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Observation of Entanglement of a Single Photon with a Trapped Atom

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We report the observation of entanglement between a single trapped atom and a single photon at a wavelength suitable for low-loss communication over large distances, thereby achieving a crucial step towards long range quantum networks. To verify the entanglement, we introduce a single atom state analysis. This technique is used for full state tomography of the atom-photon qubit pair. The detection efficiency and the entanglement fidelity are high enough to allow in a next step the generation of entangled atoms at large distances, ready for a final loophole-free Bell experiment.
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Observation of Entanglement of a Single Photon with a Trapped Atom
Ju
¨
rgen Volz,
1,
*
Markus Weber,
1,†
Daniel Schlenk,
1
Wenjamin Rosenfeld,
1
Johannes Vrana,
1
Karen Saucke,
1
Christian Kurtsiefer,
2
and Harald Weinfurter
1,3
1
Department fu
¨
r Physik, Ludwig-Maximilians-Universita
¨
tMu
¨
nchen, D-80799 Mu
¨
nchen, Germany
2
Department of Physics, National University of Singapore, Singapore
3
Max-Planck-Institut fu
¨
r Quantenoptik, 85748 Garching, Germany
(Received 13 October 2005; published 25 January 2006)
We report the observation of entanglement between a single trapped atom and a single photon at a
wavelength suitable for low-loss communication over large distances, thereby achieving a crucial step
towards long range quantum networks. To verify the entanglement, we introduce a single atom state
analysis. This technique is used for full state tomography of the atom-photon qubit pair. The detection
efficiency and the entanglement fidelity are high enough to allow in a next step the generation of entangled
atoms at large distances, ready for a final loophole-free Bell experiment.
DOI: 10.1103/PhysRevLett.96.030404 PACS numbers: 03.65.Ud, 03.67.Mn, 32.80.Qk, 42.50.Xa
Entanglement is a key element for quantum communi-
cation and information applications [1]. Demonstrations of
quantum computers with ions in linear chains nowadays
almost routinely create deterministically any desired en-
tangled state with up to eight ions [2]. The currently largest
quantum processor consisting of some tens of (not yet
distinguishable) qubits in a so-called cluster state was
implemented with neutral atoms in an optical lattice [3].
For future applications such as quantum networks or the
quantum repeater [4], it is mandatory to achieve entangle-
ment also between separated quantum processors. For this
purpose, entanglement between different quantum objects
such as atoms and photonsrecently demonstrated for
ions and photons [5]forms the interface between atomic
quantum memories and photonic quantum communication
channels [6], finally allowing the distribution of quantum
information over arbitrary distances.
Atom-photon entanglement is not only crucial for many
applications in long range quantum communication but is
also the key element to give the final answer to Einstein’s
question on the real properties of nature [7]. Together with
Podolsky and Rosen, he pointed out the inconsistencies
between quantum mechanics and their ideal of a local and
deterministic description of nature [8]. They implied that
parameters of a physical system (local hidden variables),
which might notyetbe known to us, could solve the
problem. Until now, the results of many experiments based
on Bell’s inequality [9] indicate that hidden variable theo-
ries would result in incorrect predictions and, thus, are not
a valid description of nature [10–12]. But all these tests are
subject to loopholes [11,13], and none so far could defi-
nitely rule out all alternative concepts.
Here we describe the observation of entanglement be-
tween the polarization of a single photon and the internal
state of a single neutral atom stored in an optical dipole
trap. For this purpose, we introduce a new state-analysis
method enabling full state tomography of the atomic qubit.
This now allows for the first time the direct analysis of the
entangled atom-photon state formed during the spontane-
ous emission process. Moreover, we can show that the
results achieved indeed suffice to test Einstein’s objections.
Atom-photon entanglement can be prepared best by ex-
citing an atom to a state which ideally has two decay chan-
nels ( configuration). The hyperfine structure of
87
Rb
offers a good approximation to such a level scheme
[Fig. 1(a)]. Excited to the 5
2
P
3=2
, F
0
0 hyperfine level,
the atom can spontaneously decay into the three magnetic
sublevels jm
F
0;1i of the 5
2
S
1=2
, F 1 hyperfine
ground level by emitting a photon at a wavelength of
780 nm.
