Quantum Switch for the Quantum Internet: Noiseless Communications through Noisy Channels
11 Quantum Switch for the Quantum Internet:Noiseless Communications through Noisy Channels
Marcello Caleffi,
Senior Member, IEEE,
Angela Sara Cacciapuoti,
Senior Member, IEEE
Abstract —Counter-intuitively, quantum mechanics enablesquantum particles to propagate simultaneously among multiplespace-time trajectories. Hence, a quantum information carriercan travel through different communication channels in a quan-tum superposition of different orders, so that the relative time-order of the communication channels becomes indefinite. Thisis realized by utilizing a quantum device known as quantumswitch . In this paper, we investigate, from a communication-engineering perspective, the use of the quantum switch withinthe quantum teleportation process , one of the key functionalitiesof the Quantum Internet. Specifically, a theoretical analysis isconducted to quantify the performance gain that can be achievedby employing a quantum switch for the entanglement distributionprocess within the quantum teleportation with respect to thecase of absence of quantum switch. This analysis reveals that,by utilizing the quantum switch, the quantum teleportation isheralded as a noiseless communication process with a probabilitythat, remarkably and counter-intuitively, increases with the noiselevels affecting the communication channels considered in theindefinite-order time combination.
Index Terms —Quantum Internet, Quantum Teleportation, En-tanglement, Quantum Switch, Casual Order.
I. I
NTRODUCTION
Traditionally, the transmission of quantum information isassumed to flow along classical trajectories, i.e., trajectoriesthat obey to the law of classical physics. Specifically, thequantum information carriers are usually assumed to travelalong well-defined trajectories in space-time [1].This assumption implies that, when the quantum messageis sent through a sequence of communication channels, theorder in which the channels are traversed is well-defined. Asinstance, with reference to Fig. 1, when a message m must gothrough two communication channels – let us say channels D and E – to reach the destination, either channel E is traversedafter channel D as in Fig. 1a or vice versa as in Fig. 1b.However, quantum particles can also propagate simultane-ously among multiple space-time trajectories [2]. This abilityenables in principle the possibility for a quantum particle toexperience a set of evolutions in a superposition of alternativeorders . In other words, quantum mechanics enables com-munication channels to be combined in time in a quantum M. Caleffi and A. S. Cacciapuoti are with the Department of ElectricalEngineering and Information Technology (DIETI), University of NaplesFederico II, Naples, 80125 Italy (e-mail: marcello.caleffi@unina.it; [email protected]).The authors are also with the Laboratorio Nazionale di ComunicazioniMultimediali, National Inter-University Consortium for Telecommunications(CNIT), Naples, 80126 Italy. Indeed, a quantum particle can also experience a set of alternativeevolutions by propagating simultaneously along multiple paths [3]. superposition of different orders as in Fig. 1c, with the relativeorder of the communication channels becoming indefinite.This “exotic” communication scenario – realized througha novel quantum device called quantum switch [4] – ariseswhen the temporal order of the communication channels iscontrolled by a quantum degree of freedom, represented by a control qubit .The utilization of a quantum switch provides significantadvantages for a number of problems, ranging from quantumcomputation [4]–[6] and quantum information processing [7],[8] through non-local games [9] to communication complexity[10], [11]. And multiple physical implementations of the quan-tum switch have been proposed and experimentally realizedwith photons [12]–[14] – with the control qubit representedby polarization or orbital angular momentum degrees of free-doms.Even more interesting from a communication-engineeringperspective, the quantum switch has been recently applied toquantum communications. Specifically, the quantum channelcapacity when the message traverses noisy channels in asuperposition of alternative orders has been investigated the-oretically [1], [2], [16], [19] and experimentally [20], [21].And the results are remarkable [1]: a quantum superpositionof two alternative orders of noisy channels can behave as aperfect quantum communication channel, even if no quantuminformation can be sent throughout either of the componentchannels individually .In this paper, inspired by these recent works, we investigatethe use of the quantum switch within the quantum teleporta-tion process , one of the key functionalities of the QuantumInternet, as recently surveyed in [22].Specifically, quantum teleportation [22]–[24] constitutes