Teleportation simulation of bosonic Gaussian channels:Strong and uniform convergence

# Teleportation simulation of bosonic Gaussian channels: Strong and uniform convergence

Stefano Pirandola Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, UK    Riccardo Laurenza Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, UK    Samuel L. Braunstein Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, UK
###### Abstract

We consider the Braunstein-Kimble protocol for continuous variable teleportation and its application for the simulation of bosonic channels. We discuss the convergence properties of this protocol under various topologies (strong, uniform, and bounded-uniform) clarifying some typical misinterpretations in the literature. We then show that the teleportation simulation of an arbitrary single-mode Gaussian channel is uniformly convergent to the channel if and only if its noise matrix has full rank. The various forms of convergence are then discussed within adaptive protocols, where the simulation error must be propagated to the output of the protocol by means of a “peeling” argument, following techniques from PLOB [arXiv:1510.08863]. Finally, as an application of the peeling argument and the various topologies of convergence, we provide complete rigorous proofs for recently-claimed strong converse bounds for private communication over Gaussian channels.

## I Introduction

Quantum teleportation Tele1 (); Tele2 (); telereview (); oldrev (); teleCV2 () is a fundamental operation in quantum information theory NielsenChuang (); CVbook (); RMP () and quantum Shannon theory HolevoBOOK (); Hayashi (). It is a central tool for simulating quantum channels with direct applications to quantum/private communications TQC () and quantum metrology ReviewMETRO (). In a seminal paper, Bennett et al. ref1 () showed how to simulate Pauli channels and reduce quantum communication protocols into entanglement distillation. Similar ideas can be found in a number of other investigations ref2 (); ref3 (); ref4 (); ref5 (); ref6 (); ref7 (); ref8 (); ref9 (); ref10 (); ref11 (); ref12 (); ref13 (); ref14 (); ref15 () (see Ref. (TQC, , Sec. IX) for a detailed discussion of the literature on channel simulation). More recently, in 2015, Pirandola-Laurenza-Ottaviani-Banchi (PLOB) PLOB () showed how to transform these precursory ideas into a completely general formulation.

PLOB showed how to simulate an arbitrary quantum channel (in arbitrary dimension) by means of local operations and classical communication (LOCC) applied to the channel input and a suitable resource state. For instance, this approach allowed one to deterministically simulate the amplitude damping channel for the very first time. The LOCC simulation of a quantum channel is then exploited in the technique of teleportation stretching PLOB (), where an arbitrary adaptive protocol (i.e., based on the use of feedback) is simplified into a simpler block version, where no feedback is involved.

Teleportation stretching is a very flexible technique whose combination with suitable entanglement measures (such as the relative entropy of entanglement REE1 (); REE2 (); REE3 ()) and other functionals (such as the quantum Fisher information qfi1 (); qfi2 (); qfi3 (); qfi4 (); qfi5 ()) has recently led to the discovery of a number of results. For instance, PLOB established the two-way assisted quantum/private capacities of various fundamental channels, such as the lossy channel, the quantum-limited amplifier, dephasing and erasure channels PLOB (). In particular, the PLOB bound of bits per use of a lossy channel with transmissivity sets the ultimate limit of point-to-point quantum communications or, equivalently, a fundamental benchmark for quantum repeaters Briegel (); Rep2 (); Rep3 (); Rep4 (); Rep5 (); Rep6 (); Rep7 (); Rep8 (); Rep9 (); Rep10 (); Rep12 (); Rep13 (); Rep13bis (); Rep14 (); Rep15 (); Rep16 (); Rep17 (); Rep18 (); Rep19 (). In the setting of quantum metrology, Ref. PirCo () used teleportation stretching to show that parameter estimation with teleportation-covariant channels cannot beat the standard quantum limit, establishing the adaptive limits achievable in many scenarios. Other results were established for quantum networks netPAPER (), such as a quantum version of the max flow/min cut theorem. See also Refs. next1 (); next2 (); next3 (); next4 () for other studies.

