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arXiv:1604.00325

Elementary quantum gates in different bases

Sergey A. Podoshvedov

Published Apr 1, 2016• 0 citations
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Abstract

We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their displaced properties. Superposed coherent states deterministically displace target state on equal modulo but opposite on sign values. Registration of the single photon in auxiliary mode (probabilistic operation) results in constructive interference and gives birth to entangled hybrid state corresponding to outcome of elementary quantum gates. The method requires minimal number of resource and works in realistic scenario

DOI
10.48550/arxiv.1604.00325

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We introduce transformation matrix connecting sets of the displaced states with different displacement amplitudes. Arbitrary pure one-mode state can be represented in new basis of the displaced number (Fock) states ( representation) by multiplying the transposed transformation matrix on a column vector of initial state. Analytical expressions of the representation of superposition of vacuum and single photon and two-mode squeezed vacuum (TMSV) are obtained. On the basis of the developed mathematical formalism, we consider the mechanism of interaction between qubits which is based on their displaced properties. Superposed coherent states deterministically displace target state on equal modulo but opposite on sign values. Registration of the single photon in auxiliary mode (probabilistic operation) results in constructive interference and gives birth to entangled hybrid state corresponding to outcome of elementary quantum gates. The method requires minimal number of resource and works in realistic scenario
We develop a new method of representation of quantum states in terms of the displaced number states. We call it representation, where is an amplitude of the base displaced states. In particular, representation was obtained for set of the displaced number states with different amplitude of the displacement, two-mode squeezed vacuum (TMSV) and superposition of vacuum and single photon. The treatment is employed for building of elementary linear operators, in particular, two-qubit control-sign gate, Hadamard matrix between two two-dimensional Hilbert spaces. Main idea underlying the method is extraction of the displaced number state from initial state. Coherent qubit with large amplitude taken simultaneously two modes shifts two-mode target qubit on phase plane and constructive interference appears after measurement of single photon. The results are exact, any approximations are not exploited. We show implementation of the gates with almost unity fidelity of the output states is possible in realistic scenario.
We consider novel implementation of quantum teleportation protocol of unknown qubit by superposition of displacement operators with equal modulo but opposite in sign amplitudes. Entangled hybrid state with coherent components of small amplitude is used as quantum channel. Coherent components of the quantum channel displace the teleported qubit at the same absolute but opposite in sign values, so that any information about amount by which the qubit is displaced is lost despite the fact that one of these events has definitely occurred. Alice unambiguously distinguishes her measurement outcomes and Bob obtains at his disposal two states of a single photon in superposition state of two modes. The superposition of the displacement operators is nonunitary, so success probability of the protocol depends on the amplitude parameters of the teleported qubit. Remarkably, the success probability of the teleportation can be made arbitrary high for some amplitude values of the teleported state by means of initial amplitude modulation of unknown qubit and bits of classical information for Bob to identify the states. The scheme is realized by linear optics methods and works without photon number resolving measurement.
We consider novel implementation of quantum teleportation protocol of unknown qubit with entangled hybrid state. Coherent components of the entangled hybrid state displace the teleported qubit at the same absolute but opposite in sign values, so that any information about amount by which the qubit is displaced is lost despite the fact that one of these events has definitely occurred. Alice unambiguously distinguishes her measurement outcomes and Bob obtains at his disposal states of a single photon in superposition of two modes with controllable amplitude distortions. We consider the implementation of this protocol with a third participant who off-line prepares amplitude modulated unknown qubits and hand them to Alice for teleportation. In this interpretation, the success probability of the teleportation depends on the absolute value of the amplitude of the unknown qubit. In particular, highly unbalanced unknown qubits can be teleported with success probability close to one. We also consider some ways of demodulating amplitude modulated states by the receiver in order to increase the success probability of the teleportation of unknown qubits.
We develop theory of realization of hybrid entanglement between discrete-variable (single photon) and continuous-variable states (coherent states). A highly transmissive beam splitter (HTBS) is used for interaction of the ingredients. Mechanism for generating the hybrid entangled states is based on the simultaneous displacement of the discrete-variable state on equal in absolute value but opposite in sign displacement amplitudes by coherent components, in such a way that all the information about the displacement amplitudes is lost with subsequent registration of photons in the auxiliary mode. Conditions for the balanced hybrid entangled states generation are found. Our approach follows along lines experimentally demonstrated in Nat. Photonics 8, 570-574 (2014).
We propose a new approach to generate entangled states, both hybrid and consisting exclusively of continuous variable (CV) states. A single-mode squeezed vacuum is mixed with a delocalized single photon on an arbitrary beam splitter with subsequent registration of measurement outcomes in auxiliary mode. The entangled hybrid states consisting of CV and discrete variable (DV) states are generated whenever any event is measured in auxiliary mode. Negativity is used as a measure of entanglement. Under certain initial conditions, the conditional state becomes as entangled as possible. New types of CV states, either even or odd depending on the parity of the Fock states forming the superpositions, are introduced. If the conditional entangled hybrid state is mixed with a single-mode squeezed vacuum with the subsequent registration of measurement results in DV mode, then the new state becomes CV entangled, i.e., its orthogonal components are already CV states. The entanglement synthesis can be expanded to implement a high-complexity quantum network.
We propose an approach that enables to use it for construction of rotations of coherent states, in particular, it gives a possibility to construct Hadamard gate for the coherent states. Our approach is based on representation of arbitrary one-mode pure state in free-travelling fields, in particular superposition of coherent states (SCSs), in terms of displaced number states with arbitrary amplitude of displacement. Studied optical scheme is based on alternation of photon additions and displacement operators (in general case, photon additions and displacements are required) with seed coherent state to generate both even and odd displaced squeezed SCSs (DSSCSs) regardless of number of used photon additions. It is shown the studied optical scheme is sensitive to seed coherent state provided that other parameters are invariable. Output states approximate with high fidelity either even squeezed SCS or odd SCS shifted relative each other by some value. It enables to construct local rotation operator for coherent states, in particular, Hadamard gate being mainframe element for quantum computation with coherent states. The effects deteriorating quality of output states are considered.