apriceless strategy for “transmitting” qubits [25], [26], withouteither the physical transfer of the particle storing the qubitor the violation of the quantum mechanics principles. Torealize the marvels of the quantum teleportation two resourcesare needed. One resource is classic: two classical bits mustbe transmitted from the source to the destination. The otherresource is quantum: a pair of maximally-entangled qubitsmust be generated and shared between the two parties. As aconsequence, the entanglement generation/distribution processplays a crucial role within the Quantum Internet.Unfortunately, the quantum entanglement is a very fragileresource, easily degraded by noise [22], [27]. And any entan-glement degradation maps into a degradation of the teleported Indeed, a superposition of alternative orders provides advantages also interms of classical channel capacity, as investigated in [15]–[18]. a r X i v : . [ qu a n t - ph ] J u l (a) Message m traverses channel E afterchannel D , resulting in the transformation E(D( m )) . (b) Message m traverses channel E beforechannel D , resulting in the transformation D(E( m )) . (c) Message m traverses the channels D and E in a superposition of the two alternativeorders D → E and
E → D . Fig. 1: Pictorial representation of temporal trajectories: (a)-(b) a message traversing two channels in a well-defined temporalorder; (c) a message traversing two channels in a superposition of different temporal orders.quantum information. Nevertheless, as shown through thepaper, the deleterious effects of noisy communication channelson the entanglement distribution can be significantly reducedby exploiting the quantum superposition of two alternativetime-orders provided by a quantum switch.In this context we conduct a theoretically analysis thatquantifies the gain that can be achieved by employing a quan-tum switch for the entanglement distribution in the quantumteleportation with respect to the case of absence of quantumswitch. More in details, we derive closed-form expressionsthat link the teleported qubit at Bob’s side to the degradationsexperienced by the entangled pair during the distributionprocess. And stemming from these expressions, we evalu-ate the average fidelity achievable by utilizing the quantumswitch. The theoretical analysis reveals that the possibilityof a quantum particle to experience a set of evolutions in asuperposition of alternative orders is key to enhance the fidelityof the teleported qubit. Specifically, by utilizing the quantumswitch, the quantum teleportation is heralded as a noiselesscommunication process with a probability that, remarkably andcounter-intuitively, increases with the noise levels affectingthe communication channels considered in the indefinite-ordertime combination.The rest of the paper is organized as follows. In Sec. II weprovide some preliminaries about the quantum switch. Then inSec. III we first discuss the quantum teleportation process andthe crucial role played by the entanglement generation anddistribution process within the Quantum Internet, and thenwe introduce a practical communication system model forentanglement distribution through quantum switch. In Sec. IVwe present some preliminaries on the entanglement distribu-tion process realized through a quantum switch, whereas inSec. V we conduct the theoretical analysis of the quantumteleportation in presence of quantum switch. Finally in Sec. VIwe conclude the paper by highlighting some challenges andopen problems arising with the quantum switch. II. Q
UANTUM S WITCH : P
RELIMINARIES
As mentioned in Section I, the quantum switch is a novelquantum device allowing a quantum particle to experience aset of evolutions in a superposition of alternative orders [1],[15]. In this “exotic” communication scenario, the relativeorder of the communication channels becomes indefinite, sincethe channel temporal order is governed by a quantum degreeof freedom, which can be represented without any loss ingenerality by a qubit | ϕ c (cid:105) , named control qubit .More in details, whenever the control qubit is initialized toone of the basis states, say | ϕ c (cid:105) = | (cid:105) , the quantum switchenables the message m to experience the classical trajectory D → E – representing channel E being traversed after channel D – as shown in Fig. 1a. Similarly, whenever the controlqubit is initialized to the other basis state, say | ϕ c (cid:105) = | (cid:105) ,the quantum switch enables the message m to experience thealternative classical trajectory E → D – representing channel E being traversed before channel D – as shown in Fig. 1b.Conversely, whenever the control qubit is initialized toa superposition of the basis states, such as | ϕ c (cid:105) = | + (cid:105) ,the message m experiences a quantum trajectory – i.e., itexperiences a superposition of the two alternative evolutions D → E and
