August 2004
Shared quantum states provide secure quantum
networks.
By Barry Sanders, University of Calgary and Macquarie
University
Quantum information science (QIS) is creating a revolution
in the way we think about information processing,
communication, and computation, even as optics provides some
of the most important technologies for testing proposals and
demonstrating proofs of principle.1 In particular,
a recent experimental breakthrough at the Australian National
University (Canberra, Australia) allows optical quantum
networking that is resilient against malicious parties and
component failures.2 The experiment involves
tripartite (three-beam, or three-mode in our terminology)
entanglement. It builds on the technology of optical quantum
teleportation to deliver security and reliability in quantum
information networks such as distributed quantum computers and
quantum communications networks.
The importance of QIS lies in the fact that it so
profoundly challenges long-cherished beliefs about information
security, communications, and computation. Computer science
theories assess the computability and complexity of
computational problems, but these theories are generally based
on Boolean logic and the binary representation of information.
Communication theory applies to the transmission of binary
strings down channels, and information security concerns
encoding and decoding of bit strings. QIS provides an entirely
new alternative to the use of digital strings as the
representation of information and Boolean gates for
processing.
Quantum theory allows us to replace binary data, or bits,
with quantum bits, or qubits; the qubit can be in an arbitrary
superposition a0|0> + a1|1>, with |0> the logical 0
state (for example, the H = horizontal polarization state of a
single photon) and |1> the logical 1 state (for example,
the V = vertical polarization state of a single photon). The
coefficients ai are complex, and the logical states can thus
co-exist in a single qubit. The capacity to create
superpositions provides unconditional security for quantum
cryptography; the Heisenberg uncertainty principle guarantees
that an eavesdropper will disturb the superposition, and
therefore be detectable, provided that the basis states of
|0> and |1> (such as horizontal/vertical versus
left/right circular polarization of light) are randomly
selected. As long as noise tolerance conditions are met, the
shared key remains unconditionally secure and can be used for
cryptography. Quantum cryptography is a successful quantum
information technology and is available commercially.
More importantly, two qubits can co-exist in the
superposition a00|00> + a01|01> + a10|10> +
a11|11>, and multipartite superposition (entangled) states
can be obtained for arbitrarily many qubits. Thus, in quantum
information processing, all inputs can simultaneously be
supplied to the device. Provided that a qubit can be
"rotated," i.e., a0|0> + a1|1> → a0 ′|0> + a1
′|1>, and a two-qubit gate such as a controlled NOT, i.e.,
a00|00> + a01|01> + a10|10> + a11|11> → a00|00>
+ a01|01> + a10|11> + a11|10>, can be achieved, our
system can demonstrate universal quantum computation.
Optical quantum information processing has been remarkably
successful. For example, commercial quantum key distribution
requires an optical approach because photons are relatively
easy to prepare and transmit (via fibers or through the
atmosphere), and researchers have made rapid progress toward
demonstrating components of the linear optical quantum
computer such as the controlled NOT gate.3,4
Although competing technologies exist for QIS, optical
technology offers advantages with respect to transmission, low
de-coherence, source and detector availability, and accurate
theoretical descriptions. A major disadvantage is the weak
nonlinearity, but non-deterministic operation provides an
effective approach to overcome this hurdle.
Continuous-Variable QIS
There are, in fact, two approaches to optical QIS. Above,
we summarized the polarization-encoded single-photon qubit
method, but another approach has been successful as well.
Continuous-variable QIS encodes the quantum information into
the amplitude or phase quadratures of the field mode.
Classically, the amplitude and the phase can be fixed, but for
a quantum field, the state can exist in a superposition of
amplitude states, thus providing a vehicle for quantum
information. More importantly, we can achieve entanglement by
creating quantum correlations between the amplitude states of
two or more modes (beams).
The continuous-variable approach to optical QIS offers
advantages over its single-photon counterpart for several
reasons. The typical input state is a coherent state, which
requires a stable laser source. Such fields are easier to
create than single photons, which will ultimately be needed
for single-photon-based QIS. Bipartite, or two-mode,
entanglement can be achieved by standard squeezing technology.
Our group has successfully created entangled light beams by
the method outlined below.
A neodymium-doped yttrium aluminum garnet (Nd:YAG) laser
operating at 1064 nm provides a source for the signal field to
be shared and for the local oscillator used in the
reconstruction of the signal state. We obtain a portion of the
source field by splitting the 1064-nm beam and doubling the
frequency of one portion to provide a 532-nm pump beam for a
pair of hemilithic magnesium oxide:lithium niobate
(MgO:LiNbO3) optical parametric amplifiers (OPAs). The OPAs
are seeded with 1064 nm light, and the OPAs are pumped to
produce squeezed vacuum states, which are mixed at a
beamsplitter to produce the entanglement resource.
