This review covers the latest developments in continuous-variable quantum-state tomography of optical fields and photons, placing a special emphasis on its practical aspects and applications in quantum-information technology. Optical homodyne tomography is reviewed as a method of reconstructing the state of light in a given optical mode. A range of relevant practical topics is discussed, such as state-reconstruction algorithms (with emphasis on the maximum-likelihood technique), the technology of time-domain homodyne detection, mode-matching issues, and engineering of complex quantum states of light. The paper also surveys quantum-state tomography for the transverse spatial state (spatial mode) of the field in the special case of fields containing precisely one photon.

https://arxiv.org/pdf/quant-ph/0511044.pdf

Continuous-variable optical quantum-state tomography

The superposition principle is a fundamental tenet of quantum mechanics, allowing a quantum system to be 'in two places at the same time'. The preparation and use of superposed states forms the basis of quantum computation and simulation. Max Hofheinz and colleagues now demonstrate the technically challenging preparation and measurement of arbitrary quantum states in an electromagnetic resonator. States with different numbers of photons are superposed in a completely controlled and deterministic manner. The superposition principle is a fundamental tenet of quantum mechanics, allowing a quantum system to be 'in two places at the same time'. Here, the preparation and measurement of arbitrary quantum states in an electromagnetic resonator is demonstrated; states with different numbers of photons are superposed in a completely controlled and deterministic manner. The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be ‘in two places at the same time’, because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation1. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions2, microwave resonators3,4,5 or optical cavities6) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm7 that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator8. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator’s full density matrix.

Synthesizing arbitrary quantum states in a superconducting resonator

Recently, there has been much interest in a new kind of ``unspeakable'' quantum information that stands to regular quantum information in the same way that a direction in space or a moment in time stands to a classical bit string: the former can only be encoded using particular degrees of freedom while the latter are indifferent to the physical nature of the information carriers. The problem of correlating distant reference frames, of which aligning Cartesian axes and synchronizing clocks are important instances, is an example of a task that requires the exchange of unspeakable information and for which it is interesting to determine the fundamental quantum limit of efficiency. There have also been many investigations into the information theory that is appropriate for parties that lack reference frames or that lack correlation between their reference frames, restrictions that result in global and local superselection rules. In the presence of these, quantum unspeakable information becomes a new kind of resource that can be manipulated, depleted, quantified, etc. Methods have also been developed to contend with these restrictions using relational encodings, particularly in the context of computation, cryptography, communication, and the manipulation of entanglement. This paper reviews the role of reference frames and superselection rules in the theory of quantum-information processing.

https://arxiv.org/pdf/quant-ph/0610030.pdf

Reference frames, superselection rules, and quantum information

Single-photon-added coherent states are the result of the most elementary amplification process of classical light fields by a single quantum of excitation. Being intermediate between a single-photon Fock state (fully quantum-mechanical) and a coherent (classical) one, these states offer the opportunity to closely follow the smooth transition between the particle-like and the wavelike behavior of light. We report the experimental generation of single-photon-added coherent states and their complete characterization by quantum tomography. Besides visualizing the evolution of the quantum-to-classical transition, these states allow one to witness the gradual change from the spontaneous to the stimulated regimes of light emission.

http://science.sciencemag.org/content/sci/306/5696/660.full.pdf

Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light

Matter and energy cannot be teleported (that is, transferred from one place to another without passing through intermediate locations). However, teleportation of quantum states (the ultimate structure of objects) is possible1: only the structure is teleported—the matter stays at the source side and must be already present at the final location. Several table-top experiments have used qubits2,3,4,5,6,7 (two-dimensional quantum systems) or continuous variables8,9,10 to demonstrate the principle over short distances. Here we report a long-distance experimental demonstration of probabilistic quantum teleportation. Qubits carried by photons of 1.3 µm wavelength are teleported onto photons of 1.55 µm wavelength from one laboratory to another, separated by 55 m but connected by 2 km of standard telecommunications fibre. The first (and, with foreseeable technologies, the only) application of quantum teleportation is in quantum communication, where it could help to extend quantum cryptography to larger distances11,12,13.

Long-distance teleportation of qubits at telecommunication wavelengths

An electromagnetic field quadrature measurement, performed on one of the modes of the nonlocal single-photon state alpha|1,0>-beta|0,1>, collapses it into a superposition of the single-photon and vacuum states in the other mode. We use this effect to implement remote preparation of arbitrary single-mode photonic qubits conditioned on observation of a preselected quadrature value. The preparation efficiency of the resulting qubit can be higher than that of the initial single photon.

http://arxiv.org/pdf/quant-ph/0308127.pdf

Remote preparation of a single-mode photonic qubit by measuring field quadrature noise.

We employ the quantum state of a single photon entangled with the vacuum (|GROUPA|GROUPB − |GROUPA|GROUPB), generated by a photon incident upon a symmetric beam splitter, to teleport single-mode quantum states of light by means of the Bennett protocol. The teleportation of coherent states results in the truncation of their Fock expansion to the first two terms. We analyze the teleported ensembles by means of homodyne tomography and obtain fidelities of up to 99 per cent for low-source state amplitudes. This work is an experimental realization of the quantum scissors device proposed by Pegg, Phillips and Barnett (Phys. Rev. Lett., 81 (1998) 1604).

https://iopscience.iop.org/article/10.1209/epl/i2003-00504-y/pdf

Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon

Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state $|1〉$ with coherent states on a high-reflection beam splitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals nonclassical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. To our knowledge, this is the first time complete tomographic reconstruction has been performed on a highly nonclassical optical state.

Synthesis and tomographic characterization of the displaced Fock state of light

A single photon, delocalized over two optical modes, is characterized by means of quantum homodyne tomography. The reconstructed four-dimensional density matrix extends over the entire Hilbert space and thus reveals, for the first time, complete information about the dual-rail optical quantum bit as a state of the electromagnetic field. The experimental data violate the Bell inequality albeit with a loophole similar to the detection loophole in photon counting experiments.

https://arxiv.org/pdf/quant-ph/0312135.pdf

Homodyne tomography characterization and nonlocality of a dual-mode optical qubit.

Heralded single photons are prepared at a rate of 100 kHz via conditional measurements on polarizationnondegenerate biphotons produced in a periodically poled potassium-titanyl phosphate crystal. The singlephoton Fock state is characterized using high-frequency pulsed optical homodyne tomography with a fidelity of (57.6±0.1)%. The state preparation and detection rates allowed us to perform on-the-fly alignment of the apparatus based on real-time analysis of the quadrature measurement statistics.

S. A. Babichev

Papers

Remote preparation of a single-mode photonic qubit by measuring field quadrature noise.

Quantum scissors: Teleportation of single-mode optical states by means of a nonlocal single photon

Synthesis and tomographic characterization of the displaced Fock state of light

Homodyne tomography characterization and nonlocality of a dual-mode optical qubit.

Instant single-photon Fock state tomography