If the emitted photon is
-polarized, the atom will be
in the state jm
F
1i. If the photon is linearly
-polarized, the atom will be in the state jm
F
0i, and
for
polarization we find the state jm
F
1i. Since the
emitted photons are collected along the quantization axis,
-polarized light (emitted into a different spatial mode) is
not collected. Therefore, only spontaneous decay into the
states jm
F
1i is observed. As long as these emission
processes are indistinguishable in all other degrees of free-
dom, one obtains a coherent superposition of the two decay
possibilities, i.e., the maximally entangled state
j
i
1

2
p
jm
F
1ij
ijm
F
1ij
i: (1)
Here in each of the terms the first ket describes the state of
the atom and the second one the polarization of the photon.
The quantum mechanical phase of the atom-photon state is
well defined and follows from the Clebsch-Gordan coef-
ficients of the transitions.
In our experiment, atoms are cooled from a shallow
magneto-optical trap (MOT) into an optical dipole trap
located in the center of the MOT [14]. For the dipole trap
waist size of 3:5 m, a collisional blockade mechanism
ensures that only single atoms are captured [15].
When a single atom is loaded into the trap and its fluo-
rescence is registered, the sequence entangling the atom
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with a photon is started by pumping it into the F 1,
m
F
0 state. Next, a 30 ns optical pulse excites the
atom to the F
0
0 level, from which it will decay back to
F 1. Photons emitted along the quantization axis are
collected and guided via a single mode optical fiber to a
single photon polarization analyzer to determine the state
of the photonic qubit [see Fig. 1(b)]. Because the emitted
photon is detected with an overall efficiency of
ph
5
10
4
, the whole process has to be repeated approximately
2000 times to observe the photon. The repetition rate of
this process is 1:25 10
5
s
1
. However, in the atomic
state detection, the atom is lost with a probability of 0.5
(see below). Therefore, the mean time to load an atom into
the dipole trap limits the total rate for the generation of
entangled atom-photon pairs to 0:2s
1
.
Once the emitted photon is detected, the state analysis of
the atom is initiated. Standard spectroscopy techniques
probing only the populations of the states jm
F
1i
and jm
F
1i are not sufficient to confirm entanglement.
Instead, a projection onto general superposition states is
required. We thus apply a state selective stimulated Raman
adiabatic passage (STIRAP) technique [16], which allows
one to transfer an arbitrary superposition state j i
sinjm
F
1ie
i
cosjm
F
1i adiabatically to
the F 2 ground level (Fig. 2). Because of the selection
rules of atomic dipole transitions, the orthogonal quantum
state does not couple to the STIRAP light field
1
and
remains in the F 1 level. The angles and in this
process are defined by the relative amplitude and phase of
the
and
polarization components of the STIRAP
laser
1
, respectively. In essence, the polarization of the
STIRAP laser defines which superposition state is trans-
ferred, thus allowing a full tomographic analysis of the
atomic state without the necessity to perform any state
manipulation on the atomic qubit.
After the STIRAP pulse, the atom is in a superposition
of the hyperfine ground levels F 1 and F 2, which
now can be distinguished by standard methods. We apply a
detection laser pulse (resonant to the closed transition F
2 ! F
0
3), removing atoms in the F 2 level from the
trap. Finally, to read out the atomic state, the cooling lasers
of the MOT are switched on and atomic fluorescence is
measured for 30 ms to decide whether the atom is still in
the trap or not. Thereby, we obtain the binary result of the
projective atomic state measurement on the state j i and
the orthogonal state j
?
i. For the results shown in Fig. 3,
we repeated the experimental cycle approximately
300 times per data point from which we obtain the proba-
bility of the atom to remain in F 1 with a statistical error
of 2%.
To verify the entanglement of the generated atom-
photon state, we perform ^
x
( =4, 0) as well
as ^
y
( =4, =2) state analysis of the atomic
qubit for different polarization measurements of the photon
(Fig. 3, ^
i
are the spin-1=2 Pauli operators). Thereby, the
probability of the atom to be transferred by the STIRAP
pulse sequence, or the probability to remain in the F 1
FIG. 2 (color online). Experimental procedure for the atomic
state detection. To analyze the atomic state, a two-photon
STIRAP-process state selectively transfers a superposition of
the states jm
F
1i and jm
F
1i to the F 2 hyperfine
level. To read out the atomic qubit, a hyperfine-level selective
detection pulse is applied before standard fluorescence detection.
