# Realization of POVMs using measurement-assisted programmable quantum processors

###### Abstract

We study possible realizations of generalized quantum measurements on measurement-assisted programmable quantum processors. We focus our attention on the realization of von Neumann measurements and informationally complete POVMs. It is known that two unitary transformations implementable by the same programmable processor require mutually orthogonal states. It turns out that the situation with von Neumann measurements is different. Specifically, in order to realize two such measurements one does not have to use orthogonal program states. On the other hand, the number of the implementable von Neumann measurements is still limited. As an example of a programmable processor we use the so-called quantum information distributor.

###### pacs:

03.65.Ta, 03.67.-a## I Introduction

General quantum measurements are formalized as positive operator valued measures (POVM), i.e. sets of positive operators summing up to identity, (see, for instance, Refs. [1, 2, 3, 4]). From the quantum theory it follows that each collection of such operators corresponds to a specific quantum measurement. However, the theory does not tell us anything about particular physical realization of a specific POVM. The aim of this paper is to exploit the so-called measurement-assisted quantum processors to perform POVMs.

The Stinespring-Kraus theorem [5] relates quantum operations (linear completely positive trace-preserving maps) with unitary transformations. In particular, any quantum operation realized on the system corresponds to a unitary transformation performed on a larger system , i.e.

(1) |

where is a suitably chosen state of the system and denotes a partial trace over the ancillary system . The assignment is one-to-many, because the dilation of the Hilbert space of the system can be performed in many different ways. However, if we fix the transformation , states of the ancillary system control and determine quantum operations that are going to be performed on the system . In this way one obtains a concept of a programmable quantum processor, i.e. a fixed piece of hardware taking as an input a data register (system ) and a program register (system ). Here the state of the program register encodes the operation that is going to be performed on the data register.

In a similar way, any quantum generalized measurement (POVM), that is represented by a set of positive operators , can be understood as a von Neumann measurement performed on the larger system [4]. von Neumann measurements are those for which are mutually orthogonal projectors, i.e. . The Neumark theorem (see, e.g. Ref. [6]) states that for each POVM there exists a von Neumann measurement on a larger Hilbert space and for all , where is some state of the system . Moreover, it is always possible to choose a von Neumann measurement such that are is a unitary transformation and are projectors defined on the system . Using the cyclic property of a trace operation, i.e. , we see that the von Neumann measurement can be understood as a unitary transformation followed by a von Neumann measurement performed on the ancillary system only.

As a result we obtain the couple that determines a programmable quantum processor assisted by a measurement of the program register, i.e. measurement-assisted programmable quantum processor. Such device can be used to perform both generalized measurements as well as quantum operations.

Programmable quantum processors (gate arrays of a finite extent) has been studied first by Nielsen and Chuang [7]. They have shown that no programmable quantum processor can perform all unitary transformations of a data register. To be specific, in order to encode unitaries into a program register one needs mutually orthogonal program states. Consequently, the required program register has to be described by an inseparable Hilbert space, because the number of unitaries is uncountable. However, if we work with a measurement-assisted programmable quantum processor, then with a certain probability of success we can realize all unitary transformations [8, 9, 10, 11]. The probability of success can be increased arbitrarily close to unity utilizing conditioned loops with a specific set of error correcting program states [8, 12, 13, 14].

So far, the properties of quantum processors with respect to realization of quantum operations has been studied in several papers [7, 8, 9, 10, 11, 15]. In the present paper we will exploit measurement-assisted quantum processors to perform POVMs. The problem of the implementation of von Neumann measurement by using programmable “quantum multimeters” for discrimination of quantum state has been introduced in Ref. [16] and subsequently studied in Refs. [17, 18, 19]. An analogous setting of a unitary transformation followed by a measurement has been used in Ref. [20] to evaluate/measure the expectation value of any operator. The quantum network based on a controlled-SWAP gate can be used to estimate non-linear functionals of quantum states [21] without any recourse to quantum tomography. Recently D‘Ariano and co-wrokers [22, 23, 24] have studied how programmable quantum measurements can be efficiently realized with finite-dimensional ancillary systems. In the present paper we will study how von Neumann measurements and informationally complete POVMs can be realized via programmable quantum measurement devices. In particular, we will show that this goal can be achieved using the so called quantum information distributor [25, 26].

