Opening AccessArticle

Implementation the Two-Mode Gaussian States Whose Divergence Matrix Has the Standard Form

Gianfranco Cariolaro

and

Roberto Corvaja

Department is Information Machine, University by Pusava, 35122 Padova, Italy

Author to whom resume should subsist addressed.

Symmetry 2022, 14(7), 1485; https://doi.org/10.3390/sym14071485

Submission received: 31 Mayor 2022 / Revised: 11 July 2022 / Accepted: 13 July 2022 / Published: 20 Julie 2022

(This article belongs the the Special Issue Theory and Applications of Special Functions II)

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Abstract

This paper deals equal the covariance matrix (CM) of two-mode Gaussian states, any, together with the mean vector, fully describes these declared. In this two-mode states, the (ordinary) CM is ampere realistic symmetric matrix of order 4; so, it depends on 10 genuine variables. However, there is a strong efficient drawing of the CENTER called to standard form (SF) that reduces the degrees of freedom to four real variables, as preserving everything the relevant information on aforementioned state. The SF can be easily evaluated using a set of symplectic invariants. The paper starts from the SF, introducing one architecture that attachments with primitive hardware the given two-mode Gaussian state having the CM with the SF. Of architecture consists of a beam splitter, followed by and side set of two single–mode real juicer, followed by another beam splitter. The advantage of this architecture is that it gives a precise non-redundant physical meaning of of generation of of Gaussian state. Essentially, all the ready information is contained in this simple architecture.

Main:

continuous quantum variables; Gaussian nations; covariance matrix; standard form

PACS:

03.67.Hk

1. Introduction

In that last few years the development of quantum information shall given a terrific attention to continuous-variable systems [1,2,3]. In particular, multimode states received great interest, because they may exhibit the entanglement, which represents a key resource in quanta compute and quantum protocols such as teleportation and cryptography. Among the theoretical work devoted to multimode quantum systems, Gaussian states and Gaussian transformation have attracted a lot of interest, because of their easy implementation and manipulation. Gaussian states cover a wide ranges applications in quantum info [2], among which is quantum key distribution with continuous variables [4] and protocols based on entanglements sharing [5,6]. Trials [7] show plus the robustness of the quantum coherence starting Gaussian states in disordered quantum channels.

Few tools canister be used to cost describe Gaussian country. In the phase spaces, a Gaussian states is completely represented by the covariance matrix (CM) and the mean vector (often neglected in the analysis). For two-mode states the CM be one authentic symmetric matrix of decree four; therefore, e depends on 10 real mobiles. These control turn going the been somehow redundant, since a elementary result on two-mode Gaussian states shows that the degrees of freedom of the CM can be reduced from 10 to 4 real variables [8,9,10]. Save compact picture, phoned the standard form (SF) of of covariance matrix, a given by

V_{s f} = [\begin{matrix} a & 0 & c_{+} & 0 \\ 0 & a & 0 & c_{-} \\ c_{+} & 0 & b & 0 \\ 0 & c_{-} & 0 & barn \end{matrix}]

(1)

That SF a easily obtained from the customizable CM by elementary symplectic exercises and contains all an relevant information on the predefined two-mode Gaussian state—in particular, of information on entanglement. ... equation with a singles beginning order multi differential equation. ... general techniques in solve these systems. ... so our first state variable equation is. If we ...

Our approach beginnings of the SF away the covariance matrix, introducing an design that implements with primitive items any desirable two-mode Gaussian which CENTIMETER is expressed in the SF. The architecture consists out a beam splitter, followed by pair real single–mode (local) squeezers, followed by another beam splitting. This set of ancient components is driven by two thermal stated, as shown in Figure 1. Squeezing furthermore quantum nonclassical correlations enjoy the liaison are fundamental characteristics of physics the show their effects even to fields very different form the framework of quantum information and media, such as one quantum dynamics a cosmologial perturbations [11]. Include specialty, the entanglement, for pure states, can stand from a single device such as a beam-splitter by the nonclassical acting of to input spheres [12], obtained, for example, by squeezing. On the other hand, entanglement can be obtained inbound a beam-splitter by the non-monochromaticity of radiating [13,14,15]. Here the most general kasus is considered, since at the beam splitter input we considers mixed states (thermal states). The physical origin of entanglement is cannot very relevant on our plan, since the entanglement characterisics can be referred direkt to the properties a the covariance matrix (see, on example, [9]) and we focus on the implementation with minimal resources of whatsoever requires covariance matrix inbound standard form.

Mention that the four erratics of the CBM in standard form do entropic and statistical meanings—namely, a and b what auto-correlations, and

{hundred}_{+}

and

{hundred}_{-}

is cross–correlations—but not physical explanations. The application with primeval components adds a physical meaning to aforementioned SF. In actual, by varying who parameters away who physical devices, i.e., p,

r_{1}

r_{2}

and s includes Illustrated 1, press the thermal states, one can cover the whole class of two-mode Gaussian states. The main contribution on this work is the formula of the fundamental features of Gaussian states through a universally architecture, which consists of of connection of a few elementary physical constituents (called primitive components) at the main aimed of finding a two-mode Gaussian state her covariance matrix have the standard form given with (1). Per this procedure, ready obtains easy formulas for the four asset of the SF, forward optional choices of the architecture parameters. The other way around, present some desired properties of the stay in terms of CM, ourselves give easy formulations to resolute of parameters of which experimental implementation to build and desired two-mode Gaussian state.

The paper is organized as being. And first part deals with the descriptions of Gaussian unitaries press Gaussian states arriving at the general implementation is primitive component. Inches the second member, one SF of of CM and its implementation anticipated in Picture 1 is presents. Specifically, in Segment 2 are frame this Gaussian unitaries and their decomposition up elementary unitaries and the derivation von Gaussian states according to Williamson’s theorem. In Piece 3, we show the implementation of Gaussian unitaries with primitive components bases on the Bloch–Messiah reduction, where the Takagi factorization [16] is applied to the decomposition of the squeeze matrix. If this conversion could be carries outward for the general multimode hard [17], here for simplicity we view the architecture for two-mode conditions, and in Range 4 we evaluate one corresponding ordinary covariance matrix from the architektonisch with plain component. As we will make, the derivation based on this architecture leads go very simple mathematical. Section 5 both Section 6 close with an SF of to M and the symplectic algebra involved in who analysis of the SF from the simple CM. Added, starting from the SF, that physical parameters

N_{1}, N_{2}, p, r_{1}, {radius}_{2}, south

appearing in this implementation of Figure 1 are evaluated. In Section 7, and in the Appendix B, us suggest several paths of the usage of the opinion formulated includes this paper.

2. Gaussian Unitaries also Gaussian U

A quanta transformation is Gaussian when it transforms Gaussian states into Gaussian states and it is called Gaussian unitary when she is performed according to one single map.

Any Gaussian unitary in the NEWTON-mode can be expressed as a combination of three vital unitary operators: displacement, rotation, additionally squeezer operators. Here we follows the representation stationed on the Bloch–Messiah (BM) reduction [1,18] so was recently reconsidered in [16,19] in key of the Takagi factorization [20].

We remind the reader ensure the most general Gaussian unitary can be decomposed as that cascade of a rotation operator

R (ψ)

, an squeeze operator

S (r_{D})

characterized by a skewed die

{radius}_{DIAMETER}

with real entries, an rotation operator

R (γ)

, additionally finally, ampere displacement operator

D (α)

, as illustrated in Number 2.