FIG. 1 (color online). (a) Preparation of atom-photon entan-
glement in
87
Rb. The excited hyperfine level with F
0
0 can
decay to three possible ground states with the magnetic quantum
numbers m
F
1, 0, or 1, by spontaneously emitting a
, ,
or
polarized photon, respectively. If the light is collected
along the quantization axis, -polarized photons are suppressed.
Thus, an effective configuration is obtained which allows the
preparation of a maximally entangled state between the photon
polarization and the orientation of the atomic magnetic moment.
(b) Scheme of the experimental setup. The dipole trap light (
856 nm, P 30 mW) is focused by a microscope objective
(NA 0:38) to a waist of 3:5 m. The photon from the sponta-
neous decay is collected with the same objective, separated from
the trapping beam by a dichroic mirror, and coupled into a single
mode optical fiber guiding it to the polarization analyzer. The
analyzer consists of a rotable half- and quarter-wave plate, a
polarizing beam splitter (PBS), and two avalanche photodiodes
(APD) for single photon detection. Triggered by the detection of
the photon in either APD
1
or APD
2
, the atomic state is analyzed
using a STIRAP light field whose polarization defines the atomic
measurement basis.
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ground level, respectively, is measured, conditioned on the
polarization measurement outcome of the photon. Varying
the photon polarization analyzer, this probability shows the
expected sinusoidal dependence for both ^
x
and ^
y
. From
the fits to the measured data, we obtain an effective visi-
bility (peak to peak amplitude) of V
atph
0:85 0:01 for
analysis in ^
x
and V
atph
0:87 0:01 for analysis in ^
y
.
This clearly proves entanglement of the generated atom-
photon state.
For the determination of the full atom-photon state, we
perform two-qubit state tomography. This involves a new
set of measurements determining correlations of all com-
binations of the operators ^
x
, ^
y
, and ^
z
on the atom and
the photon [17]. The density matrix
atph
determined this
way clearly proves the state to be of the form of (1) [see
Fig. 4(a)]. The fidelity, defined as the overlap between
j
ih
j and
atph
, in this measurement was F
0:87 0:01. The limited visibility in the atom-photon
correlations is caused mainly by errors in the atomic state
detection (5%), accidental photon detection events due to
the dark counts of the single photon detectors (3%), off-
resonant excitation to the hyperfine level 5
2
P
3=2
, F
0
1
(1%), and polarization errors of the STIRAP laser beams
(2%). Applying the Peres-Horodecki criterion [18] to the
combined density matrix proves the entanglement with a
negativity of 0:382. Figures 4(b) and 4(c) show the density
matrices of the atomic and the photonic state after tracing
over the partner qubit. Obviously, these states are in a
complete statistical mixture. However, it becomes clear
that the resulting atom-photon state is not a mixture of
all possible contributions but is instead a well defined
(ideally) pure entangled state.
In view of these results, let us now analyze the perform-
ance of a possible loophole-free Bell experiment with a
pair of entangled atoms. Crucial for such a test is a highly
efficient state analysis by spacelike separated observers. To
generate entanglement between atoms at remote locations,
they are first entangled with a photon each. The two pho-
tons are brought together and then are subject to a Bell-
state measurement, which serves to swap the entanglement
to the atoms [19]. If we use the average visibility observed
in our experiment and extrapolate results of recent two-
photon interference experiments [2022], we derive an
expected atom-atom visibility of V
atat
V
2
atph
0:74
0:01. Thus, the violation of a Clauser-Horne-Shimony-Holt
(CHSH)-type Bell inequality [23], which is achieved above
the threshold visibility of 0.71, is feasible.
We emphasize that, triggered on the detection of a
photon, every atomic state measurement yields a result.
In this sense, the atomic detection efficiency is equal to
FIG. 4 (color online). (a) Graphical representation of the real
part of the measured density matrix of the entangled atom-
photon state. The fidelity (overlap with the expected state
j
i) from this measurement is F 0:87 0:01. Insets (b)
and (c) show the single particle density matrices for the atom and
photon state, respectively, indicating that the single particles
when observed on their own are found in a completely mixed
state.