It is clear that continuous variable (CV) quantum teleportation telereview (), also known as the Braunstein-Kimble (BK) protocol Tele2 (), is central in many of the previous results and in several other important applications. The BK protocol is a tool for optical quantum communications, from realistic implementations of quantum key distribution, e.g., via swapping in untrusted relays MDI1 (); CVMDIQKD (); CVMDIQKD2 (); CVMDIQKD3 () to more ambitious goals such as the design of a future quantum Internet HybridINTERNET (); Kimble2008 (). That being said, the BK protocol is still the subject of misunderstandings by some authors. Typical misuses arise from confusing the different forms of convergence that can be associated with this protocol, an error which is connected with a specific order of the limits to be carefully considered when teleportation is performed within an infinite-dimensional Hilbert space.

In this work, we discuss and clarify the convergence properties of the BK protocol and its consequences for the simulation of bosonic channels. As a specific case, we investigate the simulation of single-mode bosonic Gaussian channels, which can be fully classified in different canonical forms HolevoCanonical (); Caruso (); HolevoVittorio () up to input/output Gaussian unitaries. We show that the teleportation simulation of a single-mode Gaussian channel uniformly converges to the channel as long as its noise matrix has full rank. This matrix is generally connected with the covariance matrix of the Gaussian state describing the environment in a single-mode symplectic dilation of the quantum channel.

Assuming various topologies of convergence (strong, uniform, and bounded-uniform), we then study the teleportation simulation of bosonic channels in adaptive protocols. Here we discuss the crucial role of a peeling argument that connects the channel simulation error, associated with the single channel transmissions, to the overall simulation error accumulated on the final quantum state at the output of the protocol. This argument is needed in order to rigorously prove strong converse upper bounds for two-way assisted private capacities. As a direct application of our analysis, we then provide various complete proofs for the strong converse bounds claimed in Wilde-Tomamichel-Berta (WTB) WildeFollowup (). In particular, we show how the bounds claimed in WTB can be rigorously proven for adaptive protocols, and how their illness (divergence to infinity) is fixed by a correct use of the BK teleportation protocol. In this regard, our study extends the one already given in Ref. TQC () to also include the topologies of strong and uniform convergence.

The paper is organized as follows. In Sec. II, we provide some preliminary notions on bosonic systems, Gaussian states, and Gaussian channels, including the classification in canonical forms HolevoCanonical (); Caruso (); HolevoVittorio (), as revisited in terms of matrix ranks in Ref. RMP (). In Sec. III, we discuss the convergence properties of the BK protocol for CV teleportation, also discussing the interplay between the different limits associated with this protocol. In Sec. IV, we consider the teleportation simulation of bosonic channels under the topologies of strong and bounded-uniform convergence. In Sec. V, we present the main result of our work, which is the necessary and sufficient condition for the uniform convergence of the teleportation simulation of a Gaussian channel. In Sec. V, we present the peeling argument for adaptive protocols, considering the various forms of convergence. Next, in Sec. VI, we present implications for quantum/private communications, showing the rigorous proofs of the claims presented in WTB. Finally, Sec. VII is for conclusions.

## Ii Preliminaries

### ii.1 Bosonic systems and Gaussian states

CV systems have an infinite-dimensional Hilbert space . The most important example of CV systems is given by the bosonic modes of the radiation field. In general, a bosonic system of modes is described by a tensor product Hilbert space and a vector of quadrature operators satisfying the commutation relations

 [^xl,^xm]=2iΩlm  (1≤l,m≤2n) , (1)

where is the symplectic form

 Ω:=n⨁k=1ω , ω:=(01−10). (2)

An arbitrary bosonic state is characterized by a density operator or, equivalently, by its Wigner representation. Introducing the Weyl operator Weyl ()

 ^D(ξ):=exp(i^xTξ),  ξ∈R2n, (3)

an arbitrary is equivalent to a characteristic function

 χ(ξ)=Tr[ρ^D(ξ)] , (4)

or to a Wigner function

 W(x)=∫R2nd2nξ(2π)2n exp(−ixTξ)χ(ξ) , (5)

where the continuous variables span the real symplectic space which is called the phase space.