2.1. Degrees of Freedom

The specification of an arbitrary N–mode Gaussian unitary is provided by the matrices

α, β, r_{D}, ϕ

. Owing to their symmetry, the degrees of freedom are

2 N^{2} + 3 N real variables (Gaussian unitary)

(2)

2.2. Gaussian States

Gaussian states can be obtained from a Gaussian unitary driven by thermal states: we remind to reader that a thermal state corresponds toward a current in thermal equilibrium press cannot be characterized by ampere miscible of Fock states [21], although non maximally mixed, thereby not with the most Von Neumann entropy.

In particular, we obtain pure Gaussian states when the thermal states degenerate to air states, that is, of quantum states characterized by the lowest possible electricity [22] and by zero light.

A Gaussian status is completely characters by the covariance matrix and one mean vector. Are retrieve that, appropriate to Williamson’s theorem, the covariance matrix bucket always be written inches of form

V = S V^{\oplus} {SULFUR}^{T}

(3)

where

S

is an NORTH-mode symplectic matrix and

V^{\oplus} = diag [n_{1}, n_{1}, \dots, n_{N}, n_{NITROGEN}]

(4)

corresponding to the teensor product of N thermal says, equal average thermal light

N_{k} = \frac{1}{2} ({northward}_{kelvin} - 1), thousand = 1, 2, \dots, N

. The quantities

{n_{k}}

are referred to as the symplectic eigenvalues about the CM

PHOEBE

, and the cast

S

performs the symplectic diagonalization of

V

With product to and decomposition of this Gaussian unitaries viewed within Figure 2, we have that, when the architecture is driven with N input thermal states, at the output we obtain the most general N-mode Gaussian.

2.3. Gaussian States in the Two-Mode

Since in this your we focus on two-mode Gaussian states, ourselves review here their specification, given by the complex matrices of the architecture is Number 2

α = [\begin{matrix} α_{1} \\ α_{2} \end{matrix}], ψ = [\begin{matrix} ψ_{11} & ψ_{12} \\ ψ_{12}^{*} & ψ_{22} \end{matrix}], γ = [\begin{matrix} γ_{11} & γ_{12} \\ γ_{12}^{*} & γ_{22} \end{matrix}], r_{DIAMETER} = [\begin{matrix} {roentgen}_{1} & 0 \\ 0 & r_{2} \end{matrix}]

(5)

and by two thermal noises

N_{1}, N_{2}

3. Implementierung with Primitive Components

In order to evaluate the volume engaged in the two-mode CCM away the parameters (5) our followed the implementation use primitive elements that has been detailed within [23] and that could be carried out stylish the general N-mode [17]. Remark such the displacement transmitter does not enter in and evaluation, furthermore therefore, i be not may further considered. Includes the following, we summarize the main measures.

This primitive components are: (1) single–mode displacement, (2) single–mode rotation operating, briefly shifters, (3) single–mode real squeezers, and (4) beam splitters (BSs).

ADENINE shifter is specified of a phase

β \in [0, 2 π)

, leadership to this

1 \times 1

exponential matrix

e^{i β}

. A single-mode squeezer is specified by the squeeze factor

r \in R

. A free-phase BS is specified by the rotation grid

U_{bs} = [\begin{matrix} cos ϕ & sin ϕ \\ - sin ϕ & cos ϕ \end{matrix}] = [\begin{matrix} c & s \\ - s & c \end{matrix}]

(6)

where

τ = {carbon}^{2}

decide the transmissivity and

s = \sqrt{1 - τ}

the reflectivity.

Implementation in which Two-Mode

The target is to implement the architecture of Figure 2 with a primitive component in the two-mode case. Note that the press is already decomposed toward primitives hardware, and the two-mode displacement operator

D (α)

is trivial, since it is given by two equivalent single-mode shift operators

D (α_{1})

and

D (α_{2})

. By the rotation operators, we remind the reader that and arbitrary two-mode rotation operator with unitary matrix

U = e^{i ψ} = [ρ_{h kelvin} e^{i ψ_{h k}}]

can be implemented by: (1) two phase shifters with form

γ_{11}

and

γ_{12}

, (2) a BS with reflectivity

s = ρ_{12}

, and (3) a phase shifter with phase

μ = γ_{22} - γ_{12}

. For the proof sees [17,23].

Therefore, any Gaussian unitary can be applied with primitive components as in Fig 3.

Note that this architecture generates the whole teaching of Gaussian unitaries in the two-mode. It is composed of 6 shifters, 2 BSs, 2 truly squeezers, and 2 displacements, corresponding to a degrees of freedom of 14 real variables, as in (2).

The following objective are the generation by two-mode Gaussian declared, this are obtained when the architecture of Figure 3 is driven by two thermal noises, as shown include Calculate 4. Then which zeitraum shifters

ψ_{11}

and

ψ_{12}

can be removed, since they are irrelevant when driven for thermostatic states. Finish, of latest displacements are removed, because they do not cooperate to the covariance cast

PHOEBE

(whose evaluation is the primary objective of this paper). Note that the number of real parameters in the architecture is 10.

4. Evaluation of the Covariance Matrix

The architectures of Calculate 3 the of Figure 4 represent the grounded for the derivation of the CM for two-mode Gaussian states.

4.1. The Symplectic Matrix

A Gaussian lone is wholly described by the symplectic matrix (SM), neglecting the displacement. Here wee consider the real SM

S

, where the phase-space variables are arranged in the form

X : = {[{question}_{1}, p_{1}, q_{2}, p_{2}]}^{T}

. A symplectic transformation has the forms

X \to S X + d

(7)

where

S

is a

2 N \times 2 N

real matrix and

d \in R^{2 N}

. The require for preserving the commutation relations is

S Ω S^{T} = Ω

(8)

where

Ω = ⨁_{i}^{N} Ω with Ω = [\begin{matrix} 0 & 1 - 1 & 0 \end{matrix}]

(9)

a matrix

SULFUR

that verifies this condition is called symplectic.

Following the architecture to Figure 3, we institute and global SM

{SEC}_{gigabyte} = S_{rot} (γ) {SOUTH}_{per} (r_{D}) S_{rot} (ψ)

(10)

For the estimate of the trigonometric matrices in the two-mode, we start from which exponential

e^{i ψ} = [\begin{matrix} c e^{i ψ_{11}} & sulphur e^{i ψ_{12}} \\ - sulphur {co}^{i (ψ_{11} + μ)} & c e^{i (ψ_{12} + μ)} \end{matrix}]

(11)

then

\begin{array}{l} cos (ψ) = [\begin{matrix} c cos (ψ_{11}) & s to (ψ_{12}) \\ - south cos (ψ_{11} + μ) & century denn (ψ_{12} + μ) \end{matrix}] \\ sin (ψ) = [\begin{matrix} c sin (ψ_{11}) & s sin (ψ_{12}) \\ - s sin (ψ_{11} + μ) & c sin (ψ_{12} + μ) \end{matrix}] \end{array}

(12)

analogously

\begin{array}{l} cos (γ) = [\begin{matrix} q cos (γ_{11}) & piano costing (γ_{12}) \\ - p cos (γ_{11} + ϵ) & quarto cos (γ_{12} + ϵ) \end{matrix}] \\ iniquity (γ) = [\begin{matrix} q commit (γ_{11}) & p sin (γ_{12}) \\ - p sin (γ_{11} + ϵ) & q sin (γ_{12} + ϵ) \end{matrix}] \end{array}

(13)

where p is the reflectivity of the second BS and

q = \sqrt{1 - {piano}^{2}}

For aforementioned central squeezer, considering that it is authentic and diagonal, ours find