FIG. 3 (color online). Probability of detecting the atom in the
ground level F 1 (after the STIRAP pulse) conditioned on the
detection of the photon in detector APD
1
(--) or APD
2
(--) as
the linear polarization of the photonic qubit is rotated by an
angle . (a) The atomic qubit is measured in ^
x
and (b) in ^
y
,
whereas the photonic qubit is projected onto the states
1=

2
p
j
ie
2i
j
i.
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one. In certain cases, as, e.g., the loss of the atom from the
trap, a wrong measurement outcome may occur, but one
always obtains a result. Moreover, entanglement swapping
enables a so-called event-ready scheme [11,19,24]. If mea-
surement results are reported for every joint photon detec-
tion event, this scheme is independent of any additional
assumptions and, thus, is not subject to any detection
related loopholes at all. To close at the same time the
locality loophole, the atoms have to be spacelike separated
with respect to the measurement time of the atomic states.
The minimum distance of the atoms is determined by the
duration of the atomic state detection. In detail, the atomic
state collapses by scattering photons from the detection
laser for 350 ns. Together with the STIRAP process, this
yields an overall measurement time of less than 0:5 s and
requires a separation of the atoms of 150 m. Thus, we
expect the generation of one entangled atom-atom pair per
minute [25]. A loophole-free violation of a CHSH-type
Bell’s inequality [23] by 3 standard deviations would
require approximately 7000 atom pairs at the expected
visibility of 0.74. This would be feasible within a total
measurement time of 12 days.
In this Letter, we presented a successful implementation
of a source of high-fidelity entangled atom-photon pairs.
We introduced a single atom STIRAP state analysis which
does not require additional atomic state manipulations and,
thus, can be performed with increased fidelity. This al-
lowed us to perform the first full state tomography of an
atom-photon system and proved that the spontaneous
emission of the atom results in the entangled state j
i.
In the experiment, we achieved a state fidelity of F
0:87 0:01 and a mean visibility of the atom-photon
correlations of V
atph
0:86 0:01. These methods, pos-
sibly combined with high-Q cavities to enhance the col-
lection efficiency [22,26], form the basic elements in future
quantum information experiments for building the inter-
face between quantum computers and a photonic quantum
communication channel. In addition, these tools also help
to find an answer to the long-standing question of whether
local realistic extensions of quantum mechanics can de-
scribe nature at all. The experimental demonstration of
high-fidelity entanglement provides the most important
step towards a final, loophole-free test of Bell’s inequality.
We acknowledge stimulating discussions with T. W.
Ha
¨
nsch and his group. This work was supported by the
Deutsche Forschungsgemeinschaft and the European
Commission through the EU Project QAP (IST-3-015848).
*Electronic address: juergen.volz@physik.uni-
muenchen.de
Electronic address: markus.weber@physik.uni-
muenchen.de
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... Integration of local processing and memory enhances performance of quantum repeaters over lossy channels, enables distributed quantum processing and sensing, and allows for entanglement resources to be shared within the network. In prior work with a single atomic qubit per node, entanglement between an atomic qubit and a photon [39,88,89,15] and entanglement between remote atomic qubits [90,91,92] have been realized. High-fidelity one-qubit gates [93,94] and pairwise entanglement of neighboring and nextto-nearest neighbor sites [95,96] in 1-D and 2-D arrays have also been demonstrated. ...
