9

Taking an $n$-mode simple harmonic oscillator (SHO) in a (Fock) space $\mathcal F = \bigotimes_k\mathcal H_k$, where $\mathcal H_k$ is the Hilbert space of a SHO on mode $k$. This gives the usual annihilation operator $a_k$, which act on a number state as $a_k\left|n\right> = \sqrt n\left|n-1\right>$ for $n\geq 1$ and $a_k\left|0\right> = 0$ and ...


7

For any quantum state, we have a unique density matrix $\rho$. For any $\rho$, we can do the Wigner transformation to get a unique Wigner function $P(x,p)$. For any Wigner function $P(x,p)$, we can do the Weyl transformation to get back the unique $\rho$. If the construction of the Wigner function from $\rho$ was not unique, then it would not be possible to ...


6

QML researcher at Xanadu here. Our X-series chip produce entangled states by squeezing light and then combining it at beam splitters: those 'cables' are waveguides in a chip, which when they are close enough they allow tunnelling between them and effectively couple those two modes. Note that the squeezing is necessary in this case (for example, had we used ...


6

You are right, photonic systems are described by an infinite (separable) Hilbert space---the bosonic Fock space---and their formalism makes extensive use of infinite values, both countable and uncountable. The quantum computing paradigm based on this Hilbert space is called continuous-variable (CV) quantum computing, and a lot of different protocols and ...


6

To start off, I would really suggest you to read this review on "Quantum information with continuous variables(cv)". It covers most of your questions with cv architecture. Since it is a very big review, I will try to address your questions with what I can remember from reading that paper and glancing over it again now. For discrete variables(dv), as you ...


4

Both models have their potential advantages and disadvantages. The CV model doesn't require energy intensive cooling systems. CV will also work better for continuous-valued problems. Nevertheless, since the model uses photons it brings various challenges to the table as well. Since both models (especially CV) aren't developed, your question may not be the ...


4

I assume the paper you are reading is referring to information reconciliation. Information reconciliation is a vital part of post-processing in QKD, to limit (or erase in the best-case scenario) the amount of errors/differences between the key of Alice and Bob. In that sense, it is a form of (classical) error-correction, and, broadly speaking, it works like ...


3

You mean something like $$W_{G}(\mathbf{r}) =\frac{2^{n}}{\pi^{n} \sqrt{\operatorname{Det} \sigma}} \mathrm{e}^{-(\mathbf{r}-\overline{\mathbf{r}})^{\top} \boldsymbol{\sigma}^{-1}(\mathbf{r}-\overline{\mathbf{r}})},$$ where $W_{G}(\mathbf{r})$ is the Wigner function corresponding to a Gaussian state, $\mathbf{\sigma}$ its covariance matrix, and $\overline{r}$...


3

I would say it is not exactly the same. A qudit of size $d$ has a finite number of states. A Fock state can have discretely infinite in principle. The goal of having multiple qudits, let's say $N$, is to make a larger space of size $d^N$. Rarely you consider product of Fock spaces (but it can be done). Usually, you define a Fock space that includes all ...


3

Background Often, in quantum optics, the Heisenberg picture is used, where instead of considering equations of motion of states, equations of motions of operators are looked at instead. When considering creation/annihilation operators, this is often considerably easier as the matrices that determine the evolution (assuming it can be written in terms of ...


2

direct definition is a quantum register that stores a real number defined by an observable with a spectrum consisting of ℝ Yes, qubits can be used to discretize a continuous quantum system. How? Lets say $\Phi $ is an operator on the continuous system which we want to simulate using qubits. We need a discrete register ($D$) to store the qubit values. ...


2

From the definition of the covariance matrix, $${\sigma}_{ij}=\left\langle \frac{x_i x_j+ x_j x_i}{2}\right\rangle -\langle x_i\rangle\langle x_j\rangle,$$ where we define $$\boldsymbol{x}=(q_1,p_1,\cdots, q_N, p_N)^\top.$$ Given that $2i\Omega_{jk}=[x_j,x_k],$ we learn that $${\sigma}_{ij}=\left\langle x_i x_j\right\rangle -\langle x_i\rangle\langle x_j\...


2

The idea of squeezing arises when discussing the state of a quantum harmonic oscillator (e.g. a bosonic system). Such systems differ from simpler qudit systems in that, even when only a single mode is being considered, the system is infinitely dimensional. A common way to describe these systems is via pairs of non-commuting observables, often the "position" ...


2

Here's a schematic description of quantum teleportation for general systems: Alice and Bob share a maximally entangled state. To teleport $|\psi\rangle$ to Bob, Alice performs a joint measurement of $|\psi\rangle$ and her local copy of the maximally entangled state. This measurement is entangling, in the sense that the output state after the measurement can ...


1

Recall that $a=(q+ip)/\sqrt{2}$ in some dimensionless units (Weedbrook might change the units because I think they like $\hbar=2$; I'm using $[q,p]=i$ and $[a,a^\dagger]=1$). We can thus find the equality $$a^\dagger a=(q-ip)(q+ip)/2=[q^2+p^2+i(qp-pq)]/2=\frac{q^2+p^2-1}{2}.$$ This relates the total number of photons to the trace of your matrix $\sigma$. ...


1

@gIS's comment effectively answers the question, but to provide a bit more detail Aaronson at shtetl-optimized has a nice blog post on the Gaussian Boson Sampling approach of USTC, contrasting it with Fock state Boson Sampling, wherein the presence/absence of photons correspond to something closer to a conventional digital ($\vert 0\rangle$/$\vert 1\rangle$) ...


1

So you want to calculate $\left<\beta |D\left(\alpha\right)|\beta\right>$ where $\left|\beta\right>$ is a coherent state and $D\left(\alpha\right)$ is the displacement operator. The easiest way of doing this is to take $\left<\beta |D\left(\alpha\right)|\beta\right> = \left<0|D^\dagger\left(\beta\right) D\left(\alpha\right)D\left(\beta\...


1

The link you gave says: The CV model is a natural fit for simulating bosonic systems (electromagnetic fields, harmonic oscillators, phonons, Bose-Einstein condensates, or optomechanical resonators) and for settings where continuous quantum operators – such as position & momentum – are present. Which means you can have many different different matrix ...


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