The output beams from the two OPAs feature squeezed
amplitude fluctuations of 4.5 dB below the vacuum noise limit
and are mixed with a 1:1 beamsplitter (see figure below). The
two output fields that emerge from the beamsplitter into modes
2′ and 3′ are now entangled, or quantum correlated, in
amplitude. This two-mode squeezed light is a key resource for
continuous-variable QIS, which can be combined with a signal
in mode 1.
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The two OPA outputs are mixed at a
beamsplitter to produce a two-mode squeezed state (mode
2′ and mode 3′), and mode 2′ is mixed at a beamsplitter
with the signal state (mode 1′). We can recover the
signal (mode 1′′′) from the resultant three-mode
entangled state (modes 1′′, 2′′, and 3′′) by (i) a
beamsplitter recombining modes 1′′ and 2′′ if mode 3′′
is excluded, which yields the reconstructed signal in
output beam 1′′′, and (ii) by a beamsplitter, detector,
an electronic gain, and an amplitude modulator if mode
1′′ is excluded (similar if mode 2′′ is excluded), which
yields the reconstructed signal in output beam 2′′′.
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Another advantage of continuous-variable QIS over
single-photon QIS is in detection. One approach for
single-photon QIS uses polarization beamsplitters and
single-photon counting modules to measure polarization
correlations, but the measurements are limited by linear
optics constraints; only two of the four possible maximally
entangled two-qubit states can be detected. This limitation
restricts single-photon quantum information tasks such as
teleportation to being probabilistic.5
Continuous-variable optical QIS relies instead on homodyne
detection, which measures the field quadrature, and can
measure any field quadrature by a judicious choice of local
oscillator phase. Furthermore, homodyne detection is highly
efficient—our group has demonstrated efficiencies of 0.89. The
detection advantage yields "unconditional" QIS tasks, first
demonstrated for quantum teleportation.6 The high
efficiency, low decoherence, and versatility of detecting any
field quadrature, plus the capability of producing highly
coherent fields and strongly squeezed states, makes optical
continuous-variable QIS an attractive alternative to
single-photon QIS. Of course, challenges to the
continuous-variable method remain, such as optimally encoding
quantum information and achieving universal gates, but this
can be classed as a technological challenge.7
Entangled States
Ultimately, to experimentally realize various QIS
components and tasks, researchers will require robust
entangled states over many modes; hence our interest in
processing an initial unentangled signal state into a
tripartite, or three-mode, entangled state and then processing
it further to recover the original state. States can thus be
communicated, processed, and shared in networks with
resilience against component failures and protection against
malicious parties within the network. This protocol of state
sharing via entanglement, and recovery of the original state
by disentangling, was inspired by a discrete QIS protocol for
sharing quantum secrets and adapted to the continuous-variable
case.8 In short, our group has processed a coherent
state into a three-mode entangled state, and then recovered
the original state by disentanglement.
We use the 1064-nm beam as a source for the signal field,
the OPA pump beams, and the local oscillator. It is important
that the signal, pump, and local oscillator fields are derived
from the same source to ensure that all beams are
phase-matched.
As mentioned earlier, the initial signal state is prepared
in mode 1 (beam 1) from a portion of the Nd:YAG source field
at 1064 nm that has its sideband vacuum state displaced by
amplitude and phase modulation at 6.12 MHz and is mixed at a
1:1 beam splitter with mode 2′ of the two-mode squeezed beam
described above. The resultant state is entangled between
three modes (now labeled 1′′, 2′′, and 3′′), i.e., a
continuous-variable, tripartite entangled state. This state is
important because the initial state can be recovered by
processing any two of the three modes, thus allowing for
channel breakdown or component failures in the third mode.
Suppose that mode 3′′ is disrupted. The signal state can
still be reconstructed in mode 1 by combining mode 1′′ and
mode 2′′ of the tripartite entangled state at a 1:1 beam
splitter, which, together with the aforementioned process of
preparing the tripartite entangled state, forms a Mach-Zehnder
interferometer for modes 1′′ and 2′′ and recovers the original
signal field in output mode 1′′′. We characterize the quality
of the reconstruction by the overlap of the output state with
the original state, given by F = <input|R|input>, where
|input> is the coherent state and R is the density matrix
for the output state. If F > 0.5, state recovery is firmly
in the quantum domain,9 similar to the requirement
for quantum teleportation, and F = 1 corresponds to perfect
reconstruction. For the case described above, in which mode
3′′ is excluded, the experimental result is F = 0.93.