E → D – as shown in Fig. 1c.Indeed, as an example of the quantum switch advantages,let us consider an arbitrary qubit | ϕ (cid:105) traversing two noisyquantum channels D and E , and let us assume channel D being the bit-flip channel and channel E being the phase-flipchannel . The bit-flip channel D flips the state of a qubit from | (cid:105) to | (cid:105) (and vice versa) with probability p , leaving the qubitunaltered with probability − p : D( ϕ ) = ( − p ) ϕ + pX ϕ, (1)where X denotes the X-gate in Table I. The phase-flip channel E introduces – with probability q – a relative phase-shift of π between the complex amplitudes α and β of the qubit | ϕ (cid:105) = α | (cid:105) + β | (cid:105) , leaving the qubit unaltered with probability − q : E( ϕ ) = ( − q ) ϕ + qZ ϕ, (2) where Z denotes the Z-gate in Table I. Taken individually, thequantum capacity Q(·) of each channel is [28]: Q (D) = − h b ( p ) , Q (E) = − h b ( q ) , (3)respectively, with h b ( x ) (cid:52) = − x log x −( − x ) log ( − x ) denotingthe binary entropy.When the two channels are traversed in a well-defined order,the overall quantum capacity is lower than the minimum of theindividual capacities [28] – a result referred to as bottleneckcapacity . Hence, with reference to the classical well-definedtrajectory D → E , it results: Q (D → E) ≤ min { Q (D) , Q (E)} == − max { h b ( p ) , h b ( q )} (4)and the same result holds for the classical trajectory E → D .In particular, by considering the scenario where p = q = we have that no quantum information can be sent through anyclassical trajectory traversing the channels D and E . Indeed,no quantum information can be sent either through any singleinstance of the channels.Conversely and astounding, the quantum trajectory consti-tuted by an even superposition of the two alternative evolutions D → E and
E → D behave as an ideal channel with prob-ability pq = [1], violating so the bottleneck inequality in(4).Hence, in a nutshell, a quantum superposition of twoalternative orders of noisy channels can behave as a perfectquantum communication channel, even if no quantum informa-tion can be sent throughout either of the component channelsindividually [1].III. Q UANTUM S WITCH FOR THE Q UANTUM I NTERNET
Here we apply the quantum switch to the enabling function-ality of the Quantum Internet: the entanglement generationand distribution process. Specifically, we first introduce inSec. III-A the quantum teleportation process and we highlightthe key role played by the entanglement generation anddistribution process within the Quantum Internet. Stemmingfrom this, in Sec. III-B we design a practical communicationsystem model for entanglement distribution through quantumswitch. | ψ (cid:105) H | Φ + (cid:105) X Z | ψ (cid:105) AliceBob
Fig. 2: Quantum teleportation circuit. | ψ (cid:105) denotes the qubitto be transmitted from Alice to Bob, and | Φ + (cid:105) denotes theEPR pair generated and distributed so that one qubit is storedat Alice and another qubit is stored at Bob. The symboldenotes the measurement operation and the double-linerepresents the transmission of a classical bit from Alice toBob. A. Quantum TeleportationQuantum teleportation [22]–[24] constitutes a pricelessstrategy for “transmitting” qubits [25], [26], without eitherthe physical transfer of the particle storing the qubit or theviolation of the quantum mechanics principles.To realize the marvels of the quantum teleportation tworesources are needed. One resource is classic: two classicalbits must be transmitted from the source – say Alice – to thedestination – say Bob. The other resource is quantum: a pairof maximally-entangled qubits – referred to as EPR pair inhonor of Einstein, Podolsky, and Rosen’s seminal work [29]– must be generated and shared between Alice and Bob.Once the EPR pair is distributed between Alice and Bob,Alice performs a sequence of local operations on the twoqubits at her side – namely, the qubit to be teleported and oneof the qubits forming the EPR pair – as shown in Fig. 2. Then,she transmits to Bob the output – two classical bits – of a jointmeasurement of the two qubits. Once Bob receives the twobits conveying Alice’s measurement output, he can “recover” In simple terms and oversimplifying, entanglement is a counter-intuitiveform of correlation with no counterpart in the classical domain. By measuringindividually any of the qubits forming the EPR pair, one obtains a randomoutcome. However, by comparing the results of the two independent mea-surements, one finds that they match, either directly or complementary. Inparticular, measuring one qubit of an EPR pair instantaneously changes thestatus of the second qubit, regardless of the distance dividing the two qubits[22]. For a more in-depth description of quantum entanglement and quantumteleportation, please refer to Sec. II.D and Sec. III in [22].