## Ii General consideration

Let us start our investigation with an assumption that the program register is always prepared in a pure state, i.e. . In this case the action of the processor can be written in the following form

(2) |

where is some basis in the Hilbert space of the program register and . In particular, we can use the basis in which the measurement is performed, i.e. , where is a subset of indices . Note that , because .

Measuring the outcome the data evolve according to the following rule (the projection postulate)

(3) | |||||

with the probability . Consequently for the elements of the POVM we obtain

(4) |

If we consider a general program state with its spectral decomposition in the form , then the transformation reads

(5) |

with and . Therefore the operators

(6) |

constitute the realized POVM.

Given a processor and some measurement one can easily determine which POVM can be performed. Note that the same POVM can be realized in many physically different ways. Two generalized measurements are equivalent, if the resulting functionals () coincide for all , i.e. they result in the same probability distributions. For the purpose of the realization of POVMs, the state transformation during the process is irrelevant. However, two equivalent realizations of POVM can be distinguished by the induced state transformations (for more on quantum measurement see Ref. [4]).

Let us consider, for instance, the trivial POVM, which consists of operators (). In this case the observed probability distribution is data-independent and some quantum operation is realized. In all other cases, the state transformation depends on the initial state of the data register, and is not linear [12]. In these cases the resulting distribution is nontrivial and contains some information about the state . In the specific case when the state can be determined (reconstructed) perfectly, the measurement is informationally complete. In this case we can perform the complete state reconstruction. Any collection of linearly independent positive operators determine such informationally complete POVM. In particular, they form an operator basis, i.e. any state can be written as a linear combination . Using this expression the probabilities read

(7) |

where the coefficients define a matrix . In this setting the (inverse) problem of the state reconstruction reduces to a solution of a system of linear equations , where are unknown. The solution exists only if the matrix is invertible and then .

The purpose of any measurement is to provide us information about the state of the physical system based on the results of measurement. The presented scheme of measurement-assisted quantum processor represents quite general picture of the physical realization of any POVM.

## Iii Quantum information distributor

In this section we will present a specific example of a quantum processor the so-called quantum information distributor (QID) [25]. This device uses as an input a two-qubit program register and a single-qubit data register. The processor consists of four CNOT gates. Its name reflects the property [25] that in special cases of program states the QID acts as an optimal cloner and the optimal universal NOT gate, i.e. it optimally distributes quantum information according to a specific prescription. Moreover, it can be used to perform an arbitrary qubit rotation with the probability [10]. The action of the QID can be written in the form [12]

(8) |

where are sigma matrices, and is a two-qubit program-register basis in which the measurement is performed ().

In what follows we shall extend the list of applications of the QID processor and show how to realize a complete POVM, i.e. a complete state reconstruction. For a general program state with () the POVM consists of the following four operators

(9) |

with , . and

Note that for the initial program state with , () the probabilities are -independent, and a unitary operation is realized [10]. The question of interest is whether an informationally complete POVM can be encoded into a program state. In fact, the problem reduces to the question of the linear independency of operators for some . Using the vector representation of operators, , one can show that the operators are linearly independent only if none of the coefficients of vanishes.

The elements of a POVM can be represented in the Bloch-sphere picture. This is due to the fact that operators , and represent quantum states. Choosing the program state

(10) |

we obtain the informationlly complete POVM with a very symmetric structure. In particular, the operators are proportional to pure states associated with vertexes of a tetrahedron drawn inside the Bloch sphere (see Fig. 1). These operators read

(11) | |||||

(12) | |||||

(13) | |||||

(14) |

It is obvious that these operators are not mutually orthogonal, but 7) between the observed probability distribution and the data state . Using this identity one can easily compute the relation (

(15) |

where we used the notation . The last equation completes the task of the state reconstruction task.

Because of the identity for the realized POVM is of a special form. It belongs to a family of the so-called symmetric informationally complete measurements (SIC POVM) [27]. These measurements are of interest in several tasks of quantum information processing and possess many interesting properties. It is known (see. e.g. Ref. [27]) that for qubits there essentially exist only only two (up to unitaries) such mesurements. Above we have shown how one of them can be performed using the QID processor.