The most relevant quantities that characterize the Wigner representations are the statistical moments. In particular, the first moment is the mean value

 ¯x:=⟨^x⟩=Tr(^xρ) , (6)

and the second moment is the covariance matrix (CM) , whose arbitrary element is defined by

 Vlm:=12⟨{Δ^xl,Δ^xm}⟩ , (7)

where and is the anti-commutator. The CM is a , real symmetric matrix which must satisfy the uncertainty principle

 V+iΩ≥0 , (8)

coming directly from Eq. (1). For a particular class of states, the first two moments are sufficient for a complete characterization. These are the Gaussian states which, by definition, are those bosonic states whose Wigner representation ( or ) is Gaussian, i.e.,

 χ(ξ) =exp[−12ξTVξ+i¯xTξ] , (9) W(x) =exp[−12(x−¯x)TV−1(x−¯x)](2π)n√detV . (10)

It is also very important to identify the quantum operations that preserve the Gaussian character of such quantum states. In the Heisenberg picture, Gaussian unitaries correspond to canonical linear unitary Bogoliubov transformations, i.e., affine real maps of the quadratures

 (S,d):^x→S^x+d , (11)

that preserve the commutation relations of Eq. (1). It is easy to show that such a preservation occurs when the matrix is symplectic, i.e., when it satisfies

 SΩST=Ω . (12)

By applying the map of Eq. (11) to the Weyl operator of Eq. (3), we find the corresponding transformations for the Wigner representations. In particular, the arbitrary vector of the phase space undergoes exactly the same affine map as above

 (S,d):x→Sx+d . (13)

In other words, an arbitrary Gaussian unitary acting on the Hilbert space of the system is equivalent to a symplectic affine map acting on the corresponding phase space . Notice that such a map is composed by two different elements, i.e., the phase-space displacement which corresponds to a displacement operator , and the symplectic transformation which corresponds to a canonical unitary in the Hilbert space. In particular, the phase-space displacement does not affect the second moments of the quantum state since the CM is transformed by the simple congruence

 V→SVST . (14)

Fundamental properties of the bosonic states can be easily expressed via the symplectic manipulation of their CM. In fact, according to the Williamson’s theorem Williamson (); Arnold (); Alex (), any CM can be diagonalized by a symplectic transformation. This means that there always exists a symplectic matrix such that

 SVST=diag(ν1,ν1,⋯,νn,νn) , (15)

where the set is called the symplectic spectrum and satisfies (since for symplectic ). By applying the symplectic diagonalization of Eq. (15) to Eq. (8), one can write the uncertainty principle in the simple form of RMP ()

 νk≥1  and~{} V>0 . (16)

### ii.2 Gaussian channels and canonical forms

A single-mode bosonic channel is a completely positive trace preserving (CPTP) map acting on the density matrix  of a single bosonic mode. In particular, it is Gaussian () if it transforms Gaussian states into Gaussian states. The general form of a single-mode Gaussian channel can be expressed by the following transformation of the characteristic function HolevoCanonical ()

 G:χ(ξ)→χ(Tξ)exp(−12ξTNξ+idTξ) , (17)

where is a displacement, while and are real matrices, with and

 detN≥(detT−1)2. (18)

These are the transmission matrix and the noise matrix . At the level of the first two statistical moments, the transformation of Eq. (17) takes the simple form

 ¯x→T¯x+d,  V→TVTT+N. (19)

Any single-mode Gaussian channel can be transformed into a simpler canonical form HolevoCanonical (); Caruso (); HolevoVittorio () via unitary transformations at the input and the output (see Fig. 1). In fact, for any physical there are (non-unique) finite-energy Gaussian unitaries and such that

 G(ρ)=^UB[C(^UAρ^U†A)]^U†B , (20)

where the canonical form is the CPTP map

 C:χ(ξ)→χ(Tcξ)exp(−12ξTNcξ) , (21)

characterized by zero displacement () and diagonal matrices and .

Depending on the values of the symplectic invariants , rank() and rank(), we have six different expressions for the diagonal matrices and, therefore, six inequivalent classes of canonical forms , which are denoted by and . From Ref. HolevoVittorio () we report the classification of these forms in Table 1, where , the identity matrix, and the zero matrix. In this table is the (generalized) transmissivity, while is the thermal number of the environment and is additive noise notation ().