TABLE I: Quantum Gates
Gate
Identity X(NOT) Y Z Hadamard Controlled-NOT(CNOT)
Symbol
I X Y Z H
Matrix I = (cid:20) (cid:21) X = (cid:20) (cid:21) Y = (cid:20) − ii (cid:21) Z = (cid:20) − (cid:21) H = √ (cid:20) − (cid:21) the original quantum information from the EPR qubit at hisside with a sequence of local operations that depends onAlice’s measurement, as depicted in Fig. 2. It is worthwhileto note that, since the entanglement is destroyed during theteleportation process due to the measurement process, theteleportation of another qubit requires the generation and thedistribution of a new EPR pair. Remark.
The entanglement generation/distribution processplays a key role within the Quantum Internet, since it isa fundamental pre-requisite for the transmission of quantuminformation through the quantum teleportation process.At this stage, a question arises: “how an EPR pair canbe generated and distributed between remote nodes?” In anutshell and by oversimplifying, the generation of quantum en-tanglement requires that two qubits interact each others, so thatthe state of each qubit cannot be described independently fromthe state of the other [22]. As an example, a popular schemefor entanglement generation involves carefully pointing a laserbeam toward a non-linear crystal, so that two polarization-entangled photons emerge from the crystal [30].Since Alice and Bob represents remote nodes, the entangle-ment generation occurring at one side must be complementedby the entanglement distribution functionality, which “moves”one of the entangled particles to the other side. To thismatter, there is a broad consensus on the adoption of photonsas entanglement carriers [31]. The rationale for this choicelays in the advantages provided by photons for entanglementdistribution, such as weak interaction with the environment,easy control with standard optical components as well as high-speed low-loss transmission to remote nodes.Despite the attractive features provided by photons as en-tanglement carriers, quantum entanglement is a very fragileresource and it is easily degraded by noise. More specifically,the effect of the noise is to transform the EPR pair into anon-maximally entangled pair, i.e., to degrade the amount ofentanglement shared between Alice and Bob. And any entan-glement degradation introduces an unavoidable degradation of the quantum teleportation process, which becomes noisy. Remark.
The amount of entanglement degradation introducedduring the entanglement generation/distribution process gov-erns the imperfection of the teleportation process: the less theshared pair is entangled, the more the teleported qubit at Bobwill differ from the original qubit at Alice.
B. Entanglement Distribution via Quantum Switch
With the discussions of Sec. III-A in mind, here we aim atdesigning, from a communication-engineering perspective, ascheme able to exploit the quantum switch for entanglementdistribution.More in detail, we envision the scheme depicted in Fig. 3.A pair of entangled particles is generated at Alice. Hence, onemember of the EPR pair – say | Φ + (cid:105) A – is retained at Alicewhereas the other member – say | Φ + (cid:105) B – is distributed to Bob We refer the reader to [22] for a in-depth discussion about the differentsources of imperfections affecting the quantum teleportation process. E N T AN G L E M E N T G E N E R A T I O N QUANTUMSWITCH | Φ + i B ALICE BOB
ENTANGLEMENTDISTRIBUTION ρ e = | Φ + i A | Φ + i B Fig. 3: Entanglement distribution via quantum switch. Theentanglement generation process is located at Alice, and ρ e = | Φ + (cid:105) (cid:104) Φ + | denotes the density matrix of the EPR pair | Φ + (cid:105) generated at Alice. A quantum switch is employed todistribute the entanglement-pair member | Φ + (cid:105) B to Bob.through a quantum switch by using a photon as entanglement-carrier. Remark.