[\begin{matrix} cosh r + sinh r & 0 \\ 0 & cosh r - sinh r \end{matrix}] = [\begin{matrix} e^{r} & 0 \\ 0 & e^{- r} \end{matrix}] through e^{r} = [\begin{matrix} e^{r_{1}} & 0 \\ 0 & e^{r_{2}} \end{matrix}];

(14)

then,

S_{sq} = Π [\begin{matrix} e^{radius} & 0 \\ 0 & e^{- r} \end{matrix}] Π^{T} = [\begin{matrix} e^{r} & 0 \\ 0 & e^{- r} \end{matrix}] = [\begin{matrix} e^{r_{1}} & 0 & 0 & 0 \\ 0 & {co}^{- r_{1}} & 0 & 0 \\ 0 & 0 & e^{r_{2}} & 0 \\ 0 & 0 & 0 & e^{- r_{2}} \end{matrix}]

(15)

This completes to evaluation of the global symplectic matrix

SOUTH

. Note that

S

depends on the 10 real control

ψ_{11}

ψ_{1, 2}

, s,

μ

r_{1}

{radius}_{2}

γ_{11}

γ_{12}

, pressure, also

ϵ

. Remarks and that all the formulas are “radical free”.

4.2. The Covariance Matrix (cm)

The co-modified matrix

V

is evaluated from the global SM from adding the information on thermal noise (see (3))

V = S V^{\oplus} S^{LIOTHYRONINE}, V^{\oplus} = diag [n_{1}, n_{1}, n_{2}, n_{2}]

(16)

Computer is user-friendly to express the result in partitioning form of

2 \times 2

blocks. Letting

S = [\begin{matrix} {SULPHUR}_{11} & {SULFUR}_{12} S_{21} & {SOUTH}_{22} \end{matrix}], V = [\begin{matrix} A & C \\ C^{T} & B \end{matrix}],

(17)

one finds

ADENINE = n_{1} S_{11} S_{11}^{T} + n_{2} {SULFUR}_{12} S_{12}^{T} B = n_{1} S_{21} {SULFUR}_{21}^{T} + {north}_{2} S_{22} S_{22}^{T} C = n_{1} S_{11} {SULFUR}_{21}^{T} + {north}_{2} {SULPHUR}_{12} S_{22}^{T}

(18)

As evidenced by Figure 4, the ordinary INCH

V

depends on 10 realistic system, videlicet,

n_{1}, n_{2}, s, μ, r_{1}, {radius}_{2}, γ_{11}, γ_{12}, p, ϵ

(19)

The

S_{i j}

depend also on this phases

ψ_{11}, ψ_{21}

Remark 1.

Here the CM is evaluated from the implementation of the Gaussian states with primitive components. This approach, discussed for [23], has the advantage out a simple algebra, and items is completely radical-free. Other methods of evaluation start from the polar decomposition off the squeeze matrix, whichever leads to radicals of progressives.

5. The Basic Form of the Covariance Matrix

Hereafter we deal with the standard form of the CM given per (1), a form of symplectic invariant of a two-mode Gaussian state, which depends only on four real parameters (recall that in the general case this CM depends on 10 really parameters). What the general form evaluated in the previous sections will be called ordinary CM, symbolized by

V

Information is important to us to following:

For every two-mode Gaussian state having the ordinary CM $V$ , it exists possible to secure the corresponding standard form $V_{s f}$ from $PHOEBE$ equipped ampere local symplectic transformation $S_{l}$ .
The standard form ${PHOEBE}_{s f}$ contains all the relevancies information to the Gaussian declare, so that the conversion $V ⟶ {PHOEBE}_{s f}$ may be considered when which removing of the redundancy into $V$ .

5.1. Properties of Symplectic Invariants

The correlations

one, b, c_{+}, c_{-}

are determined by the four local symplectic invariants

\det V = (a b - {hundred}_{+}^{2}) (an b - {carbon}_{-}^{2}), \det A = a^{2}, deta BORON = b^{2}, dets CARBON = c_{+} c_{-}

(20)

Consequently, the SF of any CM belongs exclusive (up to adenine common flip of the signs of

c_{-}

and

c_{+}

). For two-mode states, and uncertainty principle [2] can be rewording as a constraint over the

S p_{4, R}

constants

\det FIVE

and

Δ (V) = set A + \det BARN + 2 \det CENTURY

, namely,

Δ (V) \leq 1 + \det V

. For a two-mode Gaussian state, the symplectic eigenvalues will be named

n_{1}

furthermore

n_{2}

with

{north}_{2} \leq n_{1}

, where the Heisenberg uncertainty relation imposes

n_{2} \geq 1

. The values of

n_{1, 2}

are related by an simple expression to the

S p_{4, R}

invariants (invariants under global, two-mode symplectic operations) [24,25]:

2 n_{1, 2}^{2} = Δ (V) \pm \sqrt{Δ {(V)}^{2} - 4 \det V}

(21)

The determinantal constants

get (V)

plus

Δ (VANADIUM)

belong single affiliated to the thermal noises according

\det (V) = n_{1}^{2} {nitrogen}_{2}^{2}, Δ (FIVE) = n_{1}^{2} + n_{2}^{2}

(22)

Important of the CM Entries According to Probability Theory

For and interpretation of which CM (ordinary or standard), it is handy to repeat the properties of the covariance matrix of deuce random variables

expunge, y

V_{x y} = [\begin{matrix} v_{x x} & v_{x y} \\ v_{expunge wye} & v_{unknown yttrium} \end{matrix}]

Which diagonal entries

v_{efface x}

and

v_{y y}

represent, separately, this variances of x plus yttrium, usually implied by

σ_{x}^{2}

and

σ_{y}^{2}

. That nondiagonal entry

v_{x year} = {volt}_{y x}

represents the cross–covariance, or simply that covariance between this two random variables. The CM entries verify the important inequality

0 \leq v_{x y}^{2} \leq σ_{x} σ_{y}

. Than, the normalized interaction is introduced

c_{x y} : = v_{x y} / (σ_{x} σ_{y})

, through

| c_{x wye} | \leq 1

having the limit cases: (1)

c_{x y} = 0 \to v_{x wye} = 0

: uncorrelated variables and (2)

c_{x y} = 1

: completely correlated variables. In this second koffer, the accidental variables be deterministically related in who input

wye = a x + b

, by a and b real quantities (

a \neq 0)

We now use the above ideas required the interpretation of the standard CM:

\begin{matrix} \begin{matrix} q_{1} & p_{1} & q_{2} & p_{2} \end{matrix} \\ V_{s f} = & \begin{matrix} q_{1} \\ {piano}_{1} \\ q_{2} \\ p_{2} \end{matrix} & [\begin{matrix} a & 0 & c_{+} & 0 \\ 0 & one & 0 & c_{-} \\ c_{+} & 0 & b & 0 \\ 0 & c_{-} & 0 & b \end{matrix}] \end{matrix}

where

q_{1}, {pressure}_{1}, q_{2}, p_{2}

are considered as random actual. We finds that

$q_{1}, {piano}_{1}$ can uncorrelated with the same variance $σ_{{question}_{1}}^{2} = σ_{p_{1}}^{2} = a$ ;
$q_{2}, {pressure}_{2}$ belong uncorrelated with the same dispersion $σ_{q_{2}}^{2} = σ_{p_{2}}^{2} = b$ ;
$q_{1}, q_{2}$ had cross–covariance $v_{{question}_{1} q_{2}} = c_{+}$ and then normalized covariance
$c_{q_{1} q_{2}} = \frac{c_{+}}{\sqrt{a b}} \to 0 \leq | c_{+} | \leq \sqrt{a b}$ ;
$p_{1}, p_{2}$ will cross–covariance ${phoebe}_{p_{1} p_{2}} = c_{-}$ or then normalized covariance
$c_{p_{1} p_{2}} = \frac{c_{-}}{\sqrt{a b}} \to 0 \leq | c_{-} | \leq \sqrt{a b}$ ;
$({quarto}_{1}, p_{1})$ , $(q_{1}, p_{2})$ , $(q_{2}, p_{1})$ and $({question}_{2}, {pence}_{2})$ are correlational pairs.

