text:
\section{Introduction}

 Information and physics are closely and fascinatingly
intertwined. Their relations become even more interesting when we
leave a non-relativistic quantum mechanics for more exiting
venues. My notes are planned as a guided tour for the first steps
along that road, with  open questions and more involved mergers
left to the remarks and to the last section.

I start from a brief introduction to causality restrictions on the
distributed measurements: the limitations that are imposed by
final propagation velocity of the physical interactions. It is
followed by  the relativistic transformations of the states of
massive particles and photons, from which we can deduce what
happens to qubits which are realized as the discrete degrees of
freedom. Building on this, I discuss the distinguishability of
quantum signals, and briefly touch communication channels and the
bipartite entanglement.

I do not follow a historical order or give all of the original
references. A  review \cite{rmp} is used as the standard reference
on quantum information and relativity. The results of the ``usual"
quantum information are given without any reference: all of them
can be found in at least one of the sources \cite{nc,keyl,bgl}.
Finally, a word about units:  $\hbar=c=1$  are always assumed.

\section{Causality and distributed measurements}
 Here I present the causality constraints on  quantum measurements.
For simplicity, measurements are considered to be point-like
interventions.  First  recall the standard description of the
measurement and the induced state transformation. Consider a
system in the state $\rho$ that is subject to measurement  that is
described by a positive operator-valued measure (POVM)
$\{E_\mu\}$. The probability of the outcome $\mu$ is
\begin{equation}
p_\mu={\rm tr}\, E_\mu\rho,
\end{equation}
while the state transformation is given by some completely
positive evolution
\begin{equation}
\rho\rightarrow\rho'_\mu=\sum_mA_{\mu m}\rho A_{\mu m}^\dagger/p_{\mu},
\qquad \sum_m A_{\mu m}^\dagger A_{\mu m}=E_\mu.
\end{equation}
If the outcome is left unknown, the update rule is
 \begin{equation}
\rho\rightarrow\rho=\sum_{\mu m}A_{\mu m}\rho A_{\mu m}^\dagger.
 \end{equation}

Now consider a bipartite state  $\rho_{AB}$. The operations of
Alice and Bob are given by the operators $A_{\mu m}$ and $B_{\nu
n}$, respectively. It is easy to see that if these operators
commute,
\begin{equation}
[A_{\mu m}, B_{\nu n}]=0, \label{etcr}
\end{equation}
then the observation statistics of Bob is independent of Alice's
results and vice versa.  Indeed, the probability that Bob gets a
result $\nu$, irrespective of what Alice found, is
\begin{equation}
p_\nu=\sum_\mu{\rm tr}\,\Bigl(\,\sum_{m,n}B_{\nu n}\,A_{\mu m}\,\rho\,
  A^\dagger_{\mu m}\,B^\dagger_{\nu n}\Bigr). \label{pnu}
\end{equation}

Now make use of Eq.~(\ref{etcr}) to exchange the positions of $A_{\mu
m}$ and $B_{\nu n}$, and likewise those of $A^\dagger_{\mu m}$ and
$B^\dagger_{\nu n}$, and then we move $A_{\mu m}$ from the first
position to the last one in the product of operators in the traced
parenthesis. Since the elements of a POVM satisfy $\sum_\mu
E_\mu=\mbox{1\hskip-.25em l}$, Eq.~(\ref{pnu}) reduces to
\begin{equation}
p_\nu={\rm tr}\,\Bigl(\sum_n B_{\nu n}\,\rho\,B^\dagger_{\nu n}\Bigr),
\end{equation}
whence all the expressions involving Alice's operators $A_{\mu m}$
have totally disappeared. The statistics of Bob's result are not
affected at all by what Alice may simultaneously do somewhere
else. This proves that Eq.~(\ref{etcr}) indeed is a sufficient
condition for no instantaneous information transfer. In
particular, the local operations $A\otimes\mbox{1\hskip-.25em l}_B$ and $\mbox{1\hskip-.25em l}_A\otimes
B$ are of this form.

Note that any classical communication between distant observers
can be considered  a kind of long range interaction.  The
propagation of signals is, of course, bounded by the velocity of
light. As a result, there exists a partial time ordering of the
various interventions in an experiment, which defines the notions
earlier and later. The input parameters of an intervention are
deterministic (or possibly stochastic) functions of the parameters
of earlier interventions, but not of the stochastic outcomes
resulting from later or mutually spacelike interventions
\cite{rmp}.

Even these apparently simple notions lead to non-trivial results.
Consider a separable bipartite superoperator $T$,
\begin{equation}
T(\rho)=\sum_k M_k\rho M_k^\dag, \qquad M_k=A_k\otimes B_k,
\end{equation}
where the operators $A_k$ represent operations of Alice and $B_k$
those of Bob. Not all such superoperators can be implemented by
local transformations and classical communication (LOCC)
\cite{ben99}. This is the foundation of the ``non-locality without
entanglement''.

A classification of bipartite state transformations was introduced
in \cite{bek01}. It consists of the following categories. There
are {\it localizable\/} operations that can be implemented locally
by Alice and Bob, possibly with the help of prearranged ancillas,
but without classical comunication. Ideally, local operations are
instantaneous, and the whole process can be viewed as performed at
a definite time. A final classical output of such distributed
intervention will  be obtained at some point of the (joint) causal
future of
 Alice's and Bob's interventions. For {\it semilocalizable\/} operations, the
requirement of no communication is relaxed and one-way classical
communication is possible. It is obvious that any tensor-product
operation $T_{\rm A}\otimes T_{\rm B}$ is localizable, but it is
not a necessary condition.  For example the Bell measurements,
which distinguishes between the four standard bipartite entangled
qubit states,
\begin{equation} |\Psi^\pm\rangle := {1\over \sqrt 2}(|0\rangle |1\rangle \pm
|1\rangle |0\rangle), \qquad
 |\Phi^\pm\rangle := {1\over \sqrt
2}(|0\rangle |0\rangle \pm |1\rangle |1\rangle),
\end{equation}
are localizable.

 Other classes of bipartite operators are defined
as follows: Bob performs a local operation $T_{\rm B}$ just before
the global operation $T$. If no local operation of Alice can
reveal any information about $T_{\rm B}$, i.e., Bob cannot signal
to Alice, the operation $T$ is {\it semicausal\/}. If the
operation is semicausal in both directions, it is {\it causal\/}.
In many cases it is easier to prove causality than localizability
(see Remark 3). There is a necessary and sufficient condition for
the semicausality (and therefore, the causality) of operations
\cite{bek01}.

These definitions of causal and localizable operators appear
equivalent. It is easily proved that localizable operators are
causal. It was shown that semicausal operators are always
semilocalizable \cite{egg02}. However, there are causal operations
that are not localizable \cite{bek01}.

It is curious that while a complete Bell measurement is causal,
the two-outcome incomplete Bell measurement is not. Indeed,
consider a two-outcome PVM
\begin{equation}
E_1=|\Phi^+\rangle\langle\Phi^+|, \qquad E_2=\mbox{1\hskip-.25em l}-E_1.
\end{equation}
 If the initial state is
$|01\rangle_{\rm AB}$, then the outcome that is associated with $E_2$
always occurs and Alice's reduced density matrix after the
measurement is $\rho_{\rm A}=|0\rangle\langle 0|$. On the other hand, if
before the joint measurement Bob performs a unitary operation that
transforms the state into $|00\rangle_{\rm AB}$, then the two outcomes
are equiprobable, the resulting states after the measurement are
maximally entangled, and Alice's reduced density matrix is
$\rho_{\rm A}=\mbox{$1\over2$}\mbox{1\hskip-.25em l}$. A simple calculation shows that after this
incomplete Bell measurement two input states $|00\rangle_{\rm AB}$ and
$|01\rangle_{\rm AB}$  are distinguished by Alice with a probability of
0.75.

Here is another example of a  semicausal and semilocalizable
measurement which can be executed with one-way classical
communication from Alice to Bob.  Consider a PVM measurement,
whose complete orthogonal projectors are
\begin{equation} |0\rangle\otimes|0\rangle, \quad
|0\rangle\otimes|1\rangle, \quad |1\rangle|+\rangle, \quad |1\rangle\otimes|-\rangle,\label{examp1}
\end{equation}
where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$. The Kraus matrices are
\begin{equation}
A_{\mu j}=E_\mu\delta_{j0}\label{orthopvm},
\end{equation}
From the properties of complete orthogonal measurements
\cite{bek01}, it follows that this operation cannot be performed
without Alice talking to Bob. A protocol to realize this
measurement is the following. Alice measures her qubit in the
basis $\{|0\rangle, |1\rangle\}$, and tells her result to Bob. If Alice's
outcome was $|0\rangle$, Bob measures his qubit in the basis $\{|0\rangle,
|1\rangle\}$, and if it was $|1\rangle$, in the basis $\{|+\rangle, |-\rangle\}$.

If one allows for more complicated conditional state evolution
\cite{gr02}, then more measurements are localizable. In particular,
consider a \textit{verification} measurement, i.e.,
 the measurement yields a $\mu$-th result with
certainty, if the state prior to the classical interventions was
given by $\rho=E_\mu$, but without making any specific demand on
the resulting state $\rho'_\mu$.

It is possible to realize  a  verification measurements by means
of a shared entangled ancilla and Bell-type measurements by one of
the parties \cite{vai03}. Verification measurement  of
Eq.~(\ref{examp1}) can illustrate this construction. In addition
to the state to be tested, Alice and Bob share a Bell state
$|\Psi^-\rangle$. They do not have to coordinate their moves. Alice and
Bob perform tasks independently and convey their results to a
common center, where a final decision is made.

The procedure is based on the teleportation identity
\begin{equation}
    |\Psi\rangle_1|\Psi^-\rangle_{23} = \mbox{$1\over2$}
  \left(|\Psi^-\rangle_{12} | \Psi\rangle_{3} +
|\Psi^+\rangle_{12} |\tilde \Psi^{(z)}\rangle_{3} +
|\Phi^-\rangle_{12} |\tilde \Psi^{(x)}\rangle_{3}+
|\Phi^+\rangle_{12} |\tilde \Psi^{(y)}\rangle_{3}\right),
\label{ident}
\end{equation}
where   $|\tilde
\Psi^{(z)}\rangle$ means the state $| \Psi\rangle$
rotated  by $\pi$ around the $z$-axis, etc. The first step of this
measurement corresponds to the first step of a teleportation  of a
state of the spin from $B$ (Bob's site) to $A$ (Alice's site). Bob
and Alice do not perform the full teleportation (which  requires a
classical communication between them). Instead, Bob performs  only
the Bell measurement at his site which leads to one of the
branches of the superposition in the rhs of Eq.~(\ref{ident}).

The second step of the verification measurement is taken by Alice.
Instead of completing the teleportation protocol, she measures the
spin of her particle in the $z$ direction. According to whether
that spin is up or down, she measures the spin of her ancilla in
the $z$ or $x$ direction, respectively. This completes the
measurement and it only remains to combine the local outcomes to
get the result of the nonlocal measurement
\cite{vai03}. This method can be extended to arbitrary Hilbert
space dimensions.