For the case in which mode 1′′ of the tripartite entangled
state is disrupted, mode 2′′ and mode 3′′ combine at a 2:1
beamsplitter followed by an electro-optic feed-forward loop.
The electronic gain is characterized by g + for the amplitude
gain and g - for the phase gain. The choice of a 2:1
beamsplitter leads to g - = 3-1/2. The amplitude gain is
determined by the detected photocurrent, which is used to
modulate the amplitude of the local oscillator. This
amplitude-modulated local oscillator is in turn mixed with
mode 2′′ to recover the signal in output mode 2′′′. In our
scheme the output state is not identical to the input state
but rather related by a fixed unitary squeezing transformation
that is independent of the input state; obtaining this
equivalent state up to a known unitary transformation is
sufficient for quantum network applications.
The fidelity for recovering the state for the case in which
mode 1′′ is ignored must be much less than for ignoring mode
3′′ because of added noise in the system. Reconstructing the
state after ignoring mode 3′′ only requires the inclusion of a
beamsplitter and creation of an effective Mach-Zehnder
interferometer, but ignoring mode 1′′ means that we have to
use photodetection, feed-forward, and a local oscillator, all
with their intrinsic noise contributions. Our fidelity is 0.63
for the reconstructed state in output mode 2′′, however, which
is well within the quantum regime. As mode 1′′ could be in the
hands of a malicious party, it is not only important to have
good fidelity for the reconstructed signal state in mode 2′′′.
The party holding output mode 1′′ must be denied the state,
which means the fidelity of output mode 1′′ must be low. We
obtain a fidelity for the state in output mode 1′′ of only
0.03 so the party holding output mode 1′′ is effectively
denied access to the state.
We are not concerned with interception but rather with
members of the network being unreliable or malicious (for
example, spies). The idea of sharing states is that if a party
is discovered to be unreliable or malicious after the states
are distributed, then the reliable and/or benevolent parties
can reconstruct the original state with some of the shares of
the entangled state. Issues of interception by outsiders are
interesting but await solution.
This experiment demonstrates that the state of one field
mode can be entangled with two other modes in a three-mode
network, and the original field state can be recovered by
suitable processing of any two of the three modes. Not only is
the state recovered, thereby protecting against component
failures in a three-mode network, but the third mode is also
denied access to the signal state, thus guaranteeing
protection from a malicious party discovered within the
network. This experiment demonstrates a quantum version of the
ubiquitous secret-sharing protocol in information networks.
Although this demonstration of state sharing is performed
in a three-mode network, the protocol can be readily scaled up
to an arbitrary number of modes such that any majority can
collaborate to reconstruct the state and the remaining parties
are denied any access to the state whatsoever.10
Moreover, the state reconstruction process requires at most
two OPAs, thus providing a cost-effective scaling for
arbitrarily large networks. As quantum communication and
quantum computation technology develop, security and
resilience of quantum networks will become increasingly
important, and this state-sharing protocol will provide the
protection required in such networks. oe
Barry Sanders is iCORE professor of quantum information
science at the University of Calgary and director of the
Institute for Quantum Information Science, Calgary, Canada. He
is also adjunct professor of quantum information science at
Macquarie University and a partner in Australia's Centre of
Excellence for Quantum Computer Technology, Sydney, Australia.
Contact: 403-210-8462; 403-289-3331 (fax); bsanders@qis.ucalgary.ca.
References
- M. Nielsen and I. Chuang, Quantum Computation and
Quantum Information, Cambridge University Press, Cambridge,
UK (2000).
- A. Lance et al., "Tripartite Quantum State Sharing,"
Phys. Rev. Lett. 92, p. 177903 (2004).
- E. Knill et al., Nature 409, p. 46 (2001).
- J. O'Brien et al., "Demonstration of an all-optical
quantum controlled-NOT gate," Nature 426, p. 264 (2003).
- D. Bouwmeester et al., "Experimental Quantum
Teleportation," Nature 390, p. 575 (1997).
- A. Furusawa et al., "Unconditional Quantum
Teleportation," Science 282, p. 706 (1998).
- D. Gottesman et al., "Encoding a Qubit in an
Oscillator," Phys. Rev. A 64[1] (2001).
- R. Cleve et al., Phys. Rev. Lett. 83, p. 648 (1999).
- T. Tyc and B. C. Sanders, Phys. Rev. A 65, p. 42310
(2002).
- T. Tyc et al., J. Phys. A: Math. Gen. 36, p. 7625
(2003).