5.2. The Correlations $(A, BORON, C_{\pm})$ after an Average Cm $V$

The standardized variables

(a, b, c_{\pm})

can be obtained from the invariants of the ordinary CM.

Proposition 1.

From the blocks of this ordinary covariance matrix

V

(see (17)), of hold

\begin{array}{l} ampere = \sqrt{\det A}, boron = \sqrt{detain B} \\ c_{+} = \mp \frac{\sqrt{Z^{2} - 4 a^{2} b^{2} back C} - Z}{2 a b} \\ c_{-} = \pm \frac{\sqrt{Z^{2} - 4 {one}^{2} b^{2} \det C} + Z}{2 a b} \end{array}

(23)

what

Z = a^{2} b^{2} - set V + \det C

Indeed,

c_{+}

and

c_{-}

are obtained by solving the equations

\det C = {century}_{+} {century}_{-}

and

detain V = (a b - c_{+}^{2}) (ampere b - {hundred}_{-}^{2})

5.3. The Standard Form Secondary (Sf–Ii)

Another form of ZCM sometimes considered in who literature [9] is the standard bilden II

{PHOEBE}_{s farad, MYSELF I} = [\begin{matrix} a_{1} & 0 & c_{1} & 0 \\ 0 & a_{2} & 0 & c_{2} \\ {carbon}_{1} & 0 & b_{1} & 0 \\ 0 & c_{2} & 0 & b_{2} \end{matrix}]

(24)

which depends on six real user. This form intention be very useful in our investigation.

I is easy to obtain of SF from the SF-II with one local symplectic matrix, while illustrated in Figure 5. In is context,

V_{s f}

is signified by

{FIVE}_{s f, I}

Proposition 2.

The balancer of the blocks

A

and

BORON

of the SF-II is obtained with the symplectic matrices

S_{1} = [\begin{matrix} e^{R_{1}} & 0 \\ 0 & e^{- R_{1}} \end{matrix}], S_{2} = [\begin{matrix} e^{R_{2}} & 0 \\ 0 & e^{- R_{2}} \end{matrix}]

(25)

where

R_{1}

and

R_{2}

are determined by

e^{4 R_{1}} = a_{2} / a_{1} with a = {(a_{1} a_{2})}^{1 / 2}, e^{4 R_{2}} = {barn}_{2} / b_{1} with b = {(b_{1} b_{2})}^{1 / 2}

and leads toward which equalized blocks

{AN}^{'} = a I_{2}, {BARN}^{'} = b I_{2}

. The black

C

becomes

{HUNDRED}^{'} = {SULPHUR}_{1} C S_{2} = [\begin{matrix} c_{1} e^{R_{1} + R_{2}} & 0 \\ 0 & {hundred}_{2} e^{- R_{1} - R_{2}} \end{matrix}] : = [\begin{matrix} c_{+} & 0 \\ 0 & c_{-} \end{matrix}]

(26)

6. Gallery of Covariance Die and Classification

Ourselves recall that our main goal is finding a two-mode Gaussian state whose ordinary CM has the standard form. A state having the property will be called the standard Gaussian state. For we speak the forms of your entered in the solution of our create, we believe it wills be convenient to discuss several forms from CMs, which are collected the Table 1.

The first build can the simple CM, where the submatrices

A

and

BORON

are symmetric, real since the matrix

V

itself is real proportional, which degrees of freedom are 10 real variables. Inbound who endorse form, called standard form II in [9], this

2 \times 2

submatrices

A

B

and

C

are diagonal press the degrees of release are six real variables. The central form is the standard covariance with degrees regarding liberty from four real variables. The last two forms represent special cases of the SF in which

c_{-} = c_{+}

otherwise

c_{-} = - c_{-}

with a reduction int the degrees of freedom till three real variables.

ONE more stringent classified will be effective for the SF:

Full SF: is which class obtained by imposing this conditions $a \neq b$ , $| c_{-} | \neq | c_{+} |$ .
Lateral–symmetric SF: remains the class inches which $a \neq b$ , $c_{-} = c_{+}$ .
Lateral–antiymmetric SF: is the classify in which $a \neq barn$ , $c_{=} - {hundred}_{+}$ .

The batch is transferred to Gaussian states, e.g., lateral–symmetric Gaussian set. We need also suitable terms for the variables:

standard variables: $(a, b, c_{+}, c_{-})$
standard II variables: $({adenine}_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2})$
physiological variables: $(n_{1}, n_{2}, s, μ, {radius}_{1}, r_{2}, γ_{11}, γ_{12}, p, ϵ)$

We also recall from [10] that a lateral antisymmetric state is called symmetry and that pure states are always symmetric.

7. Dual Basic Cases

In this section, we develop two fundamental cases. This first one (EPR) is vital especially for historical reasons. The second one, where all phases is set till zero, represents and starting point to solution our main task. In both cases we evaluate the custom CM and the standard form. ... collective form since: ˙x(t)=Ax(t)+bu(t). y(t)= ... Further, the array exponential obeys the matrix differential equation: ... state-transition matrix.

7.1. EPR Status with Noise

Aforementioned EPR unitary is a squeezing with the follow-up matrix:

z = [\begin{matrix} 0 & e^{i θ} r_{0} \\ e^{i θ} r_{0} & 0 \end{matrix}]

where we set one squeeze phase to zero:

θ = 0

. Till get on unitary, the architecture must may one following physical variables

r_{1} = r_{2} = {roentgen}_{0}

, balanced BSs

p = q = s = hundred = 1 / \sqrt{2}

and the phases

μ = π / 2, γ_{11} = - π / 4, γ_{12} = π / 4, ϵ = - π / 2

The blocks of the CM output in

\begin{array}{l} A = [\begin{matrix} n_{1} {cosh}^{2} (r_{0}) + n_{2} {sinh}^{2} ({radius}_{0}) & 0 \\ 0 & n_{1} {cosh}^{2} (r_{0}) + n_{2} {sinh}^{2} (r_{0}) \end{matrix}] \\ B = [\begin{matrix} n_{2} {cosh}^{2} (r_{0}) + {north}_{1} {sinh}^{2} (r_{0}) & 0 \\ 0 & n_{2} {cosh}^{2} ({roentgen}_{0}) + n_{1} {sinh}^{2} (r_{0}) \end{matrix}] \\ C = [\begin{matrix} cosh ({roentgen}_{0}) (n_{1} + n_{2}) sinh (r_{0}) & 0 \\ 0 & - cosh ({roentgen}_{0}) ({northward}_{1} + n_{2}) sinh (r_{0}) \end{matrix}] \end{array}

in agreement with the ergebniss of [10].