\subsection*{Remarks}
\noindent 1.~ Measurements in quantum field theory are discussed in
\cite{rmp,rm1, rm2}.\medskip

\noindent 2.~ An algebraic field theory approach to statistical independence and to
related topics is presented in \cite{sf}.\medskip

\noindent 3.~ To check the causality of an operation
$T$ whose outcomes are the states $\rho_\mu=T_\mu(\rho)/p_\mu$
with probabilities $p_\mu={\rm tr}\, T_\mu(\rho)$, $\sum_\mu p_\mu=1$ it
is enough to consider the corresponding superoperator
\begin{equation}
T'(\rho):=\sum_\mu  T_\mu(\rho)
\end{equation}
Indeed, assume that Bob's action prior to the global operation
lead to one of the two different states $\rho_1$ and $\rho_2$.
Then the states $T'(\rho_1)$ and $T'(\rho_2)$ are distinguishable
 if and only if some of the pairs of
states $T_\mu(\rho_1)/p_{\mu1}$ and $T_\mu(\rho_2)/p_{\mu2}$ are
distinguishable. Such probabilistic distinguishability shows that
the operation $T$ is not semicausal.
\medskip

\noindent 3.~ Absence of the superluminal communication makes possible to evade
the theorems on the impossibility of a bit commitment.
  In particular
the protocol RBC2 allows a bit commitment to be indefinitely
maintained with unconditionally security against all classical
attacks, and at least for some finite amount of time against
quantum attacks \cite{k99,k03}. \medskip

\noindent 4.~ In these notes I am not going to deal with the relativistic
localization POVM. Their properties (and difficulties in their
construction) can be found in \cite{rmp}. An exhaustive survey of
the spatial localization of photons is given in
\cite{keller}. Here we only note in passing that if $E({\cal O})$ is an
operator that corresponds to the detection of an event in a
spacetime region ${\cal O}$, since they are not thought to be
implemented by physical operations confined to that spacetime
area, the condition $[E({\cal O}_1),E({\cal O}_2)]=0$ is not required
\cite{tol, maz}.

\section{Quantum Lorentz transformations}
There is no elementary particle that is called ``qubit". Qubits
are realized by  particular degrees of freedom of more or less
complicated systems. To decide how qubits transform (e.g., under
Lorentz transformations) it may be necessary to consider again the
entire system. In the following our qubit will be either a spin of
a massive particle or a polarization of a photon. A quantum
Lorentz transformation connects the  description of a quantum
state $|\Psi\rangle$ in two reference frames that are connected by a
Lorentz transformation $\Lambda$ (i.e., their coordinate axes are
rotated with respect to each other and the frames  have a fixed
relative velocity). Then $|\Psi'\rangle=U(\Lambda)|\Psi\rangle$, and the
unitary $U(\Lambda)$ is represented  on Fig.~1 below. The purpose
of this section is to explain the elements of this quantum
circuit.

From the mathematical point of view the single-particle states
belong to some irreducible representation of the Poincar\'{e}
group. An  introductory discussion of these representations and
their relations with
 states and quantum fields may be found, e.g., in
\cite{wkt,wei}. Within each
particular irreducible representation there are six commuting
operators. The eigenvalues of two of them are invariants that
label the representation by defining the mass $m$ and the
intrinsic spin $j$. The basis states are labelled by three
components of the momentum ${\bf p}$ and the spin operator $\Sigma_3$.
Hence a generic state is given by
\begin{equation}
|\Psi\rangle=\sum_\sigma\!\int \!d\mu(p)\psi_\sigma(p)|p,\sigma\rangle.
\end{equation}
In this formula $d\mu(p)$ is the Lorentz-invariant measure,
\begin{equation}
d\mu(p)=\frac{1}{(2\pi)^3}\frac{d^3{\bf p}}{2E({\bf p})},
\end{equation}
where the energy $E({\bf p})=p^0=\sqrt{{\bf p}^2+m^2}$. The improper
momentum and spin eigenstates are $\delta$-normalized,
\begin{equation}
\6p,\sigma|q,\sigma'\rangle=(2\pi)^3(2E({\bf p}))\delta^{(3)}({\bf p}-{\bf q})\delta_{\sigma\sigma'},
\end{equation}
and are complete on the one-particle space, which is
${\cal H}={\mathbb C}^{2j+1}\otimes L^2({\mathbb R}^3,d\mu(p))$ for spin-$j$ fields.

\vspace{-6.cm}
\begin{figure}[htbp]
\epsfxsize=0.50\textwidth
\centerline{\epsffile{fig1terno.eps}} \vspace*{-6.1cm}
\caption{\small{
Relativistic state transformation as a quantum circuit: the gate
$D$ which represents the matrix $D_{\xi\sigma}[W(\Lambda,p)]$ is
controlled by both the classical information and the momentum $p$,
which is itself subject to the classical information $\Lambda$.}}
\end{figure}

To find the transformation law  we have to be more concrete about
the spin operator. The operator $\Sigma_3(p)$ is a function of the
generators of the Poincar\'{e} group. One popular option is
helicity,
 $\Sigma_3={\bf J}\cdot{\bf P}/|{\bf P}|$, which is applicable for both massive and massless particles.
 For massive particles we use the $z$-component of the rest-frame (or
 Wigner spin, that we now describe in the next section.

 \subsection{Massive particles}

The construction involve picking a reference 4-momentum $k$, which
for massive particles is taken to be $k_R=(m,\bf{0})$. The Wigner
spin  ${\bf S}(p)$ is defined to  coincide with the non-relativistic
spin ${\bf S}$ in particle's rest frame. The state of a particle at
rest is labelled  $|k_R,\sigma\rangle$,
\begin{equation}
{\bf S}^2|k_R,\sigma\rangle=j(j+1)|k_R,\sigma\rangle,\qquad
{\bf S}_3|k_R,\sigma\rangle=\sigma|k_R,\sigma\rangle.
\end{equation}
The spin states of arbitrary momenta are defined as follows. The
standard rotation-free boost that brings $k_R$ to an arbitrary
momentum $p$, $p^\mu=L(p)^\mu_{~\nu}\/ k^\nu$ is given by
\begin{equation}
L(p)=\left(\begin{array}{cccc}
\frac{E}{m} & \frac{p_1}{m} & \frac{p_2}{m} & \frac{p_3}{m}\\
\frac{p_1}{m} & \mbox{\small 1}+ \frac{p_1^2}{m(m+E)} & \frac{p_1 p_2}{m(m+E)} &
\frac{p_1 p_3}{m(m+E)} \\
\frac{p_2}{m} & \frac{p_2 p_1}{m(m+E)} & \mbox{\small 1}+
\frac{p_2^2}{m(m+E)}&\frac{p_2 p_3}{m(m+E)} \\
\frac{p_3}{m} & \frac{p_3 p_1}{m(m+E)} & \frac{p_3 p_2}{m(m+E)} & \mbox{\small 1}+ \frac{p_3^2}{m(m+E)}
\end{array} \label{lp}
\right).
\end{equation}
The Wigner spin ${\bf S}(p)$ and the one-particle basis states are
defined by
\begin{equation}
|p,\sigma\rangle\equiv U[L(p)]|k_R,\sigma\rangle,
\qquad{\bf S}_3(p)|p,\sigma\rangle=\sigma|p,\sigma\rangle. \label{basis}
\end{equation}

In deriving the transformation rules we begin with the momentum
eigenstates.  Using the group representation property and
Eqs.~(\ref{basis}) the transformation  is written as
\begin{equation}
U(\Lambda)=U[L(\Lambda p)]U[L^{-1}(\Lambda p)\Lambda.
L(p)]U[L^{-1}(p)]
\end{equation}
The element of the Lorentz group
\begin{equation}
W(\Lambda,p)\equiv L^{-1}(\Lambda p) \Lambda L(p), \label{wigw}
\end{equation}
leaves $k_R$ invariant, $k_R=Wk_R$. Hence it belongs to the
stability subgroup (or Wigner little group) of $k_R$. For
$k_R=(m,\bf{0})$ it is a rotation. Pressing on
\begin{equation}
U(\Lambda)|p,\sigma\rangle=U[L(\Lambda p)]U[W(\Lambda,p)]|k_R,\sigma\rangle,
\end{equation}
and as a result,
\begin{equation}
U(\Lambda)|p,\sigma\rangle=\sum_\xi D_{\xi\sigma}[W(\Lambda,p)]|\Lambda
p,\xi\rangle,
\end{equation}
where $D_{\xi\sigma}$ are the matrix elements of the
representation of  the Wigner rotation $W(\Lambda,p)$.

We consider only spin-$\mbox{$1\over2$}$ particles, so $\sigma=\pm\mbox{$1\over2$}$. Any
$2\times2$ unitary matrix can be written as $\hat{D}=\exp(-i\omega
\hat{\mathbf n}\cdot \mbox{\boldmath
$\sigma$})$, where $\omega$ is a rotation angle and $\hat{\mathbf
n}$ is a rotation axis that corresponds to $W(\Lambda,p)$.

The wave functions transform according to $
\psi'_\xi(q)=\langle\xi,q|U(\Lambda)|\Psi\rangle$
so the same state in the Lorentz-transformed frame is
\begin{equation}
|\Psi'\rangle=U(\Lambda)|\Psi\rangle=\sum_{\sigma,\xi}\int_{-\infty}^\infty\!
D_{\sigma\xi}[W(\Lambda,\Lambda^{-1}p)]\psi_\xi(\Lambda^{-1}p)|\sigma,p\9d\mu(p).\label{state}
\end{equation}

For pure rotation $\EuScript{R}$ the three-dimensional (more exactly, 3D
block of 4D matrix; here and in the following we use the same
letter for a 4D and 3D matrix for $\EuScript{R}\in$SO(3)) Wigner rotation
matrix is the rotation itself,
\begin{equation}
W(\EuScript{R},p)=\EuScript{R},\qquad \forall p=(p^0,{\bf p}).
\end{equation}
 As a result, the action of
Wigner spin operators on ${\cal H}_1$ is given by than halves of Pauli
matrices that are tensored with the identity of $L^2$.

\subsection{Photons}
The single-photon states are labelled by momentum ${\bf p}$ (the
4-momentum vector is null, $E=p^0=|{\bf p}|$) and helicity
$\sigma_{\bf p}=\pm 1$, so the state with a definite momentum is given
by $\sum_{\sigma=\pm 1}\alpha_\sigma|p,\sigma_{\bf p}\rangle,$ where
$|\alpha_+|^2+|\alpha_-|^2=1$. Polarization states are also
labelled by 3-vectors $\mbox{\boldmath $\epsilon$}^\sigma_{\bf p}$,
${\bf p}\cdot\mbox{\boldmath $\epsilon$}_{\bf p}^\sigma=0$, that correspond to the two senses of
polarization of  classical electromagnetic waves. An alternative
labelling of the same state, therefore, is $\sum_{\sigma=\pm
1}\alpha_\sigma|p,\mbox{\boldmath $\epsilon$}^\sigma_{\bf p}\rangle$.

Helicity is invariant under proper Lorentz transformation, but the
basis states acquire phases.
\
 The little group element $W(\Lambda,p)=L^{-1}(\Lambda p)\Lambda
 L(p)$ is defined with respect to the standard
four-momentum $k_R=(1,0,0,1)$. The standard Lorentz transformation
is
\begin{equation}
L(p)=R(\hat{\bp})B_z(u),
\end{equation}
 where $B_z(u)$ is a pure boost along the $z$-axis with a
velocity $u$ that takes $k_R$ to $(|{\bf p}|,0,0,|{\bf p}|)$ and $R(\hat{\bp})$
is the standard rotation that carries the $z$-axis into the
direction of the unit vector $\hat{\bp}$. If $\hat{\bp}$ has polar and
azimuthal angles $\theta$ and $\phi$, the standard rotation
$R(\hat{\bp})$ is accomplished  by a rotation by $\theta$ around the
$y$-axis, that is followed by a rotation by $\phi$ around the
$z$-axis. Hence,
\begin{equation}
R(\hat{\bp})=\left(\begin{array}{ccccc}
\cos\theta\cos\phi & & -\sin\phi  & & \cos\phi\sin\theta \\
\cos\theta\sin\phi & & \cos\phi & & \sin\phi\sin\theta \\
-\sin\theta & & 0 & & \cos\theta
\end{array}\right), \label{strot}
\end{equation}
(here only the non-trivial 3D block is shown).

