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arXiv:1309.4254 · Physical Review A

Joining and splitting the quantum states of photons

Elsa Passaro, Chiara Vitelli, Nicolò Spagnolo, Fabio Sciarrino, Enrico Santamato, Lorenzo Marrucci

Published Dec 16, 2013• 10 citations
arXivHTMLPDF

Abstract

A photonic process called quantum-state joining has been recently experimentally demonstrated [Vitelli et al., Nat. Photonics 7, 521 (2013)] that corresponds to the transfer of the internal two-dimensional quantum states of two input photons, i.e., two photonic qubits, into the four-dimensional quantum state of a single photon, i.e., a photonic ququart. A scheme for the inverse process, namely, quantum state splitting, has also been proposed. Both processes can be iterated in a cascaded layout, to obtain the joining and/or splitting of more than two qubits, thus leading to a general scheme for varying the number of photons in the system while preserving its total quantum state, or quantum information content. Here, we revisit these processes from a theoretical point of view. After casting the theory of the joining and splitting processes in the more general occupation number notation, we introduce some modified schemes that are in principle unitary [not considering the implementation of the controlled-not (cnot) gates] and do not require projection and feed-forward steps. This can be particularly important in the quantum state splitting case, to obtain a scheme that does not rely on postselection. Moreover, we formally prove that the quantum joining of two photon states with linear optics requires the use of at least one ancilla photon. This is somewhat unexpected, given that the demonstrated joining scheme involves the sequential application of two cnot quantum gates, for which a linear optical scheme with just two photons and postselection is known to exist. Finally we explore the relationship between the joining scheme and the generation of clusters of multiparticle entangled states involving more than one qubit per particle.