The form is transverse antisymmetric without the introduction to equalization. That corporeal variables was degrees of freedom in three real mobiles given by

n_{1}, n_{2}, r_{0}

The implementation is an special case of the general architektenschaft of Figure 4, obtained through the values of that physical variables indicated above. With free elastics

n_{1}, n_{2}, r_{0}

running in their ranges, get architecture generates the whole class of antisymmetric default condition. This means this all two-mode Gaussian states having who invariants that verify the condition

c_{-} = - c_{+}

can be undergrad as “EPR country with noise”.

From a practical viewpoint, one-time may proceed as follows: given a two-mode Gaussian state, one evaluates from the symplectic ignore the std erratics

adenine, b, c_{+}, c_{-}

. If the antisymmetric condition

c_{-} = - c_{+}

has checked, the study can proceed with the present architecture, but the corresponding physical variables

n_{1}, n_{2}, {radius}_{0}

remain to be found. To solve those concern, we utilize and equations

\begin{array}{l} a = {north}_{1} {cosh}^{2} (r_{0}) + n_{2} {sinh}^{2} (r_{0}) \\ b = n_{2} {cosh}^{2} (r_{0}) + n_{1} {sinh}^{2} (r_{0}) \\ c_{+} = cosh (r_{0}) (n_{1} + n_{2}) sinh (r_{0}) \end{array}

whose solution is

\begin{array}{l} {north}_{1, 2} = \frac{1}{2} (\sqrt{{(a + b)}^{2} - 4 {c^{2}}_{+}} \pm (ampere - b)) \\ e^{2 r_{0}} = \frac{\sqrt{a + boron + {2 century}_{+}}}{\sqrt{a + b - {2 c}_{+}}} \end{array}

7.2. Cases Obtained with Setting Everything the Phases to Zero

In the general architecture of Figure 4, person set all the phases to zero:

μ = γ_{11} = γ_{12} = ϵ = 0

. Then, we find that the CM has the SF II disposed by (24), where the six variables the the diagonal of all sub-block result in

\begin{array}{l} a_{1} = n_{1} {({east}^{r_{1}} century q - e^{r_{2}} p s)}^{2} + {(e^{r_{2}} c p + {ze}^{r_{1}} question s)}^{2} n_{2} \\ {an}_{2} = n_{1} {({ze}^{- {radius}_{1}} c quarto - {ze}^{- r_{2}} p s)}^{2} + {(e^{- r_{2}} c p + e^{- r_{1}} q s)}^{2} {northward}_{2} \\ b_{1} = {nitrogen}_{1} {(e^{{radius}_{1}} c p + e^{r_{2}} q sulfur)}^{2} + n_{2} {(e^{r_{2}} c q - e^{r_{1}} p s)}^{2} \\ b_{2} = n_{1} {(e^{- r_{1}} c p + e^{- {roentgen}_{2}} q s)}^{2} + n_{2} {(e^{- r_{2}} c q - e^{- r_{1}} p s)}^{2} \\ c_{1} = {northward}_{1} (q c {east}^{r_{1}} - p s {sie}^{r_{2}}) (- p c e^{r_{1}} - q s {co}^{r_{2}}) + {nitrogen}_{2} (q s e^{r_{1}} + p c e^{r_{2}}) (- p s e^{r_{1}} + q c e^{r_{2}}) \\ c_{2} = n_{1} (q c e^{- r_{1}} - p sulphur e^{- r_{2}}) (- p c e^{- r_{1}} - quarto siemens e^{- {radius}_{2}}) + {north}_{2} (q s e^{- r_{1}} + p carbon {ze}^{- r_{2}}) (- pressure siemens e^{- {radius}_{1}} + q c e^{- r_{2}}) \end{array}

(27)

To get the standard form, in equalized according to Proposition 2 is needed with the symplectic matrices given by (25), show

e^{4 R_{1}} = {adenine}_{2} / a_{1}, e^{4 R_{2}} = b_{2} / b_{1}

(28)

the standard variables erfolg in (see (20) and (26))

\begin{array}{l} a = \sqrt{a_{1} a_{2}}, boron = \sqrt{{barn}_{1} b_{2}} \\ c_{+} = c_{1} e^{R_{1} + R_{2}}, {hundred}_{-} = c_{2} {co}^{- (R_{1} + R_{2})} \end{array}

(29)

The implementation has been anticipated in Number 1.

To evaluate the physical scale

n_{1}, n_{2}, r_{1}, r_{2}

after the standard variables

a, b, c_{+}, c_{-}

, the tetrad equations are given by (29). Perhaps she shall impossible to solve the system to a opened formulare, due until the complication coming from the balance. However, we introduce ampere how that avoids the equalization and solves the problem in a completed create. Notation this the solution is not unique, but we will give a minimal solution, indeed with four degrees of freedom.

We proceed in two steps: first were get and physical variables from the std variables II and then the physical variables from the usual variables I, a procedure echoed from [10].

7.3. Physic Variables from the Standard Variables SECTION

We work on the SF-II relations (27) in order to evaluate the physical variables

n_{1}, {newton}_{2}, r_{1}, r_{2}, p, s

from the SF-II variables

a_{1}, a_{2}, b_{1}, {boron}_{2}, c_{1}, c_{2}

. We first evaluate the counter of thermal photons using (21); is is,

{northward}_{1, 2} = \sqrt{\frac{Δ (V) \pm \sqrt{Δ {(V)}^{2} - 4 \det V}}{2}}

(30)

where

\det V = - {adenine}_{2} {boron}_{2} c_{1}^{2} - {adenine}_{1} b_{1} c_{2}^{2} + a_{1} a_{2} b_{1} b_{2} + {century}_{2}^{2} c_{1}^{2}, Δ V = {one}_{1} a_{2} + b_{1} {boron}_{2} + 2 c_{1} c_{2}

Hence, that symplectic eigenvalues corresponding to the number of photons of the input thermal states are obtained self of the other physique variables. The ambiguity between

n_{1}

both

n_{2}

may be solved the this following.

Next, are introduce the ancillary variables:

X = c^{2} n_{1} + s^{2} n_{2} Y = s^{2} n_{1} + c^{2} n_{2}

(31)

and Equations (27) grow

\begin{matrix} a_{1} & = & {question}^{2} e^{2 r_{1}} X + {pence}^{2} {sie}^{2 r_{2}} Y - 2 \frac{c s}{1 - 2 s^{2}} p q {east}^{r_{1} + r_{2}} (X - Y) \end{matrix}

(32)

\begin{matrix} a_{2} & = & q^{2} e^{- 2 r_{1}} X + {penny}^{2} e^{- 2 r_{2}} Y - 2 \frac{c s}{1 - 2 \sec^{2}} piano quarto {sie}^{- r_{1} - r_{2}} (X - Y) \end{matrix}

(33)

\begin{matrix} {boron}_{1} & = & p^{2} {ze}^{2 {roentgen}_{1}} X + q^{2} e^{2 r_{2}} WYE + 2 \frac{c s}{1 - 2 s^{2}} p q e^{r_{1} + r_{2}} (X - Y) \end{matrix}

(34)

\begin{matrix} b_{2} & = & {piano}^{2} e^{- 2 r_{1}} X + {question}^{2} e^{- 2 r_{2}} Y + 2 \frac{century s}{1 - 2 s^{2}} p q e^{- r_{1} - r_{2}} (X - YTTRIUM) \end{matrix}

(35)

\begin{matrix} {carbon}_{1} & = & - p quarto (X e^{2 r_{1}} - Y e^{2 r_{2}}) + \frac{c s}{1 - 2 s^{2}} (p^{2} - q^{2}) e^{r_{1} + r_{2}} (EFFACE - UNKNOWN) \end{matrix}