An arbitrary little group element for a massless particle is
decomposed according to
\begin{equation}
W(\Lambda,p)=S(\beta,\gamma)R_z(\xi),
\end{equation}
where the elements $S(\beta,\gamma)$ form a subgroup that is
isomorphic to the translations of the Euclidean plane and
$R_z(\xi)$ is a rotation around the $z$-axis. We are interested
only in the angle $\xi$, since  $\beta$ and $\gamma$ do not
correspond to the physical degrees of freedom. However, they are
important for gauge transformations. Finally, the little group
elements are represented by
\begin{equation}
D_{\sigma'\sigma}=\exp(i\xi\sigma)\delta_{\sigma'\sigma}.\label{dss}
\end{equation}

It is worthwhile to derive more explicit expressions for $\xi$. I
begin with rotations, $\Lambda=\EuScript{R}$.  Since rotations form a
subgroup of a Lorentz group, $R^{-1}(\EuScript{R}\hat{\bp})\EuScript{R} R(\hat{\bp})$ is a
rotation that leaves $\hat{\bz}$ invariant and thus is of the form
$R_z(\omega)$ for some $\omega$. A boost in $(t,z)$ plane and a
rotation around $z$-axis commute,  $[R_z,B_z]=0$, so
\begin{equation}
W(\EuScript{R},p)=R^{-1}(\EuScript{R}\hat{\bp})\EuScript{R} R(\hat{\bp})=R_z(\xi).\label{wr}
\end{equation}

Any rotation can be described by two angles that give a direction
of the  axis and the third angle that gives the amount of rotation
around that axis. If $\EuScript{R}{\bf p}={\bf q}$, we decompose the rotation
matrix as
\begin{equation}
\EuScript{R}=R_{\hat{\bq}}(\omega)R(\hat{\bq})R^{-1}(\hat{\bp}), \label{omdef}
\end{equation}
where $R_{\hat{\bq}}(\omega)$ characterizes a rotation around $\hat{\bq}$,
and $R(\hat{\bq})$ and $R(\hat{\bp})$ are the standard rotations that carry
the $z$-axis to $\hat{\bq}$ and $\hat{\bp}$, respectively. Using
Eq.~(\ref{wr}) we find that $S=\mbox{1\hskip-.25em l}$ and the two rotations are of
the same conjugacy class,
\begin{equation}
R_z(\xi)=R^{-1}(\EuScript{R}\hat{\bp})R_{\EuScript{R}\hat{\bp}}(\omega)R(\EuScript{R}\hat{\bp}),\label{rdeco}
\end{equation}
so we conclude that $\xi=\omega$.

A practical description of polarization states is given by spatial
vectors that correspond to the classical polarization directions.
Taking again $k_R$ as the reference momentum, two basis vectors of
linear polarization are $\mbox{\boldmath $\epsilon$}_{k_R}^1=(1,0,0)$ and
$\mbox{\boldmath $\epsilon$}_{k_R}^2=(0,1,0)$, while to the right and left circular
polarizations correspond $\mbox{\boldmath $\epsilon$}_{k_R}^\pm=(\mbox{\boldmath $\epsilon$}_{k_R}^1\pm
i\mbox{\boldmath $\epsilon$}_{k_R}^2)/\sqrt{2}$.

Phases of the states  obtained by the standard Lorentz
transformations $L(p)$ are set to 1. Since the standard boost
$B_z(u)$ leaves the four-vector $(0,\mbox{\boldmath $\epsilon$}^\pm_{k_R})$ invariant, we
define a polarization basis for any ${\bf p}$ as
\begin{equation}
\mbox{\boldmath $\epsilon$}^\pm_{\bf p}=\mbox{\boldmath $\epsilon$}^\pm_{\hat{\bp}}\equiv R(\hat{\bp})\mbox{\boldmath $\epsilon$}^\pm_{k_R},\label{conve}
\end{equation}
while the transformation of polarization vectors under an
arbitrary rotation $\EuScript{R}$ is given by the rotation itself. To see
the agreement between transformations of spatial vectors and
states, consider a generic state with a momentum ${\bf p}$. Its
polarization is described by the polarization vector
$\mbox{\boldmath $\alpha$}({\bf p})=\alpha_+\mbox{\boldmath $\epsilon$}^+_{\bf p}+\alpha_-\mbox{\boldmath $\epsilon$}^-_{\bf p}$, or by the state
vector $\alpha_+|{\bf p},+\rangle+\alpha_-|{\bf p},-\rangle$.  Using
Eq.~(\ref{conve}) we see that the transformation of $\mbox{\boldmath $\alpha$}({\bf p})$ is
given by
\begin{equation}
\EuScript{R}\mbox{\boldmath $\alpha$}({\bf p})=R_{\EuScript{R}\hat{\bp}}(\omega)R(\EuScript{R}\hat{\bp})R^{-1}(\hat{\bp})\mbox{\boldmath $\alpha$}({\bf p})=
R_{\EuScript{R}\hat{\bp}}(\omega)R(\EuScript{R}\hat{\bp})\mbox{\boldmath $\alpha$}(k_R)=R_{\EuScript{R}\hat{\bp}}(\omega)\mbox{\boldmath $\alpha$}(\EuScript{R}\hat{\bp}).
\end{equation}
 If ${\bf q}=\EuScript{R} {\bf p}$ the transformation results in
 $\alpha_+e^{i\omega}\mbox{\boldmath $\epsilon$}^+_{\bf q}+\alpha_-e^{-i\omega}\mbox{\boldmath $\epsilon$}^-_{\bf q}$,
and since $\omega=\xi$, it is equivalent to  the state
transformation
\begin{equation}
U(\EuScript{R})(\alpha_+|p,+\rangle+\alpha_-|p,-\rangle)=
\alpha_+e^{i\xi}|q,+\rangle+\alpha_-e^{-i\xi}|q,-\rangle.
\end{equation}

For a general Lorentz transformations the triad
$(\mbox{\boldmath $\epsilon$}^1_{\bf p},\mbox{\boldmath $\epsilon$}^2_{\bf p}, \hat{\bp})$ is rigidly rotated, but in a more
complicated fashion. To obtain the phase for a general Lorentz
transformation, we decompose the latter into two rotations and a
standard boost $B_z$ along the $z$-axis:
\begin{equation}
\Lambda=\EuScript{R}_2B_z(u)\EuScript{R}_1.
\end{equation}
It can be shown that $B_z$ alone does not lead to a phase
rotation. Therefore,
\begin{equation}
\xi=\omega_1+\omega_2,
\end{equation}
where both $\omega_1$ and $\omega_2$ are due to the rotations and
are given by Eq.~(\ref{omdef}). Note that although $B_z(u)$ alone
does not lead to a phase rotation, it can affect the value of
$\omega_2$, since it indirectly appears in the definition of
$\EuScript{R}_2$.

\subsection*{Remarks}

1.~ A comprehensive discussion of the Poincar\'{e} group in
physics can be found in \cite{barut, bog}. Useful expressions for
Wigner rotations and their applications for massive particles are
given in \cite{hal,soo,har}.

\medskip
\noindent 2.~  In this transformation I do not assume any
additional normalization factors. A condition of unitarity is
$UU^\dag=U^\dag U=\mbox{1\hskip-.25em l},$ but there also other conventions in the
literature.
\medskip

\noindent  3.~ A double infinity of the positive energy solutions of the
Dirac equation (functions $u^{(1/2)}_p$ and $u^{(-1/2)}_p$) spans
an improper basis of this space.  There is a one-to-one
correspondence between Wigner and Dirac wave functions. Basis
vectors of  Wigner and Dirac Hilbert spaces are in the one-to-one
correspondence \cite{bog},
\begin{equation}
u^{(1/2)}_p\Leftrightarrow|\mbox{$1\over2$},p\rangle,\qquad
u^{(-1/2)}_p\Leftrightarrow|\mbox{$1\over2$},p\rangle,
\end{equation}
while the wave functions are related by
\begin{eqnarray}
& &\Psi^\alpha(p)=\psi_{1/2}(p)u^{(1/2)\alpha}_p +
\psi_{-1/2}(p)u_p^{(-1/2)\alpha} \label{corr}
 \\
& &
2m\psi_\sigma(p)=\sum_{\alpha=1}^4\overline{u}^{(-\sigma)}_{\alpha
p}\Psi^{\alpha}(p)
\end{eqnarray}

\medskip

\noindent 4.~ Another approach to the construction of the Wigner rotation
$\hat{D}$ is based on the homomorphism between Lorentz group and
$SL(2)$ \cite{bog}.
\medskip

\noindent 5.~ When not restricted to a single-particle space the
Wigner spin operator is given by
\begin{equation}
{\bf S}=\mbox{$1\over2$}\sum_{\eta,\zeta}\mbox{\boldmath
$\sigma$}_{\eta\zeta}\int\!d\mu(p) (\hat{a}^\dag_{\eta
p}\hat{a}_{\zeta p}+\hat{b}^\dag_{\eta p}
\hat{b}_{\sigma p}),
\end{equation}
where $\hat{a}^\dag_{\eta p}$ creates a mode with a momentum $p$
and spin $\eta$ along the $z$-axis, etc. A comparison of different
spin operators can be found in \cite{ter3}.
\medskip

\noindent 6.~
If one works with the 4-vectors, then in the helicity gauge the
polarization vector is given by $\epsilon_p=(0,\mbox{\boldmath $\epsilon$}_{{\bf p}})$. A
formal connection between helicity states and polarization vectors
is made by first observing
 that  three spin-1 basis states can be constructed
from the components of a symmetric spinor of rank 2. Unitary
transformations of this spinor that are induced by $\EuScript{R}$ are in
one-to-one correspondence with transformations by $\EuScript{R}$ of certain
linear combinations of a spatial vector. In particular,
transformations of the helicity $\pm 1$ states induced by
rotations are equivalent to the rotations of $(\mbox{\boldmath $\epsilon$}_{k_S}^1\pm
i\mbox{\boldmath $\epsilon$}_{k_S}^2)/\sqrt{2}$ (the $z$-axis is the initial quantization
direction). While $p_\mu \epsilon_p^\mu=0$ gauge condition is
Lorentz-invariant, the spatial orthogonality is not.  The role of
gauge transformations in preserving the helicity gauge and some
useful expressions for the phase that photons acquire can be found
in
\cite{am,lpt,bga}

\section{Implications of quantum Lorentz transformations}
\subsection{Reduced density matrices}

In a relativistic system whatever is outside the past light cone
of the observer is unknown to him, but also cannot affect his
system, therefore does not lead to decoherence (here, I assume
that no particle emitted by from the outside the past cone
penetrates into the future cone). Since different observers have
different past light cones, by tracing out they exclude from their
descriptions different parts of spacetime. Therefore any
transformation law between them must tacitly assume that the part
excluded by one observer is irrelevant to the system of another.

Another consequence of relativity is that there is a hierarchy of
dynamical variables: {\it primary variables\/} have relativistic
transformation laws that depend only on the Lorentz transformation
matrix $\Lambda$ that acts on the spacetime coordinates. For
example, momentum components are primary variables. On the other
hand, {\it secondary variables\/} such as spin and polarization
have transformation laws that depend not only on $\Lambda$, but
also on the momentum of the particle. As a consequence, the
reduced density matrix for secondary variables, which may be well
defined in any coordinate system, has no transformation law
relating its values in different Lorentz frames.

Moreover, an unambiguous definition of the reduced density matrix
 is possible only if the secondary
degrees of freedom are unconstrained, and photons are the simplest
example when this definition fails. In the absence of a general
prescription, a case-by-case treatment is required. I describe a
particular construction, valid with respect to a certain class of
tests.