Thesis
A principal goal of distributed quantum processing is the ability to generate, manipulate, and transfer quantum states between distant nodes of a quantum network. These protocols generally require connecting photonic and material carriers of quantum information. In this thesis, I present investigations of two experimental realizations of light-matter interfaces that allow for engineered atom-photon interactions in free-space settings. First, we utilize reconfigurable arrays of trapped single atoms to study light scattering in lowdimensional systems. We observe noncollinear phase-matching geometries that have suppressed sensitivity to particle localization. We show that the scattered radiation can be controllably enhanced or diminished as a result of Bragg interference. Such scattering can be used for mapping collective states within an array of neutral atoms onto propagating light fields and for establishing quantum links between separated arrays. Second, we utilize ensembles of trapped Rydberg atoms to study collective many-body phenomena that arise due to strong dipole-dipole interactions. To do so, we employ a magic-wavelength optical lattice that allows for the simultaneous trapping of both ground and Rydberg levels. Using the enhanced coherence times enabled by this trapping scheme, we measure the so-called magic lattice detunings and use them to extract the 6P3/2 − nS1/2 reduced electric dipole matrix elements. Furthermore, we perform precision measurements of differential nuclear-spin dependent light shifts in the Paschen-Back regime in order to determine the hyperfine splitting of Rydberg levels. We create a quasi-two-level system in a regime of Rydberg excitation blockade for a mesoscopic ensemble of several hundred atoms. Using this system, we study Hanbury Brown-Twiss interference between the field radiated by the atoms and an input probe field with a controllable relative phase. Finally, we demonstrate coherent driving and Ramsey interference measurements of light shifts, with timescales on the order of ≃ 10 µs. Whereas the coupling producing the Rabi oscillations is enhanced, there is no corresponding enhancement for the lightshifts. These results may prove useful in applying collective qubits with Rydberg interactions to scalable quantum networking architectures.
... Quantum memories with light-matter interfaces enable the generation of heralded entanglement between distant locations. For this, the memories first emit a photon to generate entanglement between light and matter [118][119][120] or interact with incoming light in a state dependent manner [121,122]. Two distant memories can then be entangled by using, for example, heralded storage of an entangled photon pair, entanglement swapping from two light-matter pairs, or enhanced light-matter interaction by resonators. ...
Preprint
Device-independent quantum key distribution (DI-QKD) provides the gold standard for secure key exchange. Not only it allows for information-theoretic security based on quantum mechanics, but it relaxes the need to physically model the devices, hence fundamentally ruling out many quantum hacking threats to which non-DI QKD systems are vulnerable. In practice though, DI-QKD is very challenging. It relies on the loophole-free violation of a Bell inequality, a task that requires high quality entanglement to be distributed between distant parties and close to perfect quantum measurements, which is hardly achievable with current technology. Notwithstanding, recent theoretical and experimental efforts have led to the first proof-of-principle DI-QKD implementations. In this article, we review the state-of-the-art of DI-QKD by highlighting its main theoretical and experimental achievements, discussing the recent proof-of-principle demonstrations, and emphasizing the existing challenges in the field.
Preprint
We propose a scalable implementation of a quantum computer based on hardware-efficient mobile domain walls on magnetic racetracks. In our proposal, the quantum information is encoded in the chirality of the spin structure of nanoscale domain walls. We estimate that these qubits are long-lived and could be operated at sweet spots reducing possible noise sources. Single-qubit gates are implemented by controlling the movement of the walls in magnetic nanowires, and two-qubit entangling gates take advantage of naturally emerging interactions between different walls. These gates are sufficient for universal quantum computing and are fully compatible with current state-of-the-art experiments on racetrack memories. Possible schemes for qubit readout and initialization are also discussed.
Article
We study the disentanglement dynamics of two giant atoms coupled to a common one-dimensional waveguide. We focus on the non-Markovian retarded effect in the disentanglement of the two giant atoms by taking the photon transmission time into account. By solving the time-delayed equations of motion for the probability amplitudes, we obtain the evolution of the entanglement of the two giant atoms, which are initially in the maximally entangled states in the single-excitation space. It is found that the retardation-induced non-Markovianity leads to nonexponential decay and revivals of entanglement. Concretely, we consider separate-, braided-, and nested-coupling configurations, and find that the disentanglement dynamics in these configurations exhibits different features. We demonstrate that the steady-state entanglement depends on the time delay under certain conditions in these three coupling configurations. We also study the dependence of the disentanglement of the two giant atoms on both the detuning effect and the initial-state phase effect. In addition, we consider the disentanglement dynamics of the two giant atoms, which are initially in the state superposed by zero-excitation and two-excitation components. This work will pave the way for the generation of stationary entanglement between two giant atoms, which may have potential applications in the construction of large-scale quantum networks based on the giant-atom waveguide-QED systems.