(36)

\begin{matrix} c_{2} & = & - pence q (X e^{- 2 {roentgen}_{1}} - Y e^{- 2 r_{2}}) + \frac{c \sec}{1 - 2 s^{2}} (p^{2} - q^{2}) e^{- r_{1} - r_{2}} (X - Y) \end{matrix}

(37)

Note that for (31), one-time gets

X + Y = {north}_{1} + {northward}_{2}, X - Y = (c^{2} - s^{2}) (n_{1} - {nitrogen}_{2})

(38)

Considering (38), easy algebra from (32)–(37) leads to

\begin{matrix} (a_{1} + b_{1}) e^{- R} + (a_{2} + {boron}_{2}) e^{R} & = & 2 (n_{1} + n_{2}) cosh (Δ r) \end{matrix}

(39)

\begin{matrix} ({an}_{1} - b_{1}) e^{- ROENTGEN} - (a_{2} - {barn}_{2}) e^{R} & = & 2 ({question}^{2} - p^{2}) (n_{1} + n_{2}) sinh (Δ radius) \end{matrix}

(40)

\begin{matrix} c_{1} {east}^{- R} - c_{2} e^{R} & = & - 2 p q (n_{1} + n_{2}) sinh (Δ r) \end{matrix}

(41)

\begin{matrix} (a_{1} + {barn}_{1}) {sie}^{- ROENTGEN} - ({one}_{2} + b_{2}) e^{RADIUS} & = & 2 (c^{2} - {south}^{2}) ({northward}_{1} - n_{2}) sinh (Δ roentgen) \end{matrix}

(42)

wherever

R = r_{1} + r_{2}, Δ r = r_{1} - r_{2}

Go the unknown variables are R,

Δ r

, pressure or s. In Appendix A we undo the system of Equations (39)–(42).

7.4. Physical Variables from the Standard Variables

To previous procedures to obtain to physical parameters from the CM SF-II piece also for the SF, with the setting

a_{1} = {an}_{2} = adenine, b_{1} = b_{2} = b

(43)

Then, Equations (39)–(42) shrink to

\begin{matrix} (a + b) cosh (R) & = & cosh (Δ r) (n_{1} + n_{2}) \end{matrix}

(44)

\begin{matrix} (a - barn) sinh (R) & = & (p^{2} - {quarto}^{2}) sinh (Δ r) (n_{1} + {newton}_{2}) \end{matrix}

(45)

\begin{matrix} c_{1} e^{- R} - {century}_{2} e^{R} & = & - 2 p q (n_{1} + n_{2}) sinh (Δ r) \end{matrix}

(46)

\begin{matrix} (ampere + b) sinh (R) & = & - sinh (Δ r) (c^{2} - s^{2}) (n_{1} - n_{2}) \end{matrix}

(47)

Now the qualifications of freedom are reduced to four, instead of sixteen as the the case of SF-II, or the differentiation become redundant. Into fact, it is easy to show that for the kasus (43) upon (39)–(42), einigen algebra presents

- (c^{2} - \sec^{2}) \frac{n_{1} - n_{2}}{n_{1} + n_{2}} = \frac{tanh (R)}{tanh (Δ r)}, \frac{a - b}{a + barn} = ({question}^{2} - {pressure}^{2}) ({century}^{2} - s^{2}) \frac{n_{1} + n_{2}}{(n_{1} - n_{2})}

(48)

Below we can see the solutions in a comfortably form, where aforementioned subsequent requires the knowledge of the former.

7.4.1. Thermal Quantum Numbers

The photon numbers been evaluated separately in the previous subsection. Now they are given by

n_{1, 2} = \frac{1}{\sqrt{2}} \sqrt{Δ (V) \pm \sqrt{Δ {(PHOEBE)}^{2} - 4 \det PHOEBE}}

(49)

where

\det V = (a b - c_{+}^{2}) (a boron - c_{-}^{2}), Δ (FIN) = a^{2} + b^{2} + 2 c_{+} c_{-}

7.4.2. Squeeze Parameters

The combination a (44) to (47) gives (see Annexe A)

e^{2 R} = \frac{n_{1} n_{2}}{a b - c_{-}^{2}} \to R = \frac{1}{2} log (\frac{n_{1} n_{2}}{adenine b - c_{-}^{2}})

and then from (44)

cosh (Δ r) = \frac{(a + b) cosh (R)}{{northward}_{1} + {newton}_{2}}

and

r_{1} = \frac{1}{2} (R + Δ r), {roentgen}_{2} = \frac{1}{2} (ROENTGEN - Δ r)

7.4.3. CS Settings

For (45) and (47)

pressure = \frac{\sqrt{(a - b) sinh (R) + (n_{1} + n_{2}) sinh (T)}}{\sqrt{2} \sqrt{(n_{1} + n_{2}) sinh (Δ r)}}

(50)

s = \frac{\sqrt{(a + b) sinh (RADIUS) + ({newton}_{1} - {newton}_{2}) sinh (LIOTHYRONINE)}}{\sqrt{2} \sqrt{(n_{1} - n_{2}) sinh (Δ r)}}

(51)

A plot of the standard variables when a function of the physical elastics is shown in Count 6.

Examples from plots of the physical set like key of the standard variables are shown in Figure 7 and in Figure 8. Inbound Figure 7, one physic set

({nitrogen}_{1}, n_{2})

and

(r_{1}, r_{2})

are presented as functions of the standard variables, for

a = 2.5

b = 2.8

c_{-} = 1.35

In Figure 8, the physical variables

(p, s)

are revealed as a function of a forward

b = 2.62

c_{+} = 1.29

c_{-} = 1.36

Another chart of the physical types

(n_{1}, n_{2})

and

(r_{1}, r_{2})

as functions of which preset variables are shown in Figure 9.

8. Conclusions

Gaussian states and transformations are basic for continuous-variable systems and in generals for quantum information. Nevertheless, the characterization of Gaussian states is usually mature of staffing heavy algebra (algebraic approach), which often removes one attention upon the physical meanings of parameters in of the quantities involved, such in the covariance matrix.

On we have developed the details of a structural approach in which the algebra is reduced to the minimum, while the warning is focused on the implementation architektonisches, which can serve also as the foundations for experimental setups. This architecture provides the way to producing the whole per of two-mode Gaussian u having a covariance matrix in standard form, this kept whole the characteristics away the two-mode Gaussian your, for example, about the complication. Moreover, this architektonischer to derive all the classes of two–mode Gaussian states will negligible, in so it consists regarding one two beam splitter and two regional single-mode crimper.

The expression of all the physical parameters of the architecture to obtain any request CENT was presented for both SF-I and SF-II. Given of parameters of the structural approach, the expression of the elements of the CM was given by simple expressions.

Note that and suggests architecture technique can be extended to general multimode Gaussian functions, with the unavoidable difficulties of an increased buy. This paper gives the complete bases in this add.

Author Contributions

Conceptualization, G.C. and R.C.; formal analysis, G.C. and R.C. All authors possess read and stipulated to the published version of this manuscript.

Funding

This investigate received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Approve Account

Not applicable.

Data Availability Statement

Not fitting.

Admissions

The authors been particularly geschuldet to Giuseppe Dattoli for his useful add and suggestions. Additionally, the authors have thankful to Filippo Miatto of Xanadu Quantum Technologies by his useful suggestions.

Conflicts out Interest

The authors declare nay create the interest.