\subsection{Massive particles}

For a massive qubit  the usual definition of quantum entropy has
no invariant meaning. The reason is that under a Lorentz boost,
the spin undergoes a Wigner rotation, that as shown on Fig.~1 is
controlled both by the classical data and the corresponding
momentum. Even if the initial state is a direct product of a
function of momentum and a function of spin, the transformed state
is not a direct product. Spin and momentum become entangled.

Let us define a reduced density matrix,
\begin{equation}
\rho=\int d\mu(p)\psi(p)\psi^\dagger(p).
\end{equation}
It gives statistical predictions for the results of measurements
of spin components by an ideal apparatus which is not affected by
the momentum of the particle. Note that I tacitly assumed that the
relevant observable is the Wigner spin. The spin entropy is
\begin{equation}
S=-{\rm tr}\,(\rho\log\rho)=-\sum\lambda_j\log\lambda_j,
\end{equation}
where $\lambda_j$ are the eigenvalues of $\rho$.

As usual, ignoring some degrees of freedom leaves the others in a
mixed state. What is not obvious is that in the present case the
amount of mixing depends on the Lorentz frame used by the
observer.  Indeed consider another observer (Bob) who moves with a
constant velocity with respect to Alice who prepared that state.
In the Lorentz frame where Bob is at rest, the state is given by
Eq.~(\ref{state}).

As an example, take a particle prepared by Alice to be
\begin{equation}
 |\Psi\rangle=\chi\int \psi(p)|p\9d\mu(p)\label{rest},\qquad \chi={\zeta\choose\eta}
\end{equation}
where $\psi$ is concentrated near zero momentum and has a
characteristic spread $\Delta$. Spin density matrices of all the
states that are given by Eq.~(\ref{rest}) are
\begin{equation}
\rho={|\zeta|^2~~\zeta\eta^*\choose \zeta^*\eta~~ |\eta|^2},
\end{equation}
and are independent of the specific form of $\psi(p)$.  To make
calculations explicit (and simpler) I take the wave function to be
Gaussian, $\psi(p)=N\exp({\bf p}^2/2\Delta^2)$, where $N$ is a
normalization factor. Spin and momentum are not entangled, and the
spin entropy is zero. When that particle is described in Bob's
Lorentz frame, moving with velocity $v$  at the angle $\theta$
with Alice's $z$-axis, a detailed calculation shows that the the
spin entropy is positive
\cite{rmp}. This phenomenon is illustrated in Fig.~\ref{spinent}.
A relevant parameter, apart from the angle $\theta$, is in the
leading order in momentum spread,
\begin{equation}
\Gamma=\frac{\Delta}{m}\,\frac{1-\sqrt{1-v^2}}{v},
\end{equation}
where $\Delta$ is the momentum spread in Alice's frame.  The
entropy has no invariant meaning, because the reduced density
matrix $\tau$ has no covariant transformation law, except in the
limiting case of sharp momenta.  Only the complete density matrix
transforms covariantly.

\begin{figure}[htbp]
\epsfxsize=0.55\textwidth
\centerline{\epsffile{fig2terno.eps}} \vspace*{-0.1cm} \caption{\small{Dependence
of the spin entropy $S$, in Bob's frame, on the values of the
angle $\theta$ and a parameter
$\Gamma=[1-(1-v^2)^{1/2}]\Delta/mv$.}}
\label{spinent}
\end{figure}

I outline some of the steps in this derivation. First, we
calculate the rotation parameters $(\omega,
\hat{\mathbf{n}})$ of the orthogonal matrix $W(\Lambda,p)$ for a
general momentum. The rotation axis and angle are given by
\begin{equation}
\mathbf{\hat{n}=\hat{{\bf v}}\times\hat{p}},\qquad
\cos\theta=\mathbf{\hat{{\bf v}}\cdot\hat{p}},\qquad 0\leq\theta\leq\pi
\end{equation}
where $\hat{{\bf v}}$ is  boost's direction, while the leading order
term for the angle is
\begin{equation}
\omega  =
\frac{1-\sqrt{1-v^2}}{v}\frac{p}{m}\sin\theta
-\EuScript{O}\left(\frac{p^2}{m^2}\right)
\label{omeg}
\end{equation}

Without a loss of generality we can make another simplification.
We can choose our coordinate frame in such a way that  both
$\zeta$ and $\eta$ are real. The matrix
$D[W(\Lambda,\Lambda\!^{-1}p)]$ takes the form
\begin{equation}
D[W(\Lambda,p')]=\sigma_0\cos{\omega\over 2}-i\sin{\omega\over
2}(-\sin\phi \,\sigma_x+\cos\!\phi\,\sigma_y),
\end{equation}
where $(\theta,\phi)$ are the spherical angle of  ${\bf p}'$ (to be
consistent with Eq.~(\ref{state}) momentum in Alice frame carries
a prime, $p'=\Lambda^{-1}p$). The reduced density matrix in Bob's
frame is
\begin{equation}
\rho^B_{\sigma\xi}=\int \!d\mu(p)D_{\sigma\nu}D^*_{\xi\lambda}
\psi_\nu(p')\psi^*_\lambda(p').
\end{equation}
The  symmetry of $\psi(\Lambda^{-1}p)$ is cylindrical. Hence the
partial trace is taken by performing  a momentum integration in
cylindrical coordinates.  This simplification is a result of the
spherical symmetry of the original $\psi$. The two remaining
integrations are performed by first expanding in powers of
$p/\Delta$ and taking Gaussian integrals. Finally,
\begin{equation}
\rho'=\left(
\begin{array}{cc} \zeta^2(1-\Gamma^2/4)+\eta^2\Gamma^2/4 & \zeta\eta^*(1-\Gamma^2/4) \\
\zeta^*\eta(1-\Gamma^2/4) & \zeta^2\Gamma^2/4+\eta^2(1-\Gamma^2/4)
\end{array} \right).
\end{equation}

Fidelity can be used to estimate the difference between the two
density matrices. It is defined as
\begin{equation}
f=\chi^\dag\rho'\chi,
\end{equation}
and it is easy to get an analytical result for this quantity. Set
$\zeta=\cos\theta$ and $\eta=\sin\theta$. Then
\begin{equation}
f=1-\frac{\Gamma^2}{2}\left(3+\frac{\cos{4\theta}}{8}\right).
\end{equation}

Consider now a pair of orthogonal states that were prepared by
Alice, e.g. the above state with $\chi_1=(1,0)$ and
$\chi_2=(0,1)$. How well can moving Bob distinguish them? I use
the simplest criterion, namely the probability of error $P_E$,
defined as follows: an observer receives a single copy of one of
the two known states and performs any operation permitted by
quantum theory in order to decide which state was supplied. The
probability of a wrong answer for an optimal measurement is
\begin{equation}
P_E(\rho_1,\rho_2)=\mbox{$1\over2$}-\mbox{$1\over4$}\,{\rm tr}
 \sqrt{(\rho_1-\rho_2)^2}.  \label{pe}
 \end{equation}
In Alice's frame $P_E=0$.  In Bob's frame the reduced density
matrices are
\begin{equation}
\rho^B_1=\left(\begin{array}{cc}
1-\Gamma^2/4 & 0 \\
0 & \Gamma^2/4
\end{array}\right), \qquad \rho^B_2=\left(\begin{array}{cc}
\Gamma^2/4 & 0 \\
0 & 1-\Gamma^2/4,
\end{array}\right)
\end{equation}
respectively. Hence the probability of error  is
$P_E(\rho_1,\rho_2)=\Gamma^2/4$.

\subsection{Photons}\label{photons}

The relativistic effects in photons are essentially different from
those for massive particles that were discussed above. This is
because photons have only two linearly independent polarization
states. As we know, polarization is a {\it secondary variable\/}:
states that correspond to different momenta belong to distinct
Hilbert spaces and cannot be superposed (an expression such as
$|\mbox{\boldmath $\epsilon$}_{\bf k}^\pm\rangle+|\mbox{\boldmath $\epsilon$}_{\bf q}^\pm\rangle$ is meaningless if ${\bf k}\neq{\bf q}$).
The complete basis $|p,\mbox{\boldmath $\epsilon$}_{\bf p}^\pm\rangle$  does not violate this
superselection rule, owing to the orthogonality of the momentum
basis. The reduced density matrix, according to the usual rules,
should be
\begin{equation}
\rho=\int\! d\mu(p)|\psi(p)|^2|p,\mbox{\boldmath $\alpha$}({\bf p})\rangle\6p,\mbox{\boldmath $\alpha$}({\bf p})|.
\label{rhost}
\end{equation}
However, since $\xi$ in Eq.~(\ref{dss}) depends on the photon's
momentum even for ordinary rotations, this object will have no
transformation law at all. It is still possible  define an
``effective''
 density matrix adapted to a specific method of measuring
polarization \cite{pt03,aw}. I describe one  such scheme.

The labelling of polarization states by Euclidean vectors
$\mbox{\boldmath $\epsilon$}_{\bf p}^\pm$ suggests the use of a $3\times 3$ matrix with
entries labelled $x$, $y$ and $z$. Classically, they correspond to
different directions of the electric field. For example, a
component $\rho_{xx}$ would give the expectation values of
operators representing the polarization in the $x$ direction,
seemingly irrespective of the particle's momentum.

To have a momentum-independent polarization is to admit
longitudinal photons. Momentum-independent polarization states
thus consist of physical (transverse) and unphysical
(longitudinal) parts, the latter corresponding to a polarization
vector $\mbox{\boldmath $\epsilon$}^\ell=\hat{\bp}$. For example, a generalized polarization
state along the $x$-axis is
\begin{equation}
|\hat{\bx}\rangle=x_+({\bf p})|\mbox{\boldmath $\epsilon$}^+_{\bf p}\rangle+x_-({\bf p})|\mbox{\boldmath $\epsilon$}^-_{\bf p}\rangle+
x_\ell({\bf p})|\mbox{\boldmath $\epsilon$}^\ell_{\bf p}\rangle,\label{decomp}
\end{equation}
where $x_\pm({\bf p})=\hat{\bx}\cdot\mbox{\boldmath $\epsilon$}^\pm_{\bf p}$, and $x_\ell({\bf p})=
\hat{\bx}\cdot\hat{\bp}=\sin\theta\cos\phi$. It follows that
$|x_+|^2+|x_-|^2+|x_\ell|^2=1$, and we thus define
\begin{equation}
{\bf e}_x({\bf p})=\frac{x_+({\bf p})\mbox{\boldmath $\epsilon$}^+_{\bf p}+x_-({\bf p})\mbox{\boldmath $\epsilon$}^-_{\bf p}}
{\sqrt{x_+^2+x_-^2}} \label{physdir},
\end{equation}
as the polarization vector associated with the $x$ direction. It
follows from (\ref{decomp}) that $\langle\hat{\bx}|\hat{\bx}\rangle=1$ and
$\langle\hat{\bx}|\hat{\by}\rangle=\hat{\bx}\cdot\hat{\by}=0$, and likewise for the other
directions, so that
\begin{equation}
|\hat{\bx}\rangle\langle\hat{\bx}|+|\hat{\by}\rangle\langle\hat{\by}|+|\hat{\bz}\rangle\langle\hat{\bz}|=\mbox{1\hskip-.25em l}_p,\label{xyz}
\end{equation}
where $\mbox{1\hskip-.25em l}_p$ is the unit operator in momentum space.