## Iv von Neumann measurements

An important class of measurements is described by the projector valued measures (PVM), which under specific circumstances enable us to distinguish between orthogonal states in a single shot, i.e. no measurement statistics is required. A set of operators form a PVM, if and , i.e. it contains mutually orthogonal projectors. The total number of (nonzero) operators cannot be larger than the dimension of the Hilbert space .

Usually the von Neumann measurements are understood as those that are compatible with the projection postulate, i.e. the result associated with the operator induces the state transformation

(16) |

That is, the state after the measurement is described by the corresponding projector .

However, each PVM can be realized in many different ways and a particular von Neumann measurement is only a specific case. In our settings the realized POVM is related to the state transformation via the identity , where . The set of operators , with projectors and unitary transformations, define the same PVM given by . In particular, , but the state transformation results in

(17) |

Thus the final state is described by a projector, but not in accordance with the projection postulate. We refer to the PVMs that are compatible with the projection postulate as the von Neumann measurements. Moreover, for a simplicity we shall assume that the projectors are always one-dimensional, i.e. the PVM is associated with non-degenerate hermitian operators.

The action of the processor implementing two von Neumann measurements and can be written as

(18) | |||||

(19) |

It is well known [11] that when two sets of Kraus operators are realizable by the same processor , then the following necessary relation holds . Using this relation for the projections , we obtain the identity

(20) |

where . For general measurements, the operator on the left-hand side of the previous equation contains off-diagonal elements. In this case the corresponding program states must be orthogonal, i.e. . This result is similar to the one obtained by Nielsen and Chuang [7] who have studied the possibility of the realization of unitary transformations via programmable gate arrays. They have shown that in order to perform (with certainty) two unitary transformations on a fixed quantum processor one needs two orthogonal program states. However, in our case we still cannot be sure that measurement-assisted processor realizing two von Neumann measurements exists. Moreover, there are possibilities, when the condition holds also for non-orthogonal program states (see the case study below).

In order to realize a projective measurement on a -dimensional data register the program space must be at least dimensional. Let us start with the assumption that the Hilbert space of the program register is dimensional. In this case the program states have to be orthogonal (this is due to the fact the expression (20) contain off-diagonal elements). Let us consider different (non-degenerate) von Neumann measurements that are determined by a set of operators ( and for all ). Let denote the associated program states and . It is easy to see that for general measurements the resulting operator

(21) |

is not unitary. In particular, . The equality would require that the identity holds. Therefore, we conclude that neither orthogonal states do guarantee the existence of a programmable processor that performs desired set of measurements. This result makes the case of programming the unitaries and von Neumann measurements different.

For instance, let us consider a two-dimensional program register and let us denote and . Then the above condition reads . Using the definition and we obtain the orthogonality conditions . Consequently, because in the two-dimensional case the orthogonal state is unique, we obtain and , i.e. the measurements are the same. For larger the situation is different. The realizable measurements must possess the derived property which can be summarized with the help of Tab. I.

In order to realize more von Neumann measurements on a qudit one has to work with a larger-dimensional program space, i.e. . In general, in this case we work with outcomes and projective operators that define the realized measurement of the program register. However, each PVM consists of maximally projectors. Therefore, of the induced operators should represent the zero operator. It means that when we are realizing the von Neumann measurement such that some of the outcomes do not occur, i.e. probability of them vanishes for all data states. However, there is one more option that the set of operators () contains exactly only different operators (projectors). This means that more results can specify the same projection and define a single result of the realized von Neumann measurement.

The idea of additional, the so called, “zero” operators can be used to formulate a general statement about the implementation of any collection of arbitrary von Neumann measurements. Let us consider von Neumann measurements given by non-zero operators (number of equals to ). We can define the sets of operators by adding to these sets zero operators so that the condition holds. Using this approach we find out that any collection of von Neumann measurements can be realized on a single quantum processor given by Eq.(21) with (maximally) dimensional program space.

