_{2}$ populates $4\times 4$ blocks, the projector $E_{3}$ has $8\times 8$ non-zero blocks, and so on. An exception occurs if some of the blocks already satisfy the ortogonality conditions. For instance, if $\Pi_{2}$ is already orthogonal to $\Pi_{1}$, there is no need to used the adjacent blocks to obtain its orthogonality. This is also the case if the block is idempotent, since the adjacent blocks may left unused.

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 $\{\Pi_{1},\Pi_{2},\Pi_{3}\}$, $\Pi_{1}+\Pi_{2}+\Pi_{3}={\mathbb{I}_{S}}$, defined by $\Pi_{k}=\frac{2}{3}|{\psi_{k}}\rangle\langle{\psi_{k}}|,\ k=1,2,3$, where [4]

[TABLE]

i.e.

[TABLE]

The extension originally obtained by Helstrom was based on a two-dimensional auxiliary Hilbert space with basis $|{v_{1}}\rangle=(1,0)^{T}$, $|{v_{2}}\rangle=(0,1)^{T}$, and it is given by $E_{k}^{H}=|{\xi_{k}}\rangle\langle{\xi_{k}}|$, $k=1,..,4$, where

[TABLE]

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.

[TABLE]

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 report the matrix expression of $E_{1}^{H}$ for comparison with $E_{1}$ in Eq. (46)

[TABLE]

3.2 Four elements POVM

Helstrom also considered a four-elements POVM $\{\Pi_{1},\ \Pi_{2},\ \Pi_{3},\ \Pi_{4}\}$, with[4]

[TABLE]

i.e.

[TABLE]

Again, the iterative algorithm easily finds the extension since the orthogonal construction directly gives idempotent matrices, without the need of the idempotent construction.

[TABLE]

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 $\sigma_{k}$ with a probability $\{z_{k}\}$ in the set of nondegenerate and isospectral observables $\{\sigma_{k}\}$. The POVM 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

[TABLE]

where $|{\omega_{A}}\rangle$ is the ancillary state, $U$ describes the coupled evolution between the systems, and ${P_{m}}$ is the projective measurement in the ancillary system. A first example of POVM is that of a roulette obtained from the Pauli operators $\{\sigma_{1},\sigma_{3}\}$ with probabilities $\{z,1-z\},\ z\in(0,1)$, giving the elements

[TABLE]

The solution proposed uses the ancillary state $|{\omega_{A}}\rangle=\frac{1}{\sqrt{2}}\left(|{0}\rangle+e^{i\phi}|{1}\rangle\right)$, the projectors

[TABLE]

and the unitary

[TABLE]

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

[TABLE]

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,

[TABLE]

The proposed extension employs the ancillary state $|{\omega_{A}}\rangle=|{e_{1}}\rangle$, the projectors of the observable $\sigma_{3}$, i.e. $P_{1}=|{e_{1}}\rangle\langle{e_{1}}|,\ P_{-1}=|{e_{2}}\rangle\langle{e_{2}}|$, and the unitary

[TABLE]

which gives

[TABLE]

In this case the iterative algorithm is particularly easy to apply since we have diagonal POVM elements, and it gives the solution

[TABLE]

which is equivalent to (76) since in both cases we can see $\Pi_{1}$ 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 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 $\mathcal{H}_{S}\otimes\mathcal{H}_{A}$, with the system Hilbert space $\mathcal{H}_{S}$ on the left. However, given the definition of Kronecker product

[TABLE]

the opposite convention, i.e. describing the composite system by the Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{S}$, makes it easier to graphically visualize the product matrix. For instance, for a matrix given by the product of the first element of the canonical basis only one block is non-zero

[TABLE]

The standard convention would make the notation more cumbersome.

Appendix B Building the decomposition of $E_{m}$

The procedure explained in Section 2.3 suggests a recursive construction to directly obtain the decomposition $E_{m}=Z_{m}Z_{m}^{\dagger}$. In order to evaluate $Z_{m}$, we initially need a decomposition $\Pi_{m}=X_{m}^{(1)}X_{m}^{(1)\dagger}$, obtained for instance from its singular value decomposition. Then, the orthogonal construction (22) is applied with $X=X_{m}^{(i)},\ Y_{1}=Y_{i}$ to evaluate $Y_{2}=X_{m}^{(i+1)}$. This step is repeated for $i=1$ to $m$. The last block calculated, $X_{m}^{(m)}$, is defined as $Y_{m}$ and used in the idempotent construction (16) employing $Y=Y_{m}$ to get $Z=Z_{m}$.

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

[TABLE]

Notice that to obtain the term $Z_{m}$ the decomposition $X_{m}$ of $\Pi_{m}$ is used, as well as all the terms $Y_{1},\ Y_{2},\ \dots Y_{m-1}$ used in the preceding idempotent constructions. This is an efficient procedure, since the terms such as $\left(\sqrt{I-Y_{i}^{\dagger}Y_{i}}\right)^{-1}Y_{i}^{\dagger},\ i<m$ are used in the later evaluation of the projectors, without the need to evaluate them at each iteration. Notice that also in this procedure we should check whether the matrices $X_{m}^{(i)}X_{m}^{(i)\dagger}$ are orthogonal to $E_{i}$ or if $Y_{m}Y_{m}^{\dagger}$ is idempotent. In this cases, there is no need to perform the orthogonal or idempotent construction of the algorithm.