We show that the density matrix of a spin- $l$ system can be described entirely in terms of the measurement statistics of projective spin measurements along a minimum of $4l+1$ different spin directions. It is thus possible to represent the complete quantum statistics of any $N$ -level system within the spherically symmetric three-dimensional space defined by the spin vector. An explicit method for reconstructing the density matrix of a spin-1 system from the measurement statistics of five nonorthogonal spin directions is presented and the generalization to spin- $l$ systems is discussed.We show that the density matrix of a spin-$l$ system can be described entirely in terms of the measurement statistics of projective spin measurements along a minimum of $4l+1$ different spin directions. It is thus possible to represent the complete quantum statistics of any $N$-level system within the spherically symmetric three-dimensional space defined by the spin vector. An explicit method for reconstructing the density matrix of a spin-1 system from the measurement statistics of five nonorthogonal spin directions is presented and the generalization to spin-$l$ systems is discussed.

arXiv:quant-ph/0310003Physical Review AApr 23, 200436 citations

Reconstructing the density matrix of a spinsthrough Stern - Gerlach measurements

Jean-Pierre Amiet, Stefan Weigert

The Pauli problem is solved for a spin of length s prepared in an arbitrary (unnormalized) mixed state which has free real parameters. The reconstruction of its density operator is possible if one knows the probabilities of the spin components along each of directions in space. These probabilties are directly accessible through measurements performed with a Stern - Gerlach apparatus. A multipole expansion of the density operator establishes the link between the matrix elements of and the measured intensities.The Pauli problem is solved for a spin of length s prepared in an arbitrary (unnormalized) mixed state which has free real parameters. The reconstruction of its density operator is possible if one knows the probabilities of the spin components along each of directions in space. These probabilties are directly accessible through measurements performed with a Stern - Gerlach apparatus. A multipole expansion of the density operator establishes the link between the matrix elements of and the measured intensities.

Journal of Physics A Mathematical and GeneralAug 7, 199819 citations

Optimal measurements and fixed-point maps

Chirag Dhara, N. D. Hari Dass

When an optimal measurement of $({S}_{x},{S}_{y},{S}_{z})$ is made on a qubit and what we call a mutually unbiased mixture of the resulting ensembles is taken, then the post-measurement density matrix is shown to be related to the premeasurement density matrix through a simple linear relation. It is also shown that the form of this relation is the same for all quantum systems. It is shown that for a general quantum system such a relation holds only when the measurements are made in mutually unbiased bases. The post-measurement density matrix is shown to be a normalized incoherent superposition of the identity map and the pin map of Gorini and Sudarshan. A pin map is one which maps all density matrices to a fixed density matrix which in our case turns out to be the unit matrix. The result is shown to be true irrespective of whether the initial state is pure or mixed. For spin- $1∕2$ systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.When an optimal measurement of $({S}_{x},{S}_{y},{S}_{z})$ is made on a qubit and what we call a mutually unbiased mixture of the resulting ensembles is taken, then the post-measurement density matrix is shown to be related to the premeasurement density matrix through a simple linear relation. It is also shown that the form of this relation is the same for all quantum systems. It is shown that for a general quantum system such a relation holds only when the measurements are made in mutually unbiased bases. The post-measurement density matrix is shown to be a normalized incoherent superposition of the identity map and the pin map of Gorini and Sudarshan. A pin map is one which maps all density matrices to a fixed density matrix which in our case turns out to be the unit matrix. The result is shown to be true irrespective of whether the initial state is pure or mixed. For spin-$1∕2$ systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.

arXiv:quant-ph/0406169Physical Review AAug 12, 20050 citations

We show that the density matrix of a spin-

l

system can be described entirely in terms of the measurement statistics of projective spin measurements along a minimum of

4l+1

different spin directions. It is thus possible to represent the complete quantum statistics of any

N

-level system within the spherically symmetric three-dimensional space defined by the spin vector. An explicit method for reconstructing the density matrix of a spin-1 system from the measurement statistics of five nonorthogonal spin directions is presented and the generalization to spin-

l

systems is discussed.

When an optimal measurement of

({S}_{x},{S}_{y},{S}_{z})

is made on a qubit and what we call a mutually unbiased mixture of the resulting ensembles is taken, then the post-measurement density matrix is shown to be related to the premeasurement density matrix through a simple linear relation. It is also shown that the form of this relation is the same for all quantum systems. It is shown that for a general quantum system such a relation holds only when the measurements are made in mutually unbiased bases. The post-measurement density matrix is shown to be a normalized incoherent superposition of the identity map and the pin map of Gorini and Sudarshan. A pin map is one which maps all density matrices to a fixed density matrix which in our case turns out to be the unit matrix. The result is shown to be true irrespective of whether the initial state is pure or mixed. For spin-

1∕2

systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.