Let us also introduce the symplectic invariant

 r:=rank(T) rank(N)2 , (22)

that we call the rank of the Gaussian channel formsREF (); RMP (). Then, every class is simply determined by the pair according to the refined Table 2. Note that classes and have been divided into subclasses. In fact, class  includes the identity channel (for ), while class describes an attenuator (amplifier) channel for (). In common terminology the forms , and are known as phase-insensitive, because they act symmetrically on the two input quadratures. By contrast, the forms , and (conjugate of the amplifier) are all phase-sensitive. The form is an additive form. In fact it is also known as the additive-noise Gaussian channel, which is a direct generalization of the classical Gaussian channel in the quantum setting.

### ii.3 Single-mode dilation of a canonical form

All the non-additive forms admit a simple single-mode physical representation where the degrees of freedom of the input bosonic mode “” unitarily interacts with the degrees of freedom of a single environmental bosonic mode “” described by a mixed state Caruso (); HolevoVittorio () (see Fig. 2). In particular, such a physical representation can always be chosen to be Gaussian. This means that can be represented by a canonical unitary mixing the input state with a thermal state , i.e.,

 C:ρa→C(ρa)=Tre{^Uae[ρa⊗ρe(¯n)]^U†ae}, (23)

where

 (24)

with symplectic and is a thermal state with CM (see Fig. 2).

In fact, by writing in the blockform

 M=(m1m2m3m4), (25)

so that

 ^xa →^xb:=m1^xa+m2^xe , (26) ^xe (27)

one finds that Eq. (23) corresponds to the following input-output transformation for the characteristic function

 χa(ξ)→χa(mT1ξ)exp[−12(2¯n+1)∣∣mT2ξ∣∣2]. (28)

Then, by setting and

 mT2=√Nc2¯n+1O,  OT=O−1, (29)

one easily verifies that Eq. (28) has the form of Eq. (21), where the bona fide condition of Eq. (18) is assured by the symplectic nature of  NoteSympl (). In Eq. (29) the orthogonal transformation is chosen in a way to preserve the symplectic condition for . Such a condition also restricts the possible forms of the remaining blocks and , which can be fixed up to a canonical local unitary.

Altogether, any non-additive canonical form can be described by a single-mode physical representation where the type of symplectic transformation is determined by its class  while the thermal noise only characterizes the environmental state. From the point of view of the second order statistical moments, the CM of an input state  undergoes the transformation

 Va→Tre{Mae(τ,r)[Va⊕(2¯n+1)Ie]Mae(τ,r)T}, (30)

where the partial trace must be interpreted as deletion of rows and columns associated with mode .

In particular, one has the following symplectic matrices for the various forms HolevoVittorio ()

 M(0<τ<1,2)=M(C)=(√τI√1−τI−√1−τI√τI), (31)

describing a beam-splitter,

 M(τ>1,2)=M(C)=(√τI√τ−1Z√τ−1Z√τI), (32)

describing an amplifier,

 M(τ<0,2)=M(D)=(√−τZ√1−τI−√1−τI−√−τZ), (33)

describing the complementary of an amplifier. Finally HolevoVittorio ()

 M(0,0) =M(A1)=(0II0), (34) M(0,1) =M(A2)=⎛⎜⎝I+Z2IIZ−I2⎞⎟⎠, (35) M(1,1) =M(B1)=⎛⎜⎝II+Z2I−Z2−I⎞⎟⎠. (36)

### ii.4 Asymptotic dilation of the additive B2 form

The additive-noise Gaussian channel or canonical form can be dilated into a two-mode environment HolevoVittorio (). Another possibility is to describe this form by means of an asymptotic single-mode dilation. In fact, consider the dilation of the attenuator channel, which is a beam-splitter with transmissivity coupling the input mode with an environmental mode prepared in a thermal state with mean photons. In this dilation, let us consider a thermal state with so that we realize . Then, taking the limit for (so that ), we represent the canonical form as

 C[1,2,ξ](ρa)=limτ→1Tre{^UBSae(τ)[ρa⊗ρe(¯nξ,τ)]^UBSae(τ)†}. (37)

In fact it is clear that, in this way, we may realize the asymptotic transformations and .