To the direction $\hat{\bx}$ there corresponds a projection operator
\begin{equation}
P_{xx}=|\hat{\bx}\rangle\langle\hat{\bx}|\otimes \mbox{1\hskip-.25em l}_p=|\hat{\bx}\rangle\langle\hat{\bx}|\otimes \int
d\mu(k)|{\bf p}\rangle\langle{\bf p}|,
\end{equation}
 The action of
$P_{xx}$ on $|\Psi\rangle$ follows from Eq.~(\ref{decomp}) and
$\langle\mbox{\boldmath $\epsilon$}^\pm_{\bf p}|\mbox{\boldmath $\epsilon$}^\ell_{\bf p}\rangle=0$. Only the transverse part of
$|\hat{\bx}\rangle$ appears in the expectation value:
\begin{equation}
\langle\Psi|P_{xx}|\Psi\rangle=\int d\mu(p)|\psi(p)|^2|x_
 +({\bf p})\alpha_+^*({\bf p})+x_-({\bf p})\alpha_-^*({\bf p})|^2.
 \end{equation}
It is convenient to write the transverse part of $|\hat{\bx}\rangle$ as

\begin{equation}
|{\bf b}_x({\bf p})\rangle  \equiv
 (|\mbox{\boldmath $\epsilon$}^+_{\bf p}\rangle\langle\mbox{\boldmath $\epsilon$}^+_{\bf p}|+|\mbox{\boldmath $\epsilon$}^-_{\bf p}\rangle\langle\mbox{\boldmath $\epsilon$}^-_{\bf p}|)|\hat{\bx}\rangle
 = x_+({\bf p})|\mbox{\boldmath $\epsilon$}^+_{\bf p}\rangle+x_-({\bf p})|\mbox{\boldmath $\epsilon$}^-_{\bf p}\rangle.
\label{vector}
\end{equation}
Likewise define  $|{\bf b}_y({\bf p})\rangle$ and $|{\bf b}_z({\bf p})\rangle$. These three
state vectors are neither of unit length nor mutually orthogonal.

Finally, a POVM element $E_{xx}$ which is the physical part of
$P_{xx}$, namely is equivalent to $P_{xx}$ for physical states
(without longitudinal photons) is
\begin{equation}
E_{xx}=\int d\mu(k)|p,{\bf b}_x({\bf p})\rangle\6p,{\bf b}_x({\bf p})|,
\end{equation}
and likewise for the other directions. The operators $E_{xx}$,
$E_{yy}$ and $E_{zz}$ indeed form a POVM in the space of physical
states, owing to Eq.~(\ref{xyz}).

To complete the construction of the density matrix, we introduce
additional directions. Following the standard practice of state
reconstruction, we consider $P_{x+z,x+z}$, $P_{x+iz,x+iz}$ and
similar combinations. For example,
\begin{equation}
P_{x+z,x+z}=\mbox{$1\over2$}(|\hat{\bx}\rangle+|\hat{\bz}\rangle)(\langle\hat{\bx}|+\langle\hat{\bz}|)\otimes\mbox{1\hskip-.25em l}_p.
\end{equation}
The diagonal elements of the new polarization density matrix are
defined as
\begin{equation}
\rho_{mm}=\langle \Psi|E_{mm}|\Psi\rangle,\qquad m=x,y,z,
\end{equation}
and the off-diagonal elements are recovered by combinations such
as
\begin{equation}
\rho_{xz}=\langle\Psi(|\hat{\bx}\rangle\langle\hat{\bz}|\otimes\mbox{1\hskip-.25em l}_p)|\Psi\rangle=
\langle\Psi|E_{x+z,x+z}-iE_{x-iz,x-iz}+(1-i)(E_{xx}-E_{zz})/2|\Psi\rangle.
\end{equation}
Denote $|\hat{\bx}\rangle\langle\hat{\bz}|\otimes\mbox{1\hskip-.25em l}_p$ as $P_{xz}$, and its
``physical" part by $E_{xz}$. Then the effective polarization
density matrix is
\begin{equation}
\rho_{mn}=\langle\Psi|E_{mn}|\Psi\rangle=
\int d\mu(k)|f({\bf p})|^2\langle\mbox{\boldmath $\alpha$}({\bf p})|{\bf b}_m({\bf p})\rangle\langle{\bf b}_n({\bf p})|\mbox{\boldmath $\alpha$}({\bf p})\rangle
, \quad m,n,=x,y,z. \label{reduced}
\end{equation}
It is interesting to note that this derivation gives a direct
physical meaning to the naive definition of a reduced density
matrix,
\begin{equation}
\rho^{\rm naive}_{mn}=\int d\mu(p)|\phi(p)|^2\mbox{\boldmath $\alpha$}_m({\bf p})\mbox{\boldmath $\alpha$}_n^*({\bf p})
=\rho_{mn}
\end{equation}
It is possible to show that this POVM actually corresponds to a
simple photodetection model \cite{lt}.

The basis states $|p,\mbox{\boldmath $\epsilon$}_{\bf p}\rangle$ are direct products of momentum
and polarization. Owing to the transversality requirement
$\mbox{\boldmath $\epsilon$}_{\bf p}\cdot{\bf p}=0$, they remain direct products under Lorentz
transformations. All the other states have their polarization and
momentum degrees of freedom entangled. As a result, if one is
restricted to polarization measurements as described by the above
POVM, {\it there do not exist two orthogonal polarization
states\/}. In general, any measurement procedure with finite
momentum sensitivity will lead to the errors in identification, as
demonstrated as follows

Let two states $|\Phi\rangle$ and $|\Psi\rangle$  be two orthogonal
single-photon states. Their reduced polarization density matrices,
$\rho_\Phi$ and $\rho_\Psi$, respectively, are calculated using
Eq.~(\ref{reduced}). Since the states are entangled, the von
Neumann entropies of the reduced density matrices, $S=-{\rm
tr}(\rho\ln\rho)$, are positive. Therefore, both matrices are at
least of rank two. Since the overall dimension is~3, it follows
that ${\rm tr}(\rho_\Phi\rho_\Psi)>0$ and these states are not
perfectly distinguishable. An immediate corollary is that photon
polarization states cannot be cloned perfectly, because the
no-cloning theorem  forbids exact copying of unknown
non-orthogonal states.

In general, any measurement procedure with finite momentum
sensitivity will lead to the errors in identification. First I
present some general considerations and then illustrate them with
a simple example. Let us take the $z$-axis to coincide with the
average direction of propagation so that the mean photon momentum
is $k_A\hat{\bz}$. Typically, the spread in momentum is small, but not
necessarily equal in all directions. Usually the intensity profile
of laser beams has cylindrical symmetry, and we may assume that
$\Delta_x\sim\Delta_y\sim\Delta_r$ where the index $r$ means
radial. We may also assume that $\Delta_r{\gg}\Delta_z$. We then
have
\begin{equation}
f(p)\propto f_1[(p_z-k_A)/\Delta_z]\,f_2(p_r/\Delta_r).
\end{equation}
 We  approximate
\begin{equation}
\theta\approx\tan\theta\equiv p_r/p_z\approx p_r/k_A.
\label{theta}
\end{equation}
In  pictorial language, polarization planes for different momenta
are tilted by angles up to $\sim\Delta_r/k_A$, so that we expect
an error probability of the order $\Delta_r^2/k_A^2$. In the
density matrix $\rho_{mn}$ all the elements of the form
$\rho_{mz}$ should vanish when $\Delta_r\to0$. Moreover, if
$\Delta_z\to0$, the non-vanishing $xy$ block goes to the usual
(monochromatic) polarization density matrix,
\begin{equation}
\rho_{{\rm pure}}=\left(\begin{array}{ccc}
|\alpha|^2 & \beta & 0\\
\beta^* & 1-|\alpha|^2 &0\\
0 & 0 &0
\end{array}\right).
\end{equation}

As an example,  consider two states which, if the momentum spread
could be ignored, would be $|k_A\hat{\bf z},\mbox{\boldmath $\epsilon$}^\pm_{k_A\hat{\bf
z}}\rangle$. To simplify the calculations we assume a Gaussian
distribution:
\begin{equation}
f(p)=Ne^{-(p_z-k_A)^2/2\Delta_z^2}e^{-p_r^2/2\Delta_r^2},
\end{equation}
where $N$ is a normalization factor and $\Delta_z, \Delta_r\ll
k_A$. In general the spread in $p_z$ may introduce an additional
incoherence into  density matrices, in addition to the effect
caused by the transversal spread. However, when all momentum
components carry the same helicity, this spread results in
corrections of
 the higher order. In the example below we take the polarization
components to be $\mbox{\boldmath $\epsilon$}^\pm_{\bf p}\equiv R(\hat{\bp})\mbox{\boldmath $\epsilon$}^\pm_{k_S}$. That
means we have to analyze  the states
\begin{equation}
|\Psi_\pm\rangle=\int d\mu(p) f(p)|p,\mbox{\boldmath $\epsilon$}^\pm_{\bf p}\rangle\label{st},
\end{equation}
where $f(p)$ is given above.

It is enough to expand $R(\hat{\bp})$ up to second order in $\theta$.
The reduced density matrices are calculated by techniques similar
to those for massive particles, using rotational symmetry around
the $z$-axis and normalization requirements. At the leading order
in $\Omega\equiv\Delta_r/k_A$
\begin{equation}
\rho_+=\mbox{$1\over2$}(1-\mbox{$1\over2$}\Omega^2)\left(\begin{array}{ccc}
1 & -i & 0\\
i & 1 &0\\
0 & 0 &0
\end{array}\right)+\mbox{$1\over2$}\Omega^2\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{array}\right),
\end{equation}
and $\rho_-=\rho_+^*$. At the same level of precision,
\begin{equation}
P_E(\rho_+,\rho_-)=\Delta_r^2/4k_A^2.
\end{equation}
It is interesting to note that the optimal strategy for
distinguishing between these two states is a polarization
measurement in the $xy$-plane. Then the effective $2\times 2$
density matrices are perfectly distinguishable, but there is a
probability $\Omega^2/2$ that no photon will be detected at all.
The above result is valid due to the special form of the states
that we had chosen. Potential errors in the upper $2\times 2$
blocks are averaged out in the integration over~$\phi$. The effect
becomes important  when, e.g., a plain monochromatic wave
undergoes a strong focusing. Then $\Omega\approx l/f$, where $l$
is the aperture radius and $f$ is the focal length \cite{lt}.

Now let us turn to the distinguishability problem from the point
of view of a moving observer, Bob.  The probability of an error is
still given by Eq.~(\ref{pe}), but the parameters depend on the
observer's motion. Assume again  that Bob moves along the $z$-axis
with a velocity $v$.   To calculate Bob's reduced density matrix,
we must transform the complete state, and  then take a partial
trace.

Reduced density matrices of $|\Psi_\pm\rangle$ in both frames are given
by the expression
\begin{equation}
(\rho_\pm)^{A,B}_{mn}=\int d\mu(p)|f(p)^{A,B}|^2\langle
R(\hat{\bp})\mbox{\boldmath $\epsilon$}_{k_S}^\pm|{\bf b}_m({\bf p})\rangle\langle{\bf b}_n({\bf p})|R(\hat{\bp})\mbox{\boldmath $\epsilon$}_{k_S}^\pm\rangle.
\end{equation}
This is due to the following two reasons. First, $|{\bf b}_m({\bf p})\rangle$
are defined by Eq.~(\ref{vector})  in any frame, while pure boosts
preserve the orientation of the coordinate axes in 3-space, and
therefore do not affect the indices of $\rho_{mn}$. Second, phases
acquired by polarization states  cancel out, since we choose the
states $|{\bf p},\mbox{\boldmath $\alpha$}({\bf p})\rangle$ to be the helicity eigenstates.

