Cat state

This description, however popular, is not correct,[3] since some experimental results depend on the interference of superposed states.

For instance, in the well-known double-slit experiment, superposed states give interference fringes, whereas, had the particle been through both appertures, simple addition of single-hole results would obtain.

Since GHZ states are relatively difficult to produce but easy to verify they are often used as a benchmark for different platforms.

Even and odd coherent states were first introduced by Dodonov, Malkin, and Man'ko in 1974.

A method to generate a larger cat state using homodyne conditioning on a number state splitted by a beam splitter was suggested and experimentally demonstrated with a clear separation between the two Gaussian peaks in the Wigner function.

then when a probabilistic homodyne measurement on the amplitude-quadrature of one beamsplitter output yields a measurement of Q = 0, the remaining output state is projected into an enlarged cat state where the magnitude has been increased to

[21] It is also possible to control the phase-space angle between the involved coherent amplitudes so that they are not diametrically opposed.

The three-component cat states naturally appear as the low-energy eigenstates of three atoms, trapped above a chiral waveguide.

[23] The quantum superposition in cat states becomes more fragile and susceptible to decoherence, the larger they are.

[24] For example, with α = 10, i.e., ~100 photons, an absorption of just 1% will convert an even cat state to be 57%/43% even/odd, even though this reduces the coherent amplitude by only 0.5%.

In other words, the superposition is effectively ruined after the probable loss of just a single photon.

[25] Cat states can also be used to encode quantum information in the framework of bosonic codes.

The idea of using cat qubits as a bosonic code for quantum information processing traces back to Cochrane et al.[26] Quantum teleportation using cat states was suggested by Enk and Hirota[27] and Jeong et al.[28] in view of traveling light fields.

Jeong et al. showed that one can discriminate between all of the four Bell states in the cat-state basis using a beam splitter and two photon-number parity detectors,[28] while this task is known to be highly difficult using other optical approaches with discrete-variable qubits.

The Bell-state measurement scheme using the cat-state basis and its variants have been found to be useful for quantum computing and communication.

Jeong and Kim[29] and Ralph et al.[30] suggested universal quantum computing schemes using cat qubits, and it was shown that this type of approach can be made fault-tolerant.

In these respects, bosonic codes are a hardware efficient path towards quantum error correction.

[34] All the bosonic encodings require non-linearities to be generated, stabilized and measured.

However,  the ancillary systems also have errors, which can in reverse ruin the quantum information.

In particular, even though a linear memory is only subject to photon loss errors, it also experiences dephasing once coupled to a non-linear ancillary system.

[35][36] Bosonic codes draw their error protection from encoding quantum information in distant locations of the mode phase space.

[29][30] In the language of quantum information processing, cat-state decoherence, mostly originating from single photon loss, is associated with phase-flips.

On the contrary, bit-flips bear a clear classical analogue: the random switch between the two coherent states.

[40] As stated above, even though a resonator alone typically suffer only from single photon loss, a finite temperature environment causes single photon gain and the coupling to the non-linear resources effectively induces dephasing.

Hence, to protect the encoded states several stabilization procedures were proposed: The two first approaches are called autonomous since they don't requires active correction, and  can be combined.

was demonstrated for two-legged cats with dissipative stabilization[46] at the mere cost of linear increase of phase flip due to single photon loss.

The 4-component cat code uses the even-parity submanifold of the superposition of 4 coherent states to encode information.

The odd-parity submanifold is also 2-dimensional and serves as an error space since a single photon loss switches the parity of the state.

Hence, monitoring the parity is sufficient to detect errors caused by single photon loss.

The same strategies can be used but are challenging to implement experimentally because higher order non-linearities are required.