## Iii Convergence of CV teleportation

### iii.1 Braunstein-Kimble teleportation protocol

Let us review the BK protocol for CV quantum teleportation Tele2 (); telereview (). Alice and Bob share a resource state which is a two-mode squeezed vacuum (TMSV) state . Recall that this is a zero-mean Gaussian state with CM RMP ()

 Vμ=⎛⎝μI√μ2−1Z√μ2−1ZμI⎞⎠. (38)

Here the variance parameter determines both the squeezing (or entanglement) and the energy associated with the state. In particular, we may write , where is the mean number of photons in each mode, (for Alice) and (for Bob).

Then, Alice has an input bipartite state , where is an arbitrary multimode system while is a single mode that she wants to teleport to Bob. To teleport, she combines modes and in a joint CV Bell detection, whose complex outcome is classically communicated to Bob (this can be realized by a balanced beam splitter followed by two conjugate homodyne detectors telereview ()). Finally, Bob applies a displacement  on his mode , so that the output state is the teleported version of the input .

One has perfect teleportation in the limit of infinite squeezing . In other words, for any input state (with finite energy) we may write the trace norm limit

 limμ→∞∥∥ρμRa−ρRa∥∥=0, (39)

or equivalently, we may write

 limμ→∞F(ρμRa,ρRa)=1, (40)

where is the Bures fidelity. This is a well known result which has been proven in Ref. Tele2 ().

### iii.2 Strong convergence of CV teleportation

Let us denote by the overall LOCC associated with the BK protocol, as in  Fig. 3(a). The application of this LOCC onto a finite-energy TMSV state generates a teleportation channel which is not the bosonic identity channel but a point-wise (local) approximation of . In other words, for any (energy-bounded) input state , we may consider the output

 ρμRa:=IR⊗Iμa(ρRa)=IR⊗TaAB(ρRa⊗ΦμAB), (41)

and write the trace-norm limit

 limμ→∞∥IR⊗Iμa(ρRa)−ρRa∥=0. (42)

It is clear that this point-wise limit immediately implies the convergence in the strong topology Tele2 ()

 supρRalimμ→∞∥IR⊗Iμa(ρRa)−ρRa∥=0. (43)

Similarly, we may introduce the Bures distance qfi1 ()

 dB(ρ,σ):=√2[1−F(ρ,σ)], (44)

and write the previous limit as

 supρRalimμ→∞dB[IR⊗Iμa(ρRa),ρRa]=0. (45)
###### Remark 1

Let us stress that the strong convergence of the BK protocol is known since 1998. It is well-known that, for any given energy-constrained input state, if we send the squeezing of the resource state (TMSV state) to infinite, then we can perfectly teleport the input state. In Eqs. (4) and (8) of Ref. Tele2 (), there is a convolution between the Wigner function of an arbitrary normalized input state and the Gaussian kernel , where goes to zero for increasing squeezing (and ideal homodyne detectors). Taking the limit for large , the teleportation fidelity goes to as we can also see from Eq. (11) of Ref. Tele2 (). This is just a standard delta-like limit that does not really need explicit steps to be shown and fully provides the (strong) convergence of the BK protocol.

### iii.3 Bounded-uniform convergence of CV teleportation

Consider an energy-constrained alphabet of states

 DN:={ρRa | Tr(^NρRa)≤N}, (46)

where is the total number operator associated with the input mode and the reference modes . Then, we define an energy-constrained diamond distance PLOB (); TQC () between two arbitrary bosonic channels and , as

 ∥∥E−E′∥∥⋄N:=supρRa∈DN∥∥IR⊗Ea(ρRa)−IR⊗E′a(ρRa)∥∥ . (47)

See also Ref. MaximNORM (); WinterNORM () for an alternate definition of energy-constrained diamond norm. It is easy to show that, for any finite energy , one may write PLOB ()

 limμ→∞∥Iμ−I∥⋄N=0, (48)

so that the BK channel converges to the identity channel in the bounded-uniform topology. In fact, this comes from the point-wise limit in Eq. (42) combined with the fact that is a compact set HolevoCOMPACT (); HolBook (); Werner ().