It is worthwhile to underline that the assumption ofentanglement generation located at source is not restrictive.Indeed, it constitutes one of most employed schemes for prac-tical generation and distribution process as recently surveyedin [22].Unfortunately, the quantum switch is an abstract function rather than a well-defined physical device. Clearly, the naiveimplementation proposed in [32] does not fit with any practicalcommunication system model, since it envisions a sequenceof two teleportation processes sequentially applied in a su-perposition of time-orders. Furthermore, although a numberof different physical implementations have been proposed inliterature [12]–[14], [17], [18], [20], [21], these implementa-tions aimed at confirming the theoretical results rather than atdesigning a communication system block. Indeed, within thementioned implementations – realized at a laboratory scale –the communication links needed to interconnect the differentcomponents of a quantum switch were reasonably assumedideal.Conversely, we aim at modeling a practical communicationsystem where any communication link – regardless being anoptical fiber link or a free-space optical link – reasonablybehaves as a noisy channel degrading the amount of entan-glement eventually shared between Alice and Bob. Hence –within the entanglement distribution framework – we resortto the circuit realization of the quantum switch given bythe scheme in Fig. 4 and proposed in [2], which can beimplemented with existing photonic technologies.Specifically, in Fig. 4 the gate U routes the entanglement-carrier | Φ + (cid:105) B through either the upper or the lower wire,depending on the state of the control qubit | ϕ c (cid:105) . Regardlesswhether | Φ + (cid:105) B encountered channel D (upper wire) or channel E (lower wire), the SW AP gate routes the entanglement-carrierthrough the other portion of the circuit – hence realizing thetrajectories
D → E and
E → D , respectively. Eventually,regardless of the followed trajectory, gate U † routes theentanglement-carrier through the correct (upper) wire. Clearly,whenever the control qubit | ϕ c (cid:105) is in a superposition of thebasis states, we have that the entanglement-carrier experiencesthe quantum trajectory corresponding to a superposition of thetwo alternative orders D → E and
E → D . ρ e | Φ + (cid:105) A ρ QS e | Φ + (cid:105) B U D SW AP D U † ρ c E E ρ out c quantum switch Fig. 4: Circuit realization of a quantum switch for entangle-ment distribution. The entanglement-carrier | Φ + (cid:105) A is retainedat Alice, whereas the entanglement-carrier | Φ + (cid:105) B is distributedat Bob through a quantum trajectory constituted by a su-perposition of alternative orders D → E and
E → D . The U gate routes the entanglement-carrier | Φ + (cid:105) B either throughchannel D or E , depending on the state of the control qubit ϕ c . When the entanglement-carrier emerges from one channel,the SW AP gate routes it through the other channel. Finally,the gate U † recombines the paths of the entanglement-carrier. ρ QS e denotes the density matrix of the EPR pair distributedbetween Alice and Bob through the quantum switch.From Fig. 4, it becomes evident that a practical commu-nication system model for entanglement distribution throughquantum trajectories requires – along with a SW AP block –two communication links. However, by mapping the controlqubit | ϕ c (cid:105) and the entangled-carrier | Φ + (cid:105) B into differentdegrees of freedom of a single photon, it is possible to realizea quantum trajectory by transmitting a unique photon fromAlice to Bob.More in detail, in Fig. 5 we outline the sketch of apossible photonic implementation of a quantum switch forentanglement distribution – with | ϕ c (cid:105) mapped into the pho-ton’s polarization |→(cid:105) , |↑(cid:105) and | Φ + (cid:105) B mapped into anotherphoton’s degree of freedom. Whenever | ϕ c (cid:105) is initialized into asuperposition of the basis states, two photons emerge from thefirst Polarization Beam Splitter (PBS): a horizontal-polarizedphoton and a vertical-polarized photon, which are sent toBob through two different quantum communication links. Thetwo photons, during their journey through the communicationlinks, bump into a photonic SW AP gate – implemented with aPBS and a couple of Half-Wave Plates (HWPs) converting |→(cid:105) into |↑(cid:105) and vice versa – which implements the superpositionof alternative orders
D → E and
E → D . Finally, the twophotons emerging from the two paths are recombined at Bobwith a third PBS.IV. M
ODELLING E NTANGLEMENT D ISTRIBUTION VIA Q UANTUM S WITCH
A quantum switch for a one-qubit system – represented bythe density matrix ρ – is described mathematically as a higher-order transformation [1] taking ρ as input and returning asoutput P(D , E , ρ c )( ρ ) , function of the two channel D and E along with the state ρ c = | ϕ c (cid:105) (cid:104) ϕ c | of the control qubit | ϕ c (cid:105) : P(D , E , ρ c )( ρ ) = (cid:213) i , j W ij ( ρ ⊗ ρ c ) W † ij . (5)In (5), { W ij } denote the set of Kraus operators associated withthe superposed channel trajectories, given by [1], [2]: W ij = D i E j ⊗ | (cid:105) (cid:104) | + E j D i ⊗ | (cid:105) (cid:104) | . (6) with { D i } and { E j } denoting the Kraus operators associatedwith the channels D and E , respectively.Here, we extend this result to the entanglement distributionprocess. More in detail, by considering the circuital schemedepicted in Fig. 4 with photonic implementation given inFig. 5, we extend the use of the quantum switch to the caseof a two-qubit system – represented by the density matrix ρ e that is a × matrix. To this aim, we consider the two noisyquantum channels introduced in Sec. II: the bit flip channel and the phase flip channel , given in (1) and (2), respectively.By assuming without any loss of generality ρ e being the × density matrix associated with the EPR pair | Φ + (cid:105) = (| (cid:105) + | (cid:105)) /√ : ρ e (cid:52) = | Φ + (cid:105) (cid:104) Φ + | = , (7)we have the following results. Lemma 1.