 Calculation of Bob's density matrix is similar to the previous cases.
 The only
frame dependent expression in (\theequation) is
$f^B(p)=f^A(\Lambda^{-1}p)$.  A boost along the $z$-axis preserves
$k_r$ and $\phi$. On the other hand,
\begin{equation}
k^B_z \approx k_A\sqrt{\frac{1-v}{1+v}}.
\end{equation}
Since everything else in the integral remains the same, the effect
of relative motion is given by a substitution
\begin{equation}
\Omega^B=\sqrt{\frac{1+v}{1-v}}\Omega^A, \qquad
P^B_E=\frac{1+v}{1-v}P^A_E, \label{dope}
\end{equation}
so Bob can distinguish the signals either better or worse than
Alice \cite{pt03}.

\subsection*{Remarks}

\noindent 1. A modification of the spin operator \cite{cw} will allow for a
momentum-independent transformation of the spin density matrix
between two frames that are related by a fixed Lorentz
transformation $\Lambda_{12}$. Its relation to our scheme is
discussed in \cite{czcom}.
\medskip

\noindent 2. An  additional motivation for introduction of effective polarization
density matrices cames from the analysis  of one-photon scattering
\cite{aw}.
\medskip

\noindent 3. I have discussed only discrete variables. To explore the relativistic
effects with continuous variables \cite{ralph} it is convenient to
express the quantum Lorentz transformations in terms of mode
creation and annihilation operators \cite{kokbrau}.

\section{Communication channels}
What happens when Alice and Bob that are in a relative motion try
to communicate? Assume that they use qubits that were described
above. Under a general Lorentz transformation $\Lambda$ that
relates Alice's and Bob's frames, the state of this qubit will be
transformed due to three distinct effects, which are:

 (i) A Wigner rotation due to the Lorentz boost $\Lambda$, which occurs
even for momentum eigenstates. If $\Lambda$ is known, then to the
extent that the wave-packet spread can be ignored, this is
inconsequencial.

(ii) A decoherence due to the entangling of spin and momentum
under the Lorentz transformation $\Lambda$ because the particle is
not in a momentum eigenstate. Although reduced or effective
density matrices have no general transformation rule,  such rules
can be established for particular classes of experimental
procedures. We can then ask how these effective transformation
rules, $\rho'=T(\rho)$, fit into the framework of general state
transformations. E.g., for the massive qubit of Sec.~4.2 the
effective transformation is given by
\begin{equation}
\rho'=\rho(1-\frac{\Gamma^2}{4})+(\sigma_x\rho\sigma_x+\sigma_y\rho\sigma_y)
\frac{\Gamma^2}{8}. \label{chanex}
\end{equation}
If $\Lambda$ is known and it is possible to implement the
operators that were mentioned in the Remark~1 above, then this
effect is absent. Otherwise, this noise is unavoidable. Still, it
is worth to keep in mind that the motion can improve the message
fidelity, as in Eq.~(\ref{dope}).

(iii) Another kind of decoherence arises due to Bob's lack of
knowledge about the transformation relating his reference frame to
Alice's frame. Using the techniques of the decoherence-free
subspaces, it is possible to eliminate this effect completely. E.
g., for massive particles four physical qubits may be used to
encode a logical qubit, while for photons $2\rightarrow 1$
encoding is sufficient. In both cases using the block encoding it
is possible to reach an asymptotically unit efficiency \cite{bt}.

Entanglement between the ``qubit'' and spatial degrees of freedom
leads to an interesting complication of the analysis. It is known
\cite{rmp}, that the dynamics of a subsystem my be not completely
positive, there is a prior entanglement with another system and
the  dynamics is not factorizable. Since in Eq.~(\ref{dope}) and
in the discussion following Eq.~(\ref{pe}) we have seen that
distinguishability {\it can\/} be improved, we conclude that these
transformations are {\it not completely positive\/}. The reason is
that the Lorentz transformation acts not only on the
``interesting'' discrete variables, but also on the primary
momentum variables that we elected to ignore and to trace out, and
its action on the interesting degrees of freedom depends on the
``hidden'' primary ones. Of course, the complete state, with all
the variables, transforms unitarily and distinguishability is
preserved.

This technicality has one important consequence. In quantum
information theory quantum channels are described by completely
positive maps that act on qubit states. Qubits themselves are
realized as discrete degrees of freedom of various particles. If
relativistic motion is important, then not only does the vacuum
behave as a noisy quantum channel, but the very representation of
a channel by a CP map fails.

\section{Entanglement and different Lorentz observers}\label{entrel}

In this section I consider only two-particle states. Even in this
simple setting there are several possible answers to the question
what happens to the entanglement, depending on the details of the
question. Since the quantum Lorentz transformation is given by a
tensor product  $U_1(\Lambda)\otimes U_2(\Lambda)$, the overall
entanglement between the states is Lorentz-invariant.

Let us assume that the states can be approximated by momentum
eigenstates. Then, the same conclusion applies to the spin-spin
(or polarization-polarization) entanglement between the particles,
and it is possible to write an appropriate entanglement measures
that capture the effects of particle statistics and
Lorentz-invariance of the entanglement \cite{soo}. However, it
does not mean that this invariance will be observed in an
experiment, or that the violation of Bell-type inequalities that
is observed in an experiment that is performed in Alice's frame
will be observed if the same equipment is placed  in Bob's frame.

While a field-theoretical analysis shows that violations of
Bell-type inequalities are generic, there are conditions that are
imposed on the experimental procedures that are used to detect
them. Consider  the CHSH inequality. For any two spacelike
separated regions and any pairs of of operators, $a$, $b$, there
is a state $\rho$ such that the CHSH inequality is violated, i.e.,
$\zeta(a,b,\rho)>1$. With additional technical assumptions the
existence of a maximally violating state $\rho_m$ can be proved:
\begin{equation}
\zeta(a,b,\rho_m)=\sqrt{2} \label{violall},
\end{equation}
for any spacelike separated regions ${\cal O}_L$  and ${\cal O}_R$. It
follows from convexity arguments that states that maximally
violate Bell inequalities are pure. What are then the operators
that lead to the maximal violation? It was shown \cite{sw} that
the operators $A_j$ and $B_k$ that give $\zeta=\sqrt{2}$ satisfy
$A_j^2=\mbox{1\hskip-.25em l}$ and $A_1A_2+A_2A_1=0$, and likewise for $B_k$. If we
define $A_3:=-i[A_1,A_2]/2$, then these three operators have the
same algebra as Pauli spin matrices.

In principle the vacuum state may lead to the maximal violation of
Bell-type inequalities. Their observability was discussed in
\cite{vac}.

 The operators \cite{c97}
\begin{equation}
A_i=2\left[\frac{m}{p^0}{{\bf a}}_i+
\left(1-\frac{m}{p^0}\right)({\bf a}\cdot{\bf n}){\bf n}\right]\cdot{\bf S}
\equiv2\mbox{\boldmath $\alpha$}({\bf a},{\bf p})\cdot{\bf S},
\end{equation}
where ${\bf S}$ is the Wigner spin operator and ${\bf n}={\bf p}/|{\bf p}|$ appear
quite naturally as the candidates for the  measurement
description. The length of the auxiliary vector $\mbox{\boldmath
$\alpha$}$ is
\begin{equation}
|\mbox{\boldmath$\alpha$}|=
\frac{\sqrt{({\bf p}\cdot{\bf a})^2+m^2}}{p^0},
\end{equation}
so  generically $A_i^2=\mbox{\boldmath $\alpha$}^2\mbox{1\hskip-.25em l}<\mbox{1\hskip-.25em l}$, and
indeed, the degree of violation decreases with the velocity of the
observer. Nevertheless, it is always possible to compensate for a
Wigner rotation by an appropriate choice of the operators
\cite{rmp}.

Realistic situations involve wave packets. For example, a general
spin-$\mbox{$1\over2$}$  two-particle state  may be written as
\begin{equation}
|\Upsilon_{12}\rangle=\sum_{\sigma_1,\sigma_2}\int
d\mu(p_1)d\mu(p_2)g(\sigma_1\sigma_2,{\bf p}_1,{\bf p}_2)|{\bf p}_1,\sigma_1
;{\bf p}_2,\sigma_2\rangle.
 \label{entfig}
\end{equation}
For particles with well defined momenta, $g$  sharply peaks at
some values ${\bf p}_{10}$, ${\bf p}_{20}$. Again, a boost to any Lorentz
frame $S'$ will result in a unitary $U(\Lambda)\otimes
U(\Lambda)$, acting on each particle separately, thus preserving
the entanglement. Nevertheless, since  they can change
entanglement between different degrees of freedom of a single
particle, the spin-spin entanglement is frame-dependent as well.
Having investigated the reduced density matrix for
$|\Upsilon_{12}\rangle$ and made explicit calculations for the case
where $g$ is a Gaussian, as in the Sec.~4.2 above, it is possible
to show  that if two particles are maximally entangled in a common
(approximate) rest frame (Alice's frame), then the concurrence, as
seen by a Lorentz-boosted Bob, decreases when $v\to 1$. Of course,
the inverse transformation from Bob to Alice will increase the
concurrence \cite{ga}.  Thus, we see that that spin-spin
entanglement is not a Lorentz invariant quantity, exactly as spin
entropy is not a Lorentz scalar. Relativistic properties of the
polarization entanglement are even more interesting \cite{bga},
since there is no frame where polarization and momentum are
unentangled.

\section{The omissions \& perspectives}
Because of the lack of space I am only going to mention the
various fascinating areas of the interplay between quantum
information theory and relativistic physics. Quantum field theory
provides us with new situations that should be investigated. For
example, it is possible to ask all the usual questions about
entanglement, distillability, etc and their invariance\cite{rmp,
vw, dt04,eis}. So far we discussed only observers that move with
constant velocity. An accelerated observer sees Unruh radiation.
It leads to a host of interesting effects if we consider a
teleportation between a stationary and accelerated observer
\cite{am03,ivette,unruh}. Dynamical entanglement --- the one appearing in the
scattering processes or between the decay products also have been
investigated
\cite{har,man04,solano}.

Going to more exotic settings, I just mention that black hole
physics, cosmology, loop quantum gravity and string theory provide
extremely interesting scenarios where the questions of information
can and should be asked\cite{rmp,astro,flo, lit, lit2, dt05,my05}

redpajama_set_name: RedPajamaArXiv


text:
Q: Use pear DB class into custom class i'm getting an error and i think is an access problem to the class, but i'm not sure and don't know how to solve.
I create a DSN:
require_once ('DB.php');
    $dsn = array(
'phptype'  => 'mysql',
'username' => 'cmsconex_cms',
'password' => 'y10ba01',
'hostspec' => 'localhost',
'database' => 'cmsconex_cms',
);

$db =& DB::connect($dsn, $options);
if (DB::isError($db)) {
    die($db->getMessage());
}
$db->Query("SET CHARACTER SET UTF8");
$db->Query("SET NAMES UTF8");

And in a custom class:
class user {

function get_userdata($field, $value) {
        global $db;

        $sql = printf ( "SELECT %s FROM usuarios WHERE %s = '%s'", $field, $field, $value );

        $resultado = & $db->getAll ( $sql, DB_FETCHMODE_ASSOC );

        $valor = $resultado[0][$field];

        if ($value == $valor) {
            return true;
        } else {
            return $valor;
        }
    }

}

The problem is when i execute a query, for example:
SELECT usuario FROM usuarios WHERE usuario = 'test'

I get the error:
SELECT usuario FROM usuarios WHERE usuario = 'test'
Fatal error: Cannot use object of type DB_Error as array in /home/cmsconex/public_html/includes/classes/user.php on line 269

Is strange, beacause if i call this other function i get no error and everyting runs great :
function logSession($usuario_id, $ip) {
        global $db;
        $ip = ip2long ( $ip );
        $sql = "INSERT INTO accesos (id, usuario_id, fecha, pagina, ip) VALUES (0, '$usuario_id', NOW(), '', '$ip')";
        $res = & $db->query ( $sql ); // Ejecutar la consulta
    }