### iii.4 Non-uniform convergence of CV teleportation

Can we relax the energy constraint in Eq. (48)? The answer is no. As already discussed in Ref. TQC (), we have

 limμ→∞∥Iμ−I∥⋄=2, (49)

where

 ∥∥E−E′∥∥⋄ =limN→∞∥∥E−E′∥∥⋄N (50) =supρRa∥∥IR⊗Ea(ρRa)−IR⊗E′a(ρRa)∥∥ (51)

is the standard diamond distance. In fact, Ref. TQC () provided a simple proof that the BK protocol does not uniformly converge to the identity channel. For this proof, it is sufficient to take the input state to be a TMSV state with diverging energy . Then, Eq. (49) is implied by the fact that, for any -energy BK protocol, we have

 lim~μ→∞∥∥IR⊗Iμa(Φ~μRa)−Φ~μRa∥∥=2, (52)

which is equivalent to for any .

In order to show Eq. (52) we directly report the steps given in Ref. TQC () but adapted to our different notation. The first observation is that, when applied to an energy-constrained quantum state (i.e., a “point”), the -energy BK channel is locally equivalent to an additive-noise Gaussian channel (form ) with added noise

 ξ=2[μ−√μ2−1] . (53)

For instance, see Refs. GerLimited (); next3 (). Then, from the CM  of , it is easy to compute the CM of the output state yielding

 Vμ,~μ=⎛⎜ ⎜⎝~μI√~μ2−1Z√~μ2−1Z(~μ+ξ)I⎞⎟ ⎟⎠. (54)

Using the formula for the quantum fidelity between arbitrary Gaussian states banchiPRL2015 (), we compute

 F(~μ,μ):=F(ρμ,~μRa,Φ~μRa) (55) =14√1−4~μ[√4μ2−1+~μ−2μ(1+4μ~μ−2~μ√4μ2−1)].

Here we notice the expansion at any fixed . Now using the Fuchs-van de Graaf relations Fuchs ()

 2[1−F(ρ,σ)]≤∥ρ−σ∥≤2√1−F(ρ,σ)2, (56)

we get, for any finite , the following expansion

 ∥∥ρμ,~μRa−Φ~μRa∥∥≥2−O(~μ−1/2), (57)

which implies Eq. (52).

Here it is important to observe the radically different behavior of the teleportation protocol with respect to exchanging the limits in the energy of the resource state and in the energy of the input state . In fact, by taking the limit in before the one in in Eq. (55), we get

 F(~μ,μ)≃1−O(μ−1). (58)

Because of the non-commutation between these two limits

 limμ[lim~μF(~μ,μ)]≠lim~μ[limμF(~μ,μ)], (59)

we have a difference between the strong convergence in Eq. (43) and the uniform non-convergence in Eq. (49). This also means that joint limits such as

 limμ,~μF(~μ,μ),   limsupμ,~μ F(~μ,μ) (60)

are not defined. While this problem has been known since the early days of CV teleportation, technical errors related to this issue can still be found in recent literature (see the “case study” discussed in Sec. VII D).

## Iv Teleportation simulation of bosonic channels

The BK teleportation protocol is a fundamental tool for the simulation of bosonic channels (not necessarily Gaussian). Consider a teleportation-covariant bosonic channel  PLOB (). This means that, for any random displacement , we may write

 E[D(−α)ρD(α)]=VαE(ρ)V†α, (61)

where is an output unitary. If this is the case, then the bosonic channel can be simulated by teleporting the input state with a modified teleportation LOCC over the (asymptotic) Choi matrix of the channel . In particular, Eq. (61) is true for Gaussian channels, for which is just another displacement.