By Joseph Treadway, Quantum Dot Corp.
In the most common usage of the term, quantum dots
are crystalline inorganic particles 1 to 10 nm in
diameter (a few hundred to a few thousand atoms) and
optimized for fluorescence efficiency. The defining
attribute of these fluorophores, now increasingly used
in biomedical applications, is the size-dependent nature
of their emission wavelength. In the archetypical
cadmium selenide (CdSe) quantum dots, for example,
particles varying in size from 2 to 7 nm display peak
emission from 450 to 650 nm, respectively, spanning
essentially the entire visible spectrum.
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Five-color multiplexing
with quantum dots and Hep-2 cells shows
mitochondria (orange), tubilin (green), Ki-67
(nuclear protein, magenta), nuclear antigen
(cyan), and actin (red).
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The key to this unique behavior is quantum
confinement. Light excitation of bulk semiconductors
such as CdSe creates charge-transfer excited states
characterized by delocalized electron-hole pairs. The
electron-hole pair, or exciton, has a preferred
separation distance called the Bohr radius (by analogy
with the Bohr model of the hydrogen atom). In quantum
dots, the size of the particle is smaller than the Bohr
radius for the material, and extra quantum confinement
energy is required to create the exciton in this
restricted space. The quantum confinement energy
increases as the particle size decreases and thus the
emission and absorption energies characteristic of bulk
CdSe become higher (i.e., bluer) as the bulk shrinks
down to the nanometer scale. This is the reason for the
upper limit of 650-nm emission in CdSe quantum dots; 7
nm is close to the Bohr radius for CdSe, and particles
larger than that display essentially bulk and not
quantum-dot properties.
Quantum dots such as CdSe require overcoating with a
second, larger-band-gap, crystalline material such as
zinc sulfide to form a protective shell. By isolating
the core, the shell prevents dissolution, oxidation, and
other routes to decomposition. The shell also plugs any
gaps in the surface of the core lattice and removes
dangling bonds and reorganized bonds that quench the
emission from the core.
The composite core-shell structures are often
hundreds to thousands of times brighter than naked cores
and are significantly more robust. The result is a set
of fluorophores with chemically identical behavior, but
with narrow tunable emission spanning the entire
spectrum, strong absorption at every wavelength shorter
than the emission wavelength, and a negligible tendency
to bleach even after extended photolysis. Highly
engineered quantum dot structures are often thousands of
times brighter in many applications than comparably
emitting organic dyes.
The industrial-scale manufacture of quantum dots has
proven challenging. Quantum dots emitting at 480 and 500
nm, for example, differ by only about 50 cadmium and
selenium atoms, so reproducible synthetic methods
require control down to the level of tens of atoms as
the crystals are grown. Originally, syntheses were based
on pyrolysis of organometallic precursors at very high
temperatures, rendering them incompatible with volume
manufacturing.
In the past few years, the organometallic method has
been replaced with salt-based approaches that further
make use of reaction additives designed to precisely
control crystal-growth kinetics.1,2 Our
researchers have extended these methods to include other
forms of growth control such as redox activation of
precursors to create a suite of biological labeling
products based on no less than seven unique quantum dot
colors that can be safely and reproducibly synthesized
to emit to within a few nanometers of nominal.
In the past it was difficult to obtain images in even
two or three colors because traditional dyes are too
broadly emitting to be used effectively together without
compensating for spectral overlap. In addition, each
traditional dye usually requires a unique excitation
wavelength, meaning a five-color experiment could
require five co-localized laser sources. Quantum dots,
however, can easily be excited by a single source at any
wavelength up to the blue edge of the emission and emit
in narrow symmetric bands without excessive overlap.
Challenges remain. We need to develop materials other
than CdSe and cadmium telluride to cover the UV and IR
spectral regions. While dramatically more narrowly
emitting than competitive dyes, commercial quantum dots
remain about twice as broadly emitting as necessary due
to remaining inhomogeneities in particle size
distribution within each sample. Properties unique to
non-spherical dots remain largely unexploited. Finally,
mass production methods for consumer applications remain
as yet undeveloped. As far as the biomedical researcher
is concerned, however, quantum dots have arrived.
Joseph Treadway is principal scientist of Quantum
Dot Corp., Hayward, CA. Contact: 510-887-8775 ext. 4123;
510-783-9729 (fax); JATreadway@qdots.com.
References
- Z. Peng and X. Peng, J. Am. Chem. Soc. 123, p. 168
(2001).
- B. Yen, et al., Adv. Mater. 15, p. 1858 (2003).
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