Article
We theoretically investigate the dynamics of two spin qubits interacting with a magnetic medium. A systematic formal framework for this qubit-magnet hybrid system is developed in terms of the steady-state properties of the magnetic medium. Focusing on the induced dissipative coupling between the spin qubits, we show how a sizable long-lived entanglement can be established via the magnetic environment, in the absence of any coherent coupling. Moreover, we demonstrate that maximally entangled two-qubit states (Bell states) can be achieved in this scheme when complemented by proper postselection. In this situation, the time evolution of the entanglement is governed by a non-Hermitian Hamiltonian, where dynamical phases are separated by an exceptional point. The resultant Bell state is robust against weak random perturbations and does not require the preparation of a particular initial state. Our study may find applications in quantum information science, quantum spintronics, and for sensing of nonlocal quantum correlations.
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Quantum network scales the advantages of quantum communication protocols to more than two detached users, which offers great potential to realize the quantum internet. Here, a fully connected continuous‐variable (CV) quantum teleportation network architecture is presented, in which a squeezed state of light distributes each pair of entangled sideband modes to each communication link that bridges any pair of users. The quantum teleportation scheme is similar to standard two‐party ones for each communication link, without sacrificing communication capability and reliability. The demonstration based on CV‐entangled sideband modes opens an innovative possibility to implement many tasks of deterministic quantum information processing. A new scheme of fully interconnected quantum network for continuous variable optical system is developed and demonstrated. The quantum teleportation protocol between any pair of users is realized with ultra‐low losses. Exploiting the distribution scheme of entangled sideband modes from a squeezed state opens an innovative possibility to implement many deterministic quantum information processing tasks.
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We have measured the linear polarization correlation of the photons emitted in an atomic cascade of calcium. It has been shown by a generalization of Bell's inequality that the existence of local hidden variables imposes restrictions on this correlation in conflict with the predictions of quantum mechanics. Our data, in agreement with quantum mechanics, violate these restrictions to high statistical accuracy, thus providing strong evidence against local hidden-variable theories.
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We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (“qubits”). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices that are always non-negative definite). In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.
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A deterministic local hidden-variable model is presented which describes the simultaneous measurement of the spins of two spin-½ particles which emerged from the decay of a spin-zero particle. In this model the measurement of the spin of a particle has one of three possible outcomes: spin parallel to the apparatus axis, spin antiparallel to the apparatus axis, or the particle goes undetected. It is shown that agreement with the predictions of quantum theory is obtained provided the experimenter rejects the "anomalous" data in which only one particle is detected. A reasonably model-independent lower bound to the fraction of undetected particles is also computed: It is found that in 14% of the decays or more, one or both of the particles will go undetected.
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Bell's theorem represents a significant advance in understanding the conceptual foundations of quantum mechanics. The theorem shows that essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a single experimental arrangement. Moreover, the predictions by those theories must significantly differ from those by quantum mechanics. Experimental results evidently refute the theorem's predictions for these theories and favour those of quantum mechanics. The conclusions are philosophically startling: either one must totally abandon the realistic philosophy of most working scientists, or dramatically revise out concept of space-time.
Article
We study the use of entanglement purification for quantum communication over long distances. For distances much longer than the coherence length of a corresponding noisy quantum channel, the fidelity of transmission is usually so low that standard purification methods are not applicable. It is possible, however, to divide the channel into shorter segments that are purified separately and then connected by the method of entanglement swapping. This method can be much more efficient than schemes based on quantum error correction, as it makes explicit use of two-way classical communication. An important question is how the noise, introduced by imperfect local operations (that constitute the protocols of purification and the entanglement swapping), accumulates in such a compound channel, and how it can be kept below a certain noise level. To treat this problem, we first study the applicability and the efficiency of entanglement purification protocols in the situation of imperfect local operations. We then present a scheme that allows entanglement purification over arbitrary long channels and tolerates errors on the percent level. It requires a polynomial overhead in time, and an overhead in local resources that grows only logarithmically with the length of the channel.
Article
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Article
Quantum alpha-entropy inequalities equivalent to Bell's inequality for pure states are considered in the context of the local hidden variable (LHV) model and compared with Bell's inequalities. For alpha = 1,2 they are shown to be satisfied by convex combinations of product states and Werner's mixtures admitting the model. The 2-entropy inequality is proven to be stronger than Bell's inequality in the two-spin-1/(2) case. In the latter, the alpha-entropy inequalities taken as a joint condition exclude teleportation admitted in spite of the existence of the LHV model for the Werner-Popescu states.
Article
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