In order to correctly formulate this type of simulation, we need to start from an imperfect finite-energy simulation and then take the asymptotic limit for large energy. Therefore, let us consider a -energy BK protocol generating a BK channel  at the input of a bosonic channel . Let us consider the composite channel

 Eμ=E∘Iμ. (62)

As shown in Fig. 3(b), for any input state , we may write the output state as

 IR⊗Eμa(ρRa)=IR⊗EB∘TaAB(ρRa⊗ΦμAB). (63)

If the bosonic channel is teleportation covariant, then we can swap it with the displacements , up to re-defining the teleportation corrections as . On the one hand this changes the teleportation LOCC , on the other hand the resource state becomes a quasi-Choi state

 ρμE:=IA∘EB(ΦμAB). (64)

Therefore, as depicted in Fig. 3(c), we may re-write the teleportation simulation of the output as

 IR⊗Eμa(ρRa)=IR⊗~TaAB[ρRa⊗(ρμE)AB]. (65)

Now, using Eq. (62) and the monotonicity of the trace distance under CPTP maps, we may write

 ∥IR⊗Eμa(ρRa)−IR⊗Ea(ρRa)∥ =∥IR⊗Ea∘Iμa(ρRa)−IR⊗Ea∘Ia(ρRa)∥ ≤∥IR⊗Iμa(ρRa)−ρRa∥μ→∞→0, (66)

where we exploit Eq. (42) in the last step. Therefore, for any bipartite (energy-constrained) input state , we may write the point-wise limit

 limμ→∞∥IR⊗Eμa(ρRa)−IR⊗Ea(ρRa)∥=0. (67)

### iv.1 Strong convergence in the teleportation simulation of bosonic channels

The strong convergence in the simulation of (teleportation-covariant) bosonic channels (not necessarily Gaussian) is an immediate consequence of the point-wise limit in Eq. (67). In fact, because Eq. (67) holds for any bipartite (energy-constrained) input state , we may write

 supρRalimμ→∞∥IR⊗Eμa(ρRa)−IR⊗Ea(ρRa)∥=0, (68)

or similarly in terms of the Bures distance

 supρRalimμ→∞dB[IR⊗Eμa(ρRa),IR⊗Ea(ρRa)]=0. (69)

In other words, the teleportation simulation of a bosonic channel , strongly converges to it in the limit of large .

### iv.2 Bounded-uniform convergence in the teleportation simulation of bosonic channels

Consider now an energy constrained input alphabet as in Eq. (46) and the energy-constrained diamond distance defined in Eq. (47). Given an arbitrary (teleportation-covariant) bosonic channel and its teleportation simulation as in Eq. (65), we define the simulation error as PLOB (); TQC ()

 δ(μ,N):=∥Eμ−E∥⋄N . (70)

Because of the monotonicity of the trace-distance under CPTP maps, we may certainly write

 δ(μ,N) =supρRa∈DN∥IR⊗Eμa(ρRa)−IR⊗Ea(ρRa)∥ (71) ≤supρRa∈DN∥IR⊗Iμa(ρRa)−ρRa∥ (72) :=∥Iμ−I∥⋄N. (73)

Therefore, from Eq. (48) we have that, for any finite energy , we may write

 limμ→∞δ(μ,N)=0. (74)

In other words, for any (tele-covariant) bosonic channel , its teleportation simulation converges to in energy-bounded diamond norm. The question is: Can we remove the energy constraint? In the next section we completely characterize the condition that a bosonic Gaussian channel needs to satisfy in order to be simulated by teleportation according to the uniform topology (unconstrained diamond norm).

## V Uniform convergence in the teleportation simulation of bosonic Gaussian channels

Let us now consider the convergence of the teleportation simulation in the uniform topology, i.e., according to the unconstrained diamond norm (). As we already know, this is a property that only certain bosonic channels may have. The simplest counter-example is certainly the identity channel for which the teleportation simulation via the BK protocol strongly but not uniformly converges. See Eqs. (43) and (49). As we will see below, this is also a problem for many Gaussian channels, including all the channels that can be represented as Gaussian unitaries, and those that can be reduced to the canonical form via unitary transformations. The theorem below establishes the exact condition that a single-mode Gaussian channel must have in order to be simulated by teleportation according to the uniform topology.

###### Theorem 2

Consider a single-mode bosonic Gaussian channel and its teleportation simulation

 Gμ(ρ)=~TaAB[ρa⊗(ρμG)AB], (75)

where  is the LOCCof a modified BK protocol implemented over the resource state, with being a TMSV state with energy . Then, we have uniform convergence

 limμ→∞∥Gμ−G∥⋄=0, (76)

if and only if the noise matrix of the Gaussian channel has full rank, i.e., .