The global quantum state
P(D , E , ρ c )( ρ e ) at theoutput of the quantum switch depicted in Fig. 4 is given byequation (8) reported at the top of the next page, where ( i ⊕ a ) denotes the addition modulo-2 of i and a.Proof: See Appendix A. Remark.
From (8), it is possible to recognize that the effect ofthe quantum switch on the control qubit ϕ c with initial densitymatrix ρ c is to transform the control qubit into a mixed stateof the two basis states |−(cid:105) , | + (cid:105) .By exploiting Lemma 1, we can derive the expression ofthe density matrix ρ QS e of the EPR pair distributed betweenAlice and Bob at the output of the quantum switch. Corollary 1.
The density matrix ρ QS e of the EPR pair dis-tributed between Alice and Bob via a quantum switch is givenby (9) reported at the top of the next page.Proof: See Appendix B. Remark.
From Corollary 1, we have two cases. With prob-ability pq – heralded by a measurement of the control qubitcorresponding to the state |−(cid:105) – the entanglement distributionis a noiseless process. In fact, Bob receives the particle | Φ + B (cid:105) of the EPR pair without any error, being ρ QS e = ρ e asdetailed in Appendix B. As a consequence, by utilizing thequantum switch, the entanglement distribution process is aheralded noiseless communication process with probability pq .Differently, with probability − pq – heralded by a measure-ment of the control qubit corresponding to the state | + (cid:105) –the entanglement distribution is a noisy process being Bob’sparticle | Φ + B (cid:105) degraded by the noisy channels. Nevertheless, asit will be shown in Proposition 1, also in this case the quantumswitch provides a considerable gain – in terms of degradation As noted in [1], this choice is not restrictive, since other types ofdepolarizing channels are unitarily equivalent to a bit flip and a phase flipchannel. Hence the analysis can be easily extended by considering suitablepre-processing and post-processing operations. We refer the reader to [22] for a concise introduction to the density matrixformalism, whereas a in-depth description can be found in [33]. DE PBS PBS HWPHWP |→i|↑i |→i |→i | {z } communication linkcommunication link z }| { |↑i DE ALICE BOB | Φ + i B | ϕ c i PBS
MIRROR |↑i|→i
Fig. 5: Sketch of a possible photonic implementation of a quantum switch for entanglement distribution. Two quantumcommunication links – corresponding to the noisy quantum channels D and E , respectively – are available between Alice andBob. The control qubit | ϕ c (cid:105) of the quantum switch is mapped into the horizontal/vertical photon’s polarization |→(cid:105) , |↑(cid:105) , whereasthe entangled-carrier | Φ + (cid:105) B is mapped into another photon’s degree of freedom. The Polarization Beam Splitter (PBS) transmitsa horizontally-polarized photon and reflects a vertically-polarized photon, whereas the Half-Wave Plate (HWP) realizes thepolarization conversion between |→(cid:105) and |↑(cid:105) . P(D , E , ρ c )( ρ e ) = (cid:169)(cid:173)(cid:171) ( − p )( − q ) ρ e + ( − p ) q (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) ρ e (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) † ++ p ( − q ) (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † (cid:170)(cid:174)(cid:172) ⊗ | + (cid:105) (cid:104) + | ++ pq (cid:34)(cid:213) i , j (− ) ( j ⊕ ) | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j (− ) ( j ⊕ ) | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † ⊗ |−(cid:105) (cid:104)−| (8) ρ QS e = ρ e , with probability pq , ( − p )( − q ) ρ e + p ( − q ) (cid:2)(cid:205) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:3) ρ e (cid:2)(cid:205) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:3) † − pq ++ ( − p ) q [ (cid:205) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |)] ρ e [ (cid:205) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |)] † − pq otherwise (9)reduction – with respect to the case of absence of quantumswitch.Before concluding this section, we give with Corollary 2another intermediate result: the expression of the densitymatrix ρ CT e of the EPR pair distributed between Alice andBob through the classical trajectory D → E . Corollary 2.
