We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable the identification of all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If $d$ is odd, then this observable is informationally complete. But if $d$ is even, then the joint observable is not informationally complete, and one has to allow more noise in order to obtain informational completeness.We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable the identification of all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If $d$ is odd, then this observable is informationally complete. But if $d$ is even, then the joint observable is not informationally complete, and one has to allow more noise in order to obtain informational completeness.

arXiv:1111.3509Physical Review AJan 24, 201251 citations

Further Aspects of Quantum Observables

Kevin Costello, Owen Gwilliam

Cambridge University Press eBooksSep 10, 20210 citations

Data-driven inference, reconstruction, and observational completeness of quantum devices

Michele Dall’Arno, Francesco Buscemi, Alessandro Bisio +1 more

The range of a quantum measurement is the set of outcome probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of outcome distributions, infers the quantum measurement which is (i) consistent with the data, in the sense that its range contains all the distributions observed, and (ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of outcome distributions. We show that data-driven inference is able to return a unique measurement for any data set if and only if the inference adopts a (hyper)spherical state space (for example, the classical or the quantum bit). In analogy to informational completeness for quantum tomography, we define observational completeness as the property of any set of states that, when fed into any given measurement, produces a set of outcome distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.The range of a quantum measurement is the set of outcome probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of outcome distributions, infers the quantum measurement which is (i) consistent with the data, in the sense that its range contains all the distributions observed, and (ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of outcome distributions. We show that data-driven inference is able to return a unique measurement for any data set if and only if the inference adopts a (hyper)spherical state space (for example, the classical or the quantum bit). In analogy to informational completeness for quantum tomography, we define observational completeness as the property of any set of states that, when fed into any given measurement, produces a set of outcome distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.

arXiv:1812.08470Physical Review ADec 11, 20206 citations

Universal quantum observables

Giacomo Mauro D’Ariano

Physics Letters AJul 1, 200228 citations

Interacting Quantum Observables

Bob Coecke, Ross Duncan

Lecture notes in computer scienceAug 12, 2008257 citations

Quantum indirect estimation theory and joint estimate of all moments of two incompatible observables

Giacomo Mauro D’Ariano, Paolo Perinotti, Massimiliano Federico Sacchi

We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of $AB$ -informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables $A$ and $B$ . After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of $AB$ infocompleteness and the notion of joint measurement of observables $A$ and $B$ .We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e.g., estimating the ensemble average of an observable by measuring other observables. A main ingredient in this approach is that of informationally complete (infocomplete) measurements, which allow us to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of $AB$-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables $A$ and $B$. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of $AB$ infocompleteness and the notion of joint measurement of observables $A$ and $B$.

arXiv:0708.3960Physical Review AMay 9, 20081 citation

d

is odd, then this observable is informationally complete. But if

d

is even, then the joint observable is not informationally complete, and one has to allow more noise in order to obtain informational completeness.

AB

-informationally complete measurements, by which one can estimate jointly all the moments of two incompatible observables

A

and

B

. After analyzing all general properties of such measurements, we address the problem of their optimality, and we completely solve the case of qubits, showing that a ${\ensuremath{\sigma}}_{x}{\ensuremath{\sigma}}_{y}$-infocomplete measurement is less noisy than any infocomplete one. We will also discuss the relation between the concept of

AB

infocompleteness and the notion of joint measurement of observables

A

and

B