Proof.  Let us start by showing the implication

 rank(N)=2⟹Eq.~{}(???). (77)

Consider an arbitrary single-mode Gaussian channel , so that it transforms the statistical moments as in Eq. (19). As we know from Eq. (63), for any input state , we may write

 IR⊗Gμ(ρRa) =IR⊗GB∘TaAB(ρRa⊗ΦμAB) (78) =IR⊗(Ga∘Iμa)(ρRa) (79) =IR⊗Gμa(ρRa) (80)

where is the LOCC of the standard BK protocol and is the BK channel, which is locally equivalent to an additive-noise Gaussian channel ( form) with added noise as in Eq. (53). Therefore, for the Gaussian channel we may write the modified transformations

 ¯x→T¯x+d,  V→TVTT+N+ξTTT. (81)

As we can see, the transformation of the first moments is identical. By contrast, the transformation of the second moments is characterized by the modified noise matrix

 Nξ=N+ξTTT. (82)

In order words, we may write .

Because  and  have the same displacement, we can set without losing generality. Consider the unitary reduction of into the corresponding canonical form by means of two Gaussian unitaries and as in Eq. (20). Because , we may assume that these unitaries are canonical (i.e., with zero displacement), so that they are one-to-one with two symplectic transformations, and , in the phase space. To simplify the notation define the Gaussian channels

 UA(ρ):=^UAρ^U†A, UB(ρ):=^UBρ^U†B. (83)

Then we may write

 G =UB∘C∘UA, (84) Gμ (85)

Then notice that we may re-write

 Gμ=UB∘Cμ∘UA, (86)

where we have defined

 Cμ:=C∘UA∘Iμ∘U−1A. (87)

In Appendix A we prove the following.

###### Lemma 3

Consider a Gaussian channel with and . Then and have the same unitary dilation but different environmental states and , i.e., for any input state we may write

 C(ρ)=D(ρ⊗ρe),  Cμ(ρ)=D(ρ⊗ρμe), (88)

where with unitary. Furthermore

 limμ→∞F(ρμe,ρe)=1. (89)

Using this lemma in Eqs. (84) and (86) leads to

 G(ρ) =UB∘D[UA(ρ)⊗ρe], (90) Gμ(ρ) =UB∘D[UA(ρ)⊗ρμe]. (91)

Clearly these relations can be extended to the presence of a reference system , so that for any input , we may write

 IR⊗Ga(ρRa) =IR⊗UB∘D[UA(ρRa)⊗ρe], (92) IR⊗Gμa(ρRa) =IR⊗UB∘D[UA(ρRa)⊗ρμe]. (93)

As a result for any , we may bound the trace distance as follows

 ∥IR⊗Gμa(ρRa)−IR⊗Ga(ρRa)∥ (94) =∥∥IR⊗UB∘D[UA(ρRa)⊗ρμe] −IR⊗UB∘D[UA(ρRa)⊗ρe]∥ (95) (1)≤∥∥UA(ρRa)⊗ρμe−UA(ρRa)⊗ρe∥∥ (96) (2)=∥∥ρμe−ρe∥∥(3)≤2√1−F(ρμe,ρe)2, (97)

where we use: (1) the monotonicity under CPTP maps (including the partial trace) (2) multiplicity over tensor products; and (3) one of the Fuchs-van der Graaf relations. This is a very typical computation in teleportation stretching PLOB () which has been adopted by several other authors in follow-up analyses.

As we can see the upper-bound in Eq. (97) does not depend on the input state . Therefore, we may extend the result to the supremum and write

 ∥Gμ−G∥⋄ :=supρRa∥IR⊗Gμa(ρRa)−IR⊗Ga(ρRa)∥ ≤2√1−F(ρμe,ρe)2. (98)

Now, using Eq. (89), we obtain

 limμ→∞∥Gμ−G∥⋄=0, (99)

proving the result for and , i.e.,

 τ:=detT≠1rank(N)=2}⟹Eq.~{}(\ref% {theo}). (100)

Let us now remove the assumption . Note that the Gaussian channels with and are those unitarily equivalent to the form with added noise