So i think the problem is when calling a method of DB class with parameters like DB_FETCHMODE_ASSOC
Could anyone show me the light please?
Thnx so muchh.
The Complete error is:
1
DB_Error Object ( [error_message_prefix] => [mode] => 1 [level] => 1024 [code] => -2 [message] => DB Error: syntax error [userinfo] => 51 [nativecode=1064 ** You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near '51' at line 1] [backtrace] => Array ( [0] => Array ( [file] => /usr/local/lib/php/DB.php [line] => 966 [function] => PEAR_Error [class] => PEAR_Error [type] => -> [args] => Array ( [0] => DB Error: syntax error [1] => -2 [2] => 1 [3] => 1024 [4] => 51 [nativecode=1064 ** You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near '51' at line 1] ) ) [1] => Array ( [file] => /usr/lib/php/PEAR.php [line] => 531 [function] => DB_Error [class] => DB_Error [type] => -> [object] => DB_Error Object RECURSION [args] => Array ( [0] => -2 [1] => 1 [2] => 1024 [3] => 51 [nativecode=1064 ** You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near '51' at line 1] ) ) [2] => Array ( [file] => /usr/local/lib/php/DB/common.php [line] => 1903 [function] => raiseError [class] => PEAR [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( [0] => [1] => -2 [2] => [3] => [4] => 51 [nativecode=1064 ** You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near '51' at line 1] [5] => DB_Error [6] => 1 ) ) [3] => Array ( [file] => /usr/local/lib/php/DB/mysql.php [line] => 898 [function] => raiseError [class] => DB_common [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( [0] => -2 [1] => [2] => [3] => [4] => 1064 ** You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near '51' at line 1 ) ) [4] => Array ( [file] => /usr/local/lib/php/DB/mysql.php [line] => 327 [function] => mysqlRaiseError [class] => DB_mysql [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( ) ) [5] => Array ( [file] => /usr/local/lib/php/DB/common.php [line] => 1216 [function] => simpleQuery [class] => DB_mysql [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( [0] => 51 ) ) [6] => Array ( [file] => /usr/local/lib/php/DB/common.php [line] => 1663 [function] => query [class] => DB_common [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( [0] => 51 ) ) [7] => Array ( [file] => /home/cmsconex/public_html/includes/classes/user.php [line] => 266 [function] => getAll [class] => DB_common [type] => -> [object] => DB_mysql Object ( [phptype] => mysql [dbsyntax] => mysql [features] => Array ( [limit] => alter [new_link] => 4.2.0 [numrows] => 1 [pconnect] => 1 [prepare] => [ssl] => [transactions] => 1 ) [errorcode_map] => Array ( [1004] => -15 [1005] => -15 [1006] => -15 [1007] => -5 [1008] => -17 [1022] => -5 [1044] => -26 [1046] => -14 [1048] => -3 [1049] => -27 [1050] => -5 [1051] => -18 [1054] => -19 [1061] => -5 [1062] => -5 [1064] => -2 [1091] => -4 [1100] => -21 [1136] => -22 [1142] => -26 [1146] => -18 [1216] => -3 [1217] => -3 [1356] => -13 [1451] => -3 [1452] => -3 ) [connection] => Resource id #10 [dsn] => Array ( [phptype] => mysql [dbsyntax] => mysql [username] => cmsconex_cms [password] => y10ba01 [protocol] => [hostspec] => localhost [port] => [socket] => [database] => cmsconex_cms ) [autocommit] => 1 [transaction_opcount] => 0 [_db] => cmsconex_cms [fetchmode] => 1 [fetchmode_object_class] => stdClass [was_connected] => [last_query] => 51 [options] => Array ( [result_buffering] => 500 [persistent] => 1 [ssl] => [debug] => 0 [seqname_format] => %s_seq [autofree] => [portability] => 0 [optimize] => performance ) [last_parameters] => Array ( ) [prepare_tokens] => Array ( ) [prepare_types] => Array ( ) [prepared_queries] => Array ( ) [_last_query_manip] => [_next_query_manip] => [_debug] => [_default_error_mode] => [_default_error_options] => [_default_error_handler] => [_error_class] => DB_Error [_expected_errors] => Array ( ) ) [args] => Array ( [0] => 51 [1] => 2 ) ) [8] => Array ( [file] => /home/cmsconex/public_html/includes/classes/user.php [line] => 250 [function] => get_userdata [class] => user [type] => -> [object] => user Object ( ) [args] => Array ( [0] => usuario [1] => test ) ) [9] => Array ( [file] => /home/cmsconex/public_html/gestion/usuarios-nuevo.php [line] => 39 [function] => username_exists [class] => user [type] => -> [object] => user Object ( ) [args] => Array ( [0] => test ) ) ) [callback] => ) 1

A: getAll returns a DB_Error object when it fails.
 You have to print to see what the error is.
According to the doc, to retrieve the error just do:
if (PEAR::isError($resultado)) {
    die($resultado->getMessage());
}

getAll needs an array as second parameter like:
$resultado = & $db->getAll ( $sql, array(), DB_FETCHMODE_ASSOC );

redpajama_set_name: RedPajamaStackExchange


text:
<!DOCTYPE HTML>

<title>Uses of Class com.wrapper.spotify.requests.data.AbstractDataRequest (Spotify Web API Java Client 6.5.3 API)</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="dc.created" content="2021-04-15">
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Uses of Class
com.wrapper.spotify.requests.data.AbstractDataRequest

Packages that use AbstractDataRequest

Package
Description

com.wrapper.spotify.requests.data

com.wrapper.spotify.requests.data.albums

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com.wrapper.spotify.requests.data.browse.miscellaneous

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com.wrapper.spotify.requests.data.follow.legacy

com.wrapper.spotify.requests.data.library

com.wrapper.spotify.requests.data.personalization

com.wrapper.spotify.requests.data.personalization.simplified

com.wrapper.spotify.requests.data.player

com.wrapper.spotify.requests.data.playlists

com.wrapper.spotify.requests.data.search

com.wrapper.spotify.requests.data.search.simplified

com.wrapper.spotify.requests.data.search.simplified.special

com.wrapper.spotify.requests.data.shows

com.wrapper.spotify.requests.data.tracks

com.wrapper.spotify.requests.data.users_profile

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data

Modifier and Type
Class
Description

class
AbstractDataPagingCursorbasedRequest<T>

class
AbstractDataPagingRequest<T>

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.albums

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.albums

Modifier and Type
Class
Description

class
GetAlbumRequest

Get Spotify catalog information for a single album.

class
GetAlbumsTracksRequest

Get Spotify catalog information about an album's tracks.

class
GetSeveralAlbumsRequest

Get Spotify catalog information for multiple albums identified by their Spotify IDs.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.artists

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.artists

Modifier and Type
Class
Description

class
GetArtistRequest

Get Spotify catalog information for a single artist identified by their unique Spotify ID.

class
GetArtistsAlbumsRequest

Get Spotify catalog information about an artist's albums.

class
GetArtistsRelatedArtistsRequest

Get Spotify catalog information about artists similar to a given artist.

class
GetArtistsTopTracksRequest

Get Spotify catalog information about an artist's top tracks by country.

class
GetSeveralArtistsRequest

Get Spotify catalog information for several artists based on their Spotify IDs.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.browse

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.browse

Modifier and Type
Class
Description

class
GetCategoryRequest

Get a single category used to tag items in Spotify (on, for example, the Spotify player's "Browse" tab).

class
GetCategorysPlaylistsRequest

Get a list of Spotify playlists tagged with a particular category.

class
GetListOfCategoriesRequest

Get a list of categories used to tag items in Spotify (on, for example, the Spotify player's "Browse" tab).

class
GetListOfFeaturedPlaylistsRequest

Get a list of Spotify featured playlists (shown, for example, on a Spotify player's "Browse" tab).

class
GetListOfNewReleasesRequest

Get a list of new album releases featured in Spotify (shown, for example, on a Spotify player's "Browse" tab).

class
GetRecommendationsRequest

Create a playlist-style listening experience based on seed artists, tracks and genres.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.browse.miscellaneous

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.browse.miscellaneous

Modifier and Type
Class
Description

class
GetAvailableGenreSeedsRequest

Retrieve a list of available genres seed parameter values for recommendations.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.episodes

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.episodes

Modifier and Type
Class
Description

class
GetEpisodeRequest

Get Spotify catalog information for a single episode identified by its unique Spotify ID.

class
GetSeveralEpisodesRequest

Get Spotify catalog information for multiple episodes based on their Spotify IDs.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.follow

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.follow

Modifier and Type
Class
Description

class
CheckCurrentUserFollowsArtistsOrUsersRequest

Check to see if the current user is following one or more artists or other Spotify users.

class
CheckUsersFollowPlaylistRequest

Check to see if one or more Spotify users are following a specified playlist.

class
FollowArtistsOrUsersRequest

Add the current user as a follower of one or more artists or other Spotify users.

class
FollowPlaylistRequest

Add the current user as a follower of a playlist.

class
GetUsersFollowedArtistsRequest

Get the current user's followed artists.

class
UnfollowArtistsOrUsersRequest

Remove the current user as a follower of one or more artists or other Spotify users.

class
UnfollowPlaylistRequest

Remove the current user as a follower of a playlist.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.follow.legacy

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.follow.legacy

Modifier and Type
Class
Description

class
FollowPlaylistRequest

Add the current user as a follower of a playlist.

class
UnfollowPlaylistRequest

Remove the specified user as a follower of a playlist.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.library

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.library

Modifier and Type
Class
Description

class
CheckUsersSavedAlbumsRequest

Check if one or more albums is already saved in the current Spotify user's "Your Music" library.

class
CheckUsersSavedShowsRequest

Check if one or more shows is already saved in the current Spotify user's library.

class
CheckUsersSavedTracksRequest

Check if one or more tracks is already saved in the current Spotify user's "Your Music" library.

class
GetCurrentUsersSavedAlbumsRequest

Get a list of the albums saved in the current Spotify user's "Your Music" library.

class
GetUsersSavedShowsRequest

Get a list of shows saved in the current Spotify user's library.

class
GetUsersSavedTracksRequest

Get a list of the songs saved in the current Spotify user's "Your Music" library.

class
RemoveAlbumsForCurrentUserRequest

Remove one or more albums from the current user's "Your Music" library.

class
RemoveUsersSavedShowsRequest

Delete one or more shows from current Spotify user's library.

class
RemoveUsersSavedTracksRequest

Remove one or more tracks from the current user's "Your Music" library.

class
SaveAlbumsForCurrentUserRequest

Save one or more albums to the current user's "Your Music" library.

class
SaveShowsForCurrentUserRequest

Save one or more shows to current Spotify user's library.

class
SaveTracksForUserRequest

Save one or more tracks to the current user's "Your Music" library.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.personalization

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.personalization

Modifier and Type
Class
Description

class
GetUsersTopArtistsAndTracksRequest<T extends IArtistTrackModelObject>

This class only exists for theoretical purposes.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.personalization.simplified

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.personalization.simplified

Modifier and Type
Class
Description

class
GetUsersTopArtistsRequest

Get the current user's top artists based on calculated affinity.

class
GetUsersTopTracksRequest

Get the current user's top tracks based on calculated affinity.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.player

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.player

Modifier and Type
Class
Description

class
AddItemToUsersPlaybackQueueRequest

Add a track or an episode to the end of the user's current playback queue.

class
GetCurrentUsersRecentlyPlayedTracksRequest

Get tracks from the current user's recently played tracks.

class
GetInformationAboutUsersCurrentPlaybackRequest

Get information about the user's current playback state, including track, track progress, and active device.