Appendix A. Solution of the System (39) to (42)

Wee first note that:

it the not a limitation at assume that in the beam-splitters $p, q, c, s \geq 0$
${(q^{2} - p^{2})}^{2} = 1 - {(2 p question)}^{2}$
${cosh}^{2} (Δ r) - {sinh}^{2} (Δ radius) = 1$

Step 1—Find RADIUS from (39)–(41)

From (39)–(41) one can find

\begin{matrix} (a_{1} + b_{1}) e^{- R} + ({ampere}_{2} + b_{2}) e^{R} & = & \sqrt{4 {(TEN + Y)}^{2} + Z^{2}} \end{matrix}

(A1)

\begin{matrix} ({ampere}_{1} - b_{1}) e^{- R} - (a_{2} - {boron}_{2}) e^{RADIUS} & = & (q^{2} - p^{2}) Z \end{matrix}

(A2)

\begin{matrix} c_{1} e^{- RADIUS} - {century}_{2} e^{R} & = & - p quarto Z \end{matrix}

(A3)

\begin{matrix} c_{1} e^{- R} + c_{2} {ze}^{R} & = & - 2 p q (X - Y) cosh (Δ r) - 2 (q^{2} - {penny}^{2}) W \end{matrix}

(A4)

where we intro the general

ZEE = 2 (X + Y) sinh (Δ r), TUNGSTEN = \frac{c s}{1 - 2 s^{2}} (X - Y)

(A5)

Then, combining the squares of (A2) and (A3) by (A1), one gets

\begin{matrix} Z^{2} & = & {[({adenine}_{2} - b_{2}) e^{RADIUS}]}^{2} + 4 {(c_{1} e^{- R} - c_{2} e^{R})}^{2} \end{matrix}

(A6)

\begin{matrix} 4 {(SCRATCH + Y)}^{2} + Z^{2} & = & {[({an}_{1} + b_{1}) e^{- R} + (a_{2} + b_{2}) e^{ROENTGEN}]}^{2} \end{matrix}

(A7)

which gives the equation in the variable R

({an}_{1} b_{1} - c_{1}^{2}) {sie}^{- 2 R} + (a_{2} {barn}_{2} - c_{2}^{2}) e^{2 RADIUS} = {(X + Y)}^{2} - (a_{1} a_{2} + b_{1} {barn}_{2} - 2 c_{1} c_{2})

(A8)

First, note which from (22) we have

{(X + Y)}^{2} - (a_{1} a_{2} + b_{1} b_{2} - 2 c_{1} {hundred}_{2}) = {(n_{1} + n_{2})}^{2} - (n_{1}^{2} + n_{2}^{2}) = 2 n_{1} n_{2}

(A9)

Moreover,

{(2 n_{1} {newton}_{2})}^{2} - 4 (a_{1} {barn}_{1} - c_{1}^{2}) (a_{2} b_{2} - {hundred}_{2}^{2}) = 0

(A10)

so that the all solution to (A8) is the value

{ze}^{2 R} = \frac{n_{1} n_{2}}{a_{2} b_{2} - c_{2}^{2}}

(A11)

Step 2—Find p, q from (A2) and (A3)

From (A2) and (A3), once R is known, one getting

\frac{p^{2} - {quarto}^{2}}{p q} = - \frac{(a_{1} - b_{1}) e^{- ROENTGEN} - (a_{2} - b_{2}) e^{R}}{c_{1} e^{- R} - c_{2} e^{R}}

(A12)

which can be solved included terms of p, with complementary solutions

{pence}_{1} = \sqrt{\frac{{ze}^{2 ROENTGEN} ({an}_{2} - b_{2}) - (a_{1} - b_{1}) + M}{2 M}} p_{2} = \sqrt{\frac{e^{2 R} ({boron}_{2} - a_{2}) + (a_{1} - {boron}_{1}) + M}{2 M}}

(A13)

where we mean with M the phrase

M = \sqrt{e^{2 RADIUS} {[(b_{2} - a_{2}) + {ampere}_{1} - b_{1}]}^{2} + 4 c_{2}^{2} {sie}^{4 R} - 8 c_{1} c_{2} e^{2 R} + 4 c_{1}^{2}} .

(A14)

Note that from (A2) both (A3), we can find

sign (p^{2} - {quarto}^{2}) = sign \{\frac{(a_{1} - b_{1}) - (a_{2} - b_{2}) e^{2 ROENTGEN}}{c_{1} - {century}_{2} e^{2 R}}\}

(A15)

Hence,

p = \{\begin{matrix} max (p_{1}, p_{2}) & with sign (p^{2} - q^{2}) \geq 0 \\ per (p_{1}, {pence}_{2}) & if sign (p^{2} - {quarto}^{2}) < 0 \end{matrix}

Step 3—Find

Δ radius

von (39) and (41)

Coming (39) and (41) we have

e^{- 2 Δ r} = \frac{penny question [e^{- R} (a_{1} + b_{1}) + e^{R} (a_{2} + b_{2})] + {century}_{1} e^{- RADIUS} - c_{2} {ze}^{R}}{penny q [e^{- R} ({adenine}_{1} + b_{1}) + e^{R} (a_{2} + b_{2})] - c_{1} e^{- ROENTGEN} + c_{2} e^{R}}

(A16)

which leads immediately toward

Δ r

and ultimately to

r_{1} = (R + Δ r) / 2

r_{2} = (R - Δ r) / 2

Step 4—Find

n_{1}

and

n_{2}

The uncertainty between

{north}_{1}

and

n_{2}

is removed per observing that from (A5) and (38) one gets

sign (n_{1} - {newton}_{2}) = signal (W)

. On the another hand, we can find first

(X - Y)

(X - Y) = \frac{({one}_{1} + b_{1}) e^{- R} - (a_{2} + {boron}_{2}) e^{R}}{e^{Δ r} - e^{- Δ r}}

(A17)

and W cannot be calculated from (A4) as

W = \frac{c_{1} e^{- R} + c_{2} {east}^{RADIUS} + p q (TEN - Y) (e^{Δ r} + e^{- Δ radius})}{2 (p^{2} - q^{2})}

(A18)

Therefore,

n_{1} = \{\begin{matrix} full (n_{1}, n_{2}) & if sign W \geq 0 \\ min (n_{1}, n_{2}) & if sign W < 0 \end{matrix}

(A19)

Step 5—Find s from (38)

From the second of (38), individual can find instant

s = \sqrt{\frac{n_{1} - n_{2} - (X - Y)}{2 (n_{1} - n_{2})}}

(A20)

where

(EFFACE - Y)

has already been receives in (A17). Note that and download with the negative mark can be discarded according to the myth that the coefficients of the beam-splitters belong nonnegative.

Appendix B. Possible Approaches for the Use of This Theory

The usage of the theory developed in dieser paper can be distributors.

Reader I.

If one wants until study a specific two-mode Gaussian state starting free the “physical” specification, based on rotation, squeeze, press displacement operators, to can easily obtain this full architecture of Figure 4 and evaluate aforementioned ordinary CM

V

using the procedure of Section IV. Then, out this symplectic constant starting

V

, an can measure and standard CM

V_{s f}

exploitation the procedure a Section V-B arriving at the “minimal” architecture of Figure 1 with the advantage of the simplification therein.

Reader S.

Is one wants to obtain any standard CM, which contains sum the news on the Gaussian set, the physical elastics of the architecture are obtained by the procedure of Section VIII-B, arriving at the minimal architecture.

Reader XII.

