An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in
Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such
n2 vectors in
Cn for any
n .But the existence is only proved when
n≤15 and
n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in
Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1
ρ∈Mn(C) is called a density matrix (or state) if
ρ≥0 and
′b(ρ)=1.For any density matrix
ρ , we can define a state
ρ^ by
ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t
(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators
{Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if
Pi≥0(1≤i≤k) and
∑i=1kP=I.If
P1 is a projection, then
{Pi}i=1k is called a projection valued measure
(PVM) .
Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric informationally complete (SIC) POVM, is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great
importance in quantum information and quantum foundations, but often difficult to perform. We present an
experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context.
One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the
most compact set of measurements that can be used to fully describe a quantum state. We use our technique
to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the
highly symmetric nature of this measurement, such a representation has the unique property that it permits all
other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making
no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states
encoded in single photons.
Symmetric, informationally complete, positive operator-valued measurements (SIC-POVM) are fundamental in quantum information and foundations of quantum theory[1]. For example, SIC-POVMs are crucial for optimal quantum state tomography, quantum key distribution with optimal trade-off between security and key rate, device-independent certification of optimal randomness from one bit of entanglement, and information-based interpretations and operational reconstructions of quantum theory. Experimentally, photonic experiments of qubit and qutrit tomography, and quantum key distribution[2] have aimed for SIC-POVMs. However, these experiments rely on specific models for parts of the experimental setups and do not allow us to test whether the target measurements are truly SIC-POVMs in a "device independent" (DI) way that is, without relying on any modelling of the devices.
An unavoidable task in quantum information processing is how to obtain data about the state of an individual system by suitable measurements. Informationally complete measurements are relevant in quantum state tomography, quantum cryptography, quantum cloning, and other questions. Symmetric informationally complete measurements (SIC-POVMs) form an especially important class of such measurements. We formulate some novel properties and relations for general SIC-POVMs in a finite-dimensional Hilbert space. For a given density matrix and any general SIC-POVM, the so-called index of coincidence of generated probability distribution is exactly calculated. Using this result, we obtain state-dependent entropic bounds for a single general SIC-POVM. Lower entropic bounds are derived in terms of the Rényi α -entropies for and the Tsallis α -entropies for . A lower bound on the min-entropy of an SIC-POVM is separately examined. For a pair of general SIC-POVMs, entropic uncertainty relations of the Maassen–Uffink type are considered.