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.
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′′′.
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 efficientour 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.7Entangled 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
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.
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).
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.
B. Yen, et al., Adv. Mater. 15, p. 1858 (2003).
Joseph Treadway is principal scientist of Quantum Dot Corp., Hayward, CA.
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.