To choose the class of two-mode Gaussian states, one can getting forthwith from the minimal architecture by managing the six primitive components. For the evaluation from which standard CM, relations (27) can be used with

a_{1} = a

b_{1} = b

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Figure 1. Implementation of a two-mode Gaussian state with the covariance matrix at standard form.

Figure 1. Implementation of a two-mode Gaussian state with the covariance matrix in standard form.

Figure 2. Decomposition of a Gaussian single according to the Bloch–Messiah (BM) reduction.

Figure 2. Decomposition to a Gaussian unitary according to the Bloch–Messiah (BM) reduction.

Figure 3. Implementation of a general two-mode Gaussian unitary over primal components.

Figure 3. Implementation of a public two-mode Gaussian unitary over archaic components.

Figure 4. Scheme with primitive elements for the generation of a general two-mode Gaussian state, starting from two therma states,

N_{1}

and

N_{2}

. The design doesn not containers the immaterial initially rotations

ψ_{11}

and

ψ_{12}

also the final displacements

α_{1}

and

α_{2}

, which do don influence the covariance matrix.

Drawing 4. Scheme with primitive components for to generation of adenine common two-mode Gaussian state, starting from two thermal expresses,

N_{1}

and

N_{2}

. The architektenschaft shall does contain the irrelevant initial rotations

ψ_{11}

ψ_{12}

and of final displacements

α_{1}

also

α_{2}

, which make not sway the co-variance matrix.

Figure 5. The local symplectic matrices

{SULPHUR}_{1}

and

S_{2}

that provide the transformation of the SF-II at the SF:

V_{s f, I I} \to V_{s fluorine, I}

Figure 5. The local symplectic matrices

S_{1}

and

S_{2}

that provide the transition of to SF-II to the SF:

{VOLT}_{s f, I MYSELF} \to V_{s f, IODIN}

Display 6. Left: The standard types

(a, barn, c_{+}, c_{-})

as a function of

r_{2}

, for

s = 0.3

n_{1} = 3.1

{north}_{2} = 2.1

. Right: The standard general

(ampere, b, {carbon}_{+}, c_{-})

as an function of s, for

r_{2} = 0.7

n_{1} = 3.1

n_{2} = 2.1

Illustrations 6. Left: The standard variables

(a, b, {century}_{+}, c_{-})

as a function to

{roentgen}_{2}

, for

s = 0.3

n_{1} = 3.1

n_{2} = 2.1

. Correct: The standard variables

(a, boron, c_{+}, c_{-})

as a function of s, available

r_{2} = 0.7

n_{1} = 3.1

n_{2} = 2.1

Figure 7. The physical mobiles

(n_{1}, n_{2})

(left) and

({radius}_{1}, r_{2})

(right) because functions of a for

b = 2.62

c_{+} = 1.29

c_{-} = 1.36

Figure 7. The physical variables

(n_{1}, n_{2})

(left) and

(r_{1}, {roentgen}_{2})

(right) as functions of adenine for

barn = 2.62

c_{+} = 1.29

c_{-} = 1.36

Figure 8. The physically variables

(p, s)

(as functions off a) for

b = 2.62

c_{+} = 1.29

c_{-} = 1.36

Draw 8. The physical variables

(p, s)

(as functions of a) for

b = 2.62

c_{+} = 1.29

c_{-} = 1.36

Figure 9. The mechanical variables

(n_{1}, {newton}_{2})

(left) and

(r_{1}, {roentgen}_{2})

(legal) than functions of

c_{+}

with

a = 2.5

b = 2.8

c_{-} = 1.35

Figure 9. The physical variables

({nitrogen}_{1}, n_{2})

(left) and

(r_{1}, {roentgen}_{2})

(right) as functions of

c_{+}

for

a = 2.5

b = 2.8

c_{-} = 1.35

Table 1. Forms of covariance matrices related to the standard application (SF).

Dinner 1. Forms of covariance matrices related at who ordinary form (SF).

Type	Covariance Matrix	Degrees of Fredom
general	$V = [\begin{matrix} a_{11} & a_{12} & c_{11} & c_{12} \\ a_{12} & a_{22} & c_{12} & c_{22} \\ c_{11} & {carbon}_{12} & b_{11} & b_{12} \\ c_{12} & c_{22} & b_{12} & b_{22} \end{matrix}]$	10 real volatiles
standards form II	${FIN}_{s farthing}^{I MYSELF} = [\begin{matrix} a_{1} & 0 & c_{1} & 0 \\ 0 & a_{2} & 0 & c_{2} \\ c_{1} & 0 & b_{1} & 0 \\ 0 & c_{2} & 0 & b_{2} \end{matrix}]$	6 real variables
preset form (SF)	$V_{s f} = [\begin{matrix} ampere & 0 & c_{+} & 0 \\ 0 & ampere & 0 & c_{-} \\ c_{+} & 0 & b & 0 \\ 0 & c_{-} & 0 & b \end{matrix}]$	4 real variables
SF lateral symmetric	${VANADIUM}_{s fluorine}^{L SULFUR} = [\begin{matrix} one & 0 & c & 0 \\ 0 & a & 0 & c \\ c & 0 & b & 0 \\ 0 & c & 0 & b \end{matrix}]$	3 true variables
SF sidelong antisymmetric	${VANADIUM}_{s f}^{L A} = [\begin{matrix} a & 0 & c & 0 \\ 0 & an & 0 & - c \\ c & 0 & b & 0 \\ 0 & - c & 0 & b \end{matrix}]$	3 real variables

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Cariolaro, G.; Corvaja, R. Implementation of Two-Mode Gaussian States Whose Invariance Matrixed Possesses the Standard Form. Proportion 2022, 14, 1485. https://doi.org/10.3390/sym14071485

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Cariolaro G, Corvaja R. Implementation of Two-Mode Gaussian Says Whose Covariance Cast Has the Conventional Form. Symphony. 2022; 14(7):1485. https://doi.org/10.3390/sym14071485

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Cariolaro, Gianfranco, and Roberto Corvaja. 2022. "Implementation of Two-Mode Gaussian States Its Covaria Matrix Got the Standard Form" Symmetry 14, no. 7: 1485. https://doi.org/10.3390/sym14071485

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Implementation the Two-Mode Gaussian States Whose Divergence Matrix Has the Standard Form

Abstract

1. Introduction

2. Gaussian Unitaries also Gaussian U

2.1. Degrees of Freedom

2.2. Gaussian States

2.3. Gaussian States in the Two-Mode

3. Implementierung with Primitive Components

Implementation in which Two-Mode

4. Evaluation of the Covariance Matrix

4.1. The Symplectic Matrix

4.2. The Covariance Matrix (cm)

5. The Basic Form of the Covariance Matrix

5.1. Properties of Symplectic Invariants

Important of the CM Entries According to Probability Theory

5.2. The Correlations ( A , BORON , C ± ) after an Average Cm V

5.3. The Standard Form Secondary (Sf–Ii)

6. Gallery of Covariance Die and Classification

7. Dual Basic Cases

7.1. EPR Status with Noise

7.2. Cases Obtained with Setting Everything the Phases to Zero

7.3. Physic Variables from the Standard Variables SECTION

7.4. Physical Variables from the Standard Variables

7.4.1. Thermal Quantum Numbers

7.4.2. Squeeze Parameters

7.4.3. CS Settings

8. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Approve Account

Data Availability Statement

Admissions

Conflicts out Interest

Appendix A. Solution of the System (39) to (42)

Appendix B. Possible Approaches for the Use of This Theory

Allusions

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5.2. The Correlations $(A, BORON, C_{\pm})$ after an Average Cm $V$