DOI
10.1103/physreva.88.062321

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A photonic process called quantum-state joining has been recently experimentally demonstrated [Vitelli et al., Nat. Photonics 7, 521 (2013)] that corresponds to the transfer of the internal two-dimensional quantum states of two input photons, i.e., two photonic qubits, into the four-dimensional quantum state of a single photon, i.e., a photonic ququart. A scheme for the inverse process, namely, quantum state splitting, has also been proposed. Both processes can be iterated in a cascaded layout, to obtain the joining and/or splitting of more than two qubits, thus leading to a general scheme for varying the number of photons in the system while preserving its total quantum state, or quantum information content. Here, we revisit these processes from a theoretical point of view. After casting the theory of the joining and splitting processes in the more general occupation number notation, we introduce some modified schemes that are in principle unitary [not considering the implementation of the controlled-not (cnot) gates] and do not require projection and feed-forward steps. This can be particularly important in the quantum state splitting case, to obtain a scheme that does not rely on postselection. Moreover, we formally prove that the quantum joining of two photon states with linear optics requires the use of at least one ancilla photon. This is somewhat unexpected, given that the demonstrated joining scheme involves the sequential application of two cnot quantum gates, for which a linear optical scheme with just two photons and postselection is known to exist. Finally we explore the relationship between the joining scheme and the generation of clusters of multiparticle entangled states involving more than one qubit per particle.
Quantum state joining has been recently experimentally demonstrated [C. Vitelli et al., Nat. Photon. 7, 521 (2013)] which can transfer two input photonic qubits into a photonic ququart. Here, we revisit these processes from a hybrid point of view. By exploring the giant optical circular birefringence induced by quantum-dot spins in one-sided optical microcavities, we introduce some deterministic joining schemes including two quantum-dot spin joining, hybrid photon and quantum-dot spin joining, and two-photon joining. The input quantum information is represented by one photon with polarization and spatial mode degrees of freedom (DOFs). These schemes are also adapted to the inverse processes called quantum state splitting because all the joining procedures are unitary and do not require projection and feed-forward steps. The fused photon is convenient for realizing elementary logic gates such as the controlled-not (cnot) gate, swap gate, and Toffoli gate. These hybrid fusion and splitting schemes provide flexible synthesis of the quantum-dot spin and photon in quantum applications. The transmission superiority of photons and storage superiority of the quantum-dot spin may be combined for quantum network communication or quantum computations.
Full detail of a proposed experiment required for implementing and verifying a theoretical scheme for four-partite splitting and open-destination teleportation of an arbitrary two-qubit photonic state is discussed. In this proposed experiment the quantum channel is provided by a pair of decomposable generalized (G) Bell states, which offer the experimental advantage that they can be very easily generated in photonic experiments. Our experiment is based on generating a two-qubit photonic state by ultrafast spontaneous parametric downconversion in nonlinear crystal and relies on Bell-state measurements, which in this experiment are performed by an optical Bell-state analyzer that can unambiguously determine all four Bell states. In this proposed experiment unitary transformation required at the destination station is implemented using a quantum control NOT gate. We finally show that in our four-partite optical system the two-qubit photonic state originally prepared at a sending station can be experimentally split and subsequently regenerated at any one of the three distinct receiving stations.
Abstract We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon–photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups that are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability for each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that the maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to an m -tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of (1/4) m −1 .
Single photons are the smallest excitation states of the elctromagnetic field. They can be experimentally created but there is no apparatus that performs this task with 100% efficiency. Many techniques and theorems of optical quantum information theory use single photons. In order to utilize them for information processing, it is reasonable to know if they had been successfully produced. In other words: One needs heralded single photons. How to achieve this, is the main result of this diploma thesis. The task has been to improve the efficiency of already existing single photon sources. An apparatus had to be found which postprocesses the output of such sources in order to either herald the photons or at least to improve the efficiency. This apparatus is explicitly allowed to use other optical sources, for example coherent states (laser). The situation has been formally represented by the assumption that single photon sources produce a mixture of vacuum and the single photon state. It is known that experimental set-ups also create higher photon contributions. These contributions are considered to be small in corresponding publications. For some cases they even seem to vanish at all. One can distinguish two regimes concerning the techniques which have been used in this work: linear and nonlinear optics. Linear optical tools, for example the beam splitter, do not absorb photons in the ideal case, but it is not easy to implement useful transformations by their means. Absorption and weak coupling are drawbacks which arise mainly in nonlinear optical media. This work shows that the interactions two-photon absorption and cross phase modulation can be used in schemes, which enable the heralding of single photons. Already there are some linear optical quantum gates which manipulate optical states by means of single photons. Photons do not interact with each other directly, in contrast to electrons for example. Therefore quantum gates which act on more than one mode are interesting. The effective interaction of these gates is not quitecomparable to unitary dynamics. Moreover, these gates do not function in each case. Measurements required in auxiliary modes yield different results for each single case and in order to obtain the desired effect the cases have to be selected according to the measurement results. It has been investigated, how such gates perform if they have statistical mixtures as a resource instead of perfect single-photon states. Yet, no improvement of the single-photon contribution has been accomplished by means of these apparatuses. Hereby it has not been disproved that it is possible to achieve this goal using other apparatuses and techniques of linear optics. Yet, it has been shown that an enhancement is possible but then a non-vanishing two-photon contribution is appended. One of the most basic quantum gates is the conditional sign flip. The optical 8 interaction cross phase modulation acts in quite the same way considering only single photons. This effect enables the heralding of single photons in the ideal case. Within this work it has been shown that a Mach-Zehnder interferometer, a cross phase modulating medium and a photo-detector can be employed for this purpose. One could use any source of light as input to the interferometer but a strong coherent state (laser) may be preferred for the purpose of enhancing the interaction. The absorption of two-photon states is another non-linear effect. Astonishingly this effect serves to generate a heralded single-photon state given two imperfect single-photon sources. First the photon adding tool is employed which consists of a beam splitter and a photo-detector. By means of that and the statistical mixtures of two single-photon sources one creates a mixture which contains vacuum, the single-photon state and additionally the two-photon Fock state. This mixture propagates through a Mach-Zehnder interferometer. The interferometer is adjusted such that it is transparent for incoming light. In addition to the interferometer alone a two-photon absorbing medium is placed in one of its arms. For this reason there is a chance that the two-photon contribution entering the interferometer is absorbed within this medium. We cannot exclude this unfortunate case. A detector is placed in one of the two output modes. The interferometer is adjusted such that the single photon state does not leave in the mode in which the detector is placed, but always in the other mode. The detector can only click if the two-photon state propagates through the interferometer and if it is not destroyed in the absorbing medium. If the detector then receives one photon it is obvious that another one must leave in the other mode. The whole phenomenon is very similar to the concept of null measurement. Although not a main result, it has been shown here that the so-called Hong-Ou-Mandel effect can be generalized. It has been proven that the beam splitter transforms Fock states with equal Fock number into superpositions of Fock numbers which are even in each mode. Another effect on special odd Fock states is conjectured.
Photons are natural carriers of quantum information due to their ease of distribution and long lifetime. This thesis concerns various related aspects of quantum information processing with single photons. Firstly, we demonstrate N-photon entanglement generation through a generalised N X N symmetric beam splitter known as the Bell multiport. A wide variety of 4-photon entangled states as well as the N-photon W-state can be generated with an unexpected non-monotonic decreasing probability of success with N. We also show how the same setup can be used to generate multiatom entanglement. A further study of multiports also leads us to a multiparticle generalisation of the Hong-Ou-Mandel dip which holds for all Bell multiports of even number of input ports. Next, we demonstrate a generalised linear optics based photon filter that has a constant success probability regardless of the number of photons involved. This filter has the highest reported success probability and is interferometrically robust. Finally, we demonstrate how repeat-until-success quantum computing can be performed with two distant nodes with unit success probability using only linear optics resource. We further show that using non-identical photon sources, robustness can still be achieved, an illustration of the nature and advantages of measurement-based quantum computation. A direct application to the same setup leads naturally to arbitrary multiphoton state generation on demand. Finally, we demonstrate how polarisation entanglement of photons can be detected from the emission of two atoms in a Young's double-slit type experiment without linear optics, resulting in both atoms being also maximally entangled.