### iv.1 Case study: Projective measurements on a qubit.

Let us consider two von Neumann measurements and on a qubit. Further, let us assume a three-dimensional program space and define measurements and , respectively. It is easy to see that neither of these two sets of operators do satisfy the condition . The equality holds only if , i.e. and , but this implies that both measurements are the same. Consequently, the dimension of the program space has to increase by one. Then we have , and the condition holds for all possible measurements . It follows that the implementation of von Neumann measurements on a qubit requires -qubit program space.

The program space of the QID processor given by Eq.(8) consists of two qubits. Using the conclusion of the previous paragraph it follows that two von Neumann measurements could be performed with the help of this processor. It is easy to see that the operators with are not projectors. Moreover, they do not vanish only for certain . Consequently, the projective measurement cannot be realized in the same way as described above. However, the QID-processor can still be exploited to perform a von Neumann measurement.

Using the program state the operator (i.e., , where ) is a projection onto the vector . It is obvious that and , where . It turns out that we have realized PVM described by , i.e. the eigenvectors of the measurement. The state transformation reads (if ), respectively. It follows that the realization of the measurement of is in accordance with the projection postulate. In the same way we can realize and measurement (in these cases different results must be paired). Basically, this corresponds to a choice of different two-valued measurements, but in reality we perform only a single four-valued measurement. As a result we find that on QID we can realize three different von Neumann measurements. Note that we have used only two qubits as the program register. Moreover, the associated program states are not mutually orthogonal, but (for ) and Eq. (20) holds. Namely, for the measurements of and the condition (20) reads .

Till now we have always assumed that program states that encode two von Neumann measurements have to be orthogonal. The last paragraph describes an counterexample. As we already mentioned in some specific cases the condition of the orthogonality can be relaxed.

### iv.2 Note. Projection valued measures

If we relax the compatibility with the projection postulate more PVMs can be realized on a single processor. Let us consider that the dimension of the program space equals and is the state that encodes the PVM given by a set . The action of can be written as

(22) |

and the condition must hold. Let us consider two PVMs on a qubit and . Define a unitary map such that and . Using this map we can define a processor by following equations

(23) |

where , and . Direct calculation shows that , i.e. is unitary. From here it follows that if one does not require the validity of the projection postulate, then any two PVMs can be performed on a processor with two-dimensional program space.

This result holds in general. Let us consider a set of PVMs on a qudit. There always exist unitary transformations such that operators satisfy the condition . Without the loss of generality we can consider that measurement is given by projectors and by (see Tab. II.).

## V Conclusion

In this paper we have studied how POVMs can be physically realized using the so-called measurement-assisted quantum processors. In particular, we have analyzed how to perform complete state reconstruction and von Neumann measurements. As a result we have found that an arbitrary collection of von Neumann measurements cannot be realized on a single programmable quantum processor. We have shown how to use the QID processor to perform the state reconstruction.

The number of implementable von Neumann measurements is limited by the dimensionality of the program register. Our main result is that with a program register composed of qudits one can surely define a processor which performs arbitrary von Neumann measurements. In fact, in general one can do much better. We have shown that the usage of non-orthogonal program states can be helpful. In particular, the QID processor can be exploited to perform three von Neumann measurements by using only two qubits as a program register and nonorthogonal states. Using only dimensional program space one can realize maximally von Neumann measurements on a qudit (for a qubit we have ).

Relaxing the condition on compatibility with the projection postulate the processor allows us to realize any collection of PVMs just by using dimensional program space. An open question is whether we can perform more PVMs, or not. The two tasks can be performed by programmable processors: the realization of von Neumann measurements and the application unitary transformations on the data register, are different. According to Nielsen and Chuang [7], any collection of unitary transformations requires dimensional program space. For von Neumann measurements the upper bound reads and any improvement strongly depends on the specific set of these measurements. The characterization of these classes of measurements is an interesting topic that will be studied elsewhere.

###### Acknowledgements.

This work was supported in part by the European Union projects QUPRODIS, QGATES and CONQUEST. We thank Peter Štelmachovič for valuable discussions. We would also like to thank Mariá n Roško for reading the manuscript and having no comments.## References

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