class
GetUsersAvailableDevicesRequest

Get information about a user's available devices.

class
GetUsersCurrentlyPlayingTrackRequest

Get the object currently being played on the user's Spotify account.

class
PauseUsersPlaybackRequest

Pause playback on the user's account.

class
SeekToPositionInCurrentlyPlayingTrackRequest

Seeks to the given position in the user's currently playing track.

class
SetRepeatModeOnUsersPlaybackRequest

Set the repeat mode for the user's playback.

class
SetVolumeForUsersPlaybackRequest

Set the volume for the user's current playback device.

class
SkipUsersPlaybackToNextTrackRequest

Skips to next track in the user's queue.

class
SkipUsersPlaybackToPreviousTrackRequest

Skips to previous track in the user's queue.

class
StartResumeUsersPlaybackRequest

Start a new context or resume current playback on the user's active device.

class
ToggleShuffleForUsersPlaybackRequest

Toggle shuffle on or off for user's playback.

class
TransferUsersPlaybackRequest

Transfer playback to a new device and determine if it should start playing.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.playlists

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.playlists

Modifier and Type
Class
Description

class
AddItemsToPlaylistRequest

Add one or more items to a user's playlist.

class
ChangePlaylistsDetailsRequest

Change a playlist's name and public/private state.

class
CreatePlaylistRequest

Create a playlist for a Spotify user.

class
GetListOfCurrentUsersPlaylistsRequest

Get a list of the playlists owned or followed by the current Spotify user.

class
GetListOfUsersPlaylistsRequest

Get a list of the playlists owned or followed by a Spotify user.

class
GetPlaylistCoverImageRequest

Get the current image associated with a specific playlist.

class
GetPlaylistRequest

Get a playlist owned by a Spotify user.

class
GetPlaylistsItemsRequest

Get full details of the tracks or episodes of a playlist owned by a Spotify user.

class
RemoveItemsFromPlaylistRequest

Remove one or more items from a user's playlist.

class
ReorderPlaylistsItemsRequest

Reorder an item or a group of items in a playlist.

class
ReplacePlaylistsItemsRequest

Replace all the items in a playlist, overwriting its existing items.

class
UploadCustomPlaylistCoverImageRequest

Replace the image used to represent a specific playlist.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.search

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.search

Modifier and Type
Class
Description

class
SearchItemRequest

Get Spotify catalog information about artists, albums, episodes, playlists, shows or tracks that match a keyword string.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.search.simplified

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.search.simplified

Modifier and Type
Class
Description

class
SearchAlbumsRequest

Get Spotify catalog information about albums that match a keyword string.

class
SearchArtistsRequest

Get Spotify catalog information about artists that match a keyword string.

class
SearchEpisodesRequest

Get Spotify catalog information about episodes that match a keyword string.

class
SearchPlaylistsRequest

Get Spotify catalog information about playlists that match a keyword string.

class
SearchShowsRequest

Get Spotify catalog information about shows that match a keyword string.

class
SearchTracksRequest

Get Spotify catalog information about tracks that match a keyword string.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.search.simplified.special

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.search.simplified.special

Modifier and Type
Class
Description

class
SearchAlbumsSpecialRequest

Get Spotify catalog information about albums that match a keyword string.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.shows

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.shows

Modifier and Type
Class
Description

class
GetSeveralShowsRequest

Get Spotify catalog information for multiple shows based on their Spotify IDs.

class
GetShowRequest

Get Spotify catalog information for a single show identified by its unique Spotify ID.

class
GetShowsEpisodesRequest

Get Spotify catalog information about an show's episodes.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.tracks

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.tracks

Modifier and Type
Class
Description

class
GetAudioAnalysisForTrackRequest

Get a detailed audio analysis for a single track identified by its unique Spotify ID.

class
GetAudioFeaturesForSeveralTracksRequest

Get audio features for multiple tracks based on their Spotify IDs.

class
GetAudioFeaturesForTrackRequest

Get audio feature information for a single track identified by its unique Spotify ID.

class
GetSeveralTracksRequest

Get Spotify catalog information for multiple tracks based on their Spotify IDs.

class
GetTrackRequest

Get Spotify catalog information for a single track identified by its unique Spotify ID.

Uses of AbstractDataRequest in com.wrapper.spotify.requests.data.users_profile

Subclasses of AbstractDataRequest in com.wrapper.spotify.requests.data.users_profile

Modifier and Type
Class
Description

class
GetCurrentUsersProfileRequest

Get detailed profile information about the current user (including the current user's username).

class
GetUsersProfileRequest

Get public profile information about a Spotify user.

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This communication is from a debt collector. This is an attempt to collect a debt and any information obtained will be used for that purpose. Unless you notify this office within 30 days after receiving this notice that you dispute the validity of the debt or any portion thereof, this office will assume this debt is valid. If you notify this office in writing within 30 days after receiving this notice that you dispute the validity of this debt or any portion thereof, this office will obtain verification of the debt or obtain a copy of the judgment and mail you a copy of such judgment or verification. If you request from this office in writing within 30 days after receiving this notice, this office will provide you with the name and address of the original creditor, if different than the current creditor.
California: The state Rosenthal Fair Debt Collection Practices Act and the federal Fair Debt Collection Practices Act require that, except under unusual circumstances, collectors may not contact you before 8 a.m. or after 9 p.m. They may not harass you by using threats of violence or arrest, or by using obscene language. Collectors may not use false or misleading statements or call you at work if they know or have reason to know that you may not receive personal calls at work. For the most part, collectors may not tell another person, other than your attorney, or spouse, about your debt. Collectors may contact another person to confirm your location or enforce a judgment. For more information about debt collection activities, you may contact the Federal Trade Commission at 1-877-FTC-HELP or www.ftc.gov.
Colorado: FOR INFORMATION ABOUT THE COLORADO FAIR DEBT COLLECTION PRACTICES ACT, SEE WWW.COAG.GOV/CAR. A consumer has the right to request in writing that a debt collector or collection agency cease further communication with the consumer. A written request to cease communication will not prohibit the debt collector or collection agency from taking any other action authorized by law to collect the debt.
Our office is licensed under the name of Halsted Financial Services, LLC by the Colorado Department of Commerce for the state of Colorado. Local Office information: Colorado Manager, Inc. Halsted Financial Services, LLC 80 Garden Center, Suite 3, Broomfield, CO 80020 Phone: 303-920-4763 or 877-833-1629.
Illinois: Our office is licensed under the name of Halsted Financial Services, LLC for the state of Illinois.
Massachusetts: We are required by regulation of the Massachusetts attorney general to notify you of the following information. This information is not legal advice: this debt may be too old for you to be sued on it in court. If it is too old, you cannot be required to pay it through a lawsuit. Take note: you can renew the debt and the statute of limitations for the filing of a lawsuit against you if you do any of the following: make any payment on the debt, sign a paper in which you admit you owe the debt of in which you make a new promise to pay; sign a paper in which you give up or waive your right to stop the creditor from suing you in court to collect the debt. While this debt may not be enforceable through a lawsuit, it may still affect your ability to obtain credit or affect your credit score or rating.
(a) Obtain or attempt to obtain from the creditor any document that is not in the possession of the collection agency and is reasonably responsive to the dispute of the debtor, if any; and (b) If such a document is obtained, mail the document to the debtor.
(a) If the debtor pays or agrees to pay the debt or any portion of the debt, the payment or agreement to pay may be construed as: (1) An acknowledgement of the debt by the debtor; and (2) A wavier by the debtor of any applicable statute of limitations set forth in NRS 11.190 that otherwise precludes the collection of the debt; and (b) if the debtor does not understand of has questions concerning his or her legal rights or obligations relating to the debt, the debtor should seek legal advice.
(Added to NRS by 2007,2005) NRS 449.012 "Hospital" Defined. "Hospital" means an establishment for the diagnosis, care and treatment of human illness, including care available 24 hours each day from persons licensed to practice professional nursing who are under the direction of a physician, services of a medical laboratory and medical, radiological, dietary and pharmaceutical services. (Added to NRS by 1973, 1279; A 1985, 1737) If the consumer pays or agrees to pay the debt or any portion of the debt, the payment or agreement to pay may be construed as: (1) an acknowledgement of the debt by the consumer; and (2) a wavier by the consumer od any applicable statute of limitations set forth in NRS 11.190 that otherwise precludes the collection of the debt; and (3) if the consumer does not understand or has questions concerning his/her legal rights or obligations relating to the debt, the debtor shall seek legal advice.
New York City: New York City Department of Consumer Affairs License Number: 2039870-DCA.
New York: Debt collectors in accordance with the Fair Debt Collection Practices Act, 15 U.S.C. § 1692 et seq., are prohibited from engaging in abusive, deceptive, and unfair debt collection efforts, including but not limited to: the use or threat of violence; the use of obscene or profane language; and repeated phone calls made with the intent to annoy, abuse, or harass.
Debt collectors, in accordance with the Fair Debt Collection Practices Act, 15 U.S.C. § 1692 et seq., are prohibited from engaging in abusive, deceptive, and unfair debt collection efforts, including but not limited to: (i) the use or threat of violence; (ii) the use of obscene or profane language; and (iii) repeated phone calls made with the intent to annoy, abuse, or harass. (The state does not provide sample language for this disclosure.) City of Yonkers, NY: In addition to any practices prohibited under any federal, state or local law, a debt collection agency shall not contact a consumer about or seek to collect a debt on which the statute of limitations for initiating legal action has expired unless such agency first provides the consumer such information about the consumer's legal rights as the commissioner proscribes by rule.
Tennessee: Halsted Financial Services, LLC is licensed by the Collection Service Board of the Department of Commerce and Insurance.
West Virginia: The activities of collection agencies in West Virginia are regulated by the Attorney General's Consumer Protection Division, 812 Quarrier St., Charleston, WV 25301. Federal law prohibits agencies from contacting you about your debt if you send a letter requesting that all contacts stop.
Wisconsin: This collection agency is licensed by the Division of Banking in the Wisconsin Department of Financial Institutions, www.wdfi.com.


text:
using System.Text.Json;

namespace Microsoft.JSInterop.Infrastructure;

public class DotNetStreamReferenceJsonConverterTest
{
    private readonly JSRuntime JSRuntime = new TestJSRuntime();
    private readonly JsonSerializerOptions JsonSerializerOptions;

    public DotNetStreamReferenceJsonConverterTest()
    {
        JsonSerializerOptions = JSRuntime.JsonSerializerOptions;
        JsonSerializerOptions.Converters.Add(new DotNetStreamReferenceJsonConverter(JSRuntime));
    }

    [Fact]
    public void Read_Throws()
    {
        // Arrange
        var json = "{}";

        // Act & Assert
        var ex = Assert.Throws<NotSupportedException>(() => JsonSerializer.Deserialize<DotNetStreamReference>(json, JsonSerializerOptions));
        Assert.StartsWith("DotNetStreamReference cannot be supplied from JavaScript to .NET because the stream contents have already been transferred.", ex.Message);
    }

    [Fact]
    public void Write_WritesValidJson()
    {
        // Arrange
        var streamRef = new DotNetStreamReference(new MemoryStream());

        // Act
        var json = JsonSerializer.Serialize(streamRef, JsonSerializerOptions);

        // Assert
        Assert.Equal("{\"__dotNetStream\":1}", json);
    }

    [Fact]
    public void Write_WritesMultipleValidJson()
    {
        // Arrange
        var streamRef = new DotNetStreamReference(new MemoryStream());

        // Act & Assert
        for (var i = 1; i <= 10; i++)
        {
            var json = JsonSerializer.Serialize(streamRef, JsonSerializerOptions);
            Assert.Equal($"{{\"__dotNetStream\":{i}}}", json);
        }
    }
}