The density matrix ρ CT e of the EPR pair dis-tributed between Alice and Bob through the classical trajec-tory D → E is given by (10) reported at the top of the nextpage.Proof: See Appendix C.
Remark.
Indeed, the expression ρ CT e given in (10) holds forboth the classical trajectories D → E and
E → D . V. Q
UANTUM T ELEPORTATION VIA Q UANTUM S WITCH
Here, we evaluate the performance gain achievable bydistributing the entanglement via a quantum switch within thequantum teleportation process.To this aim, in the following we first collect some def-initions. Then, we prove the preliminary result reported inLemma 2, revealing the closed-form expression of the densitymatrix of the teleported qubit at Bob’s side as a function of thedensity matrix of the EPR pair shared between Alice and Bob.Such a result is mandatory to understand and to quantify howthe communication noise impairments on the EPR distributionprocess affect the teleported qubit. Finally, stemming fromthis, we prove the main result in Proposition 1.Let ρ ψ (cid:52) = | ψ (cid:105) (cid:104) ψ | be the × density matrix of the ρ CT e = ( − p )( − q ) ρ e + p ( − q ) (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † ++ ( − p ) q (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) ρ e (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) † ++ pq (cid:34)(cid:213) i , j (− ) j ⊕ | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j (− ) j ⊕ | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † (10)unknown pure quantum state | ψ (cid:105) = α | (cid:105) + β | (cid:105) = cos (cid:0) θ (cid:1) | (cid:105) + e i φ sin (cid:0) θ (cid:1) | (cid:105) that Alice wants to ”transmit” to Bob via thequantum teleportation process introduced in Sec. III-A. Inspherical coordinates, ρ ψ is equivalent to: ρ ψ = (cid:20) cos (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) e − i φ sin (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) e i φ sin (cid:0) θ (cid:1) sin (cid:0) θ (cid:1) (cid:21) = (cid:20) ρ ψ ρ ψ ρ ψ ρ ψ (cid:21) (11)To stress the generality of Lemma 2, it is convenient to intro-duce the notation ˜ ρ e to denote the density matrix of the actualEPR pair distributed between Alice and Bob. The rational ofthis choice is that Lemma 2 holds regardless of specific noiseaffecting the entanglement generation and distribution process.As instance and according to this, whenever the entanglementgeneration/distribution process is perfect, it results ˜ ρ e = ρ e given in (7). With this in mind we provide the followingdefinitions. Definition 1.
Let us denote with (cid:8) ˜ ρ e ij (cid:9) i , j = , the four sub-block matrices arising by partitioning the × density matrix ˜ ρ e of the actual EPR pair shared between Alice and Bob into × block-matrices, i.e.: ˜ ρ e = (cid:20) ˜ ρ e ˜ ρ e ˜ ρ e ˜ ρ e (cid:21) . (12) Definition 2. Aij denotes the indicator function of the telepor-tation measurement process at Alice, i.e.: Aij = (cid:40) , if Alice measures state | i j (cid:105) , otherwise . (13) Lemma 2.
The density matrix ρ t of the teleported qubit atBob’s side is equal to: ρ t = A (cid:104) (cid:16) ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17)(cid:105) ++ A (cid:104) X (cid:16) ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) X † (cid:105) ++ A (cid:104) Z (cid:16) ρ ψ ˜ ρ e − ρ ψ ˜ ρ e − ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) Z † (cid:105) ++ A (cid:104) Z X (cid:16) ρ ψ ˜ ρ e − ρ ψ ˜ ρ e − ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) ( Z X ) † (cid:105) , (14) where ρ ij ψ is given in (11) , Aij is defined in Def. 2, ˜ ρ e denotesthe density matrix of the EPR pair distributed between Aliceand Bob, and ˜ ρ e ij is defined in Def. 1.Proof: See Appendix D. Remark.
The closed-form expression (14) derived withinLemma 2 holds regardless of the particulars of the entan-glement generation and distribution process, as long as ˜ ρ e denotes the density matrix of the actual EPR pair distributedbetween Alice and Bob. Specifically, (14) holds for bothquantum trajectories arising with a quantum switch as wellas classical trajectories. Furthermore, (14) holds regardless ofthe specific noise (if any) affecting the quantum channel usedfor entanglement distribution.Indeed, whenever the entanglement generation and distribu-tion process is perfect, the sub-matrices { ˜ ρ e ij } i , j = , are givenby (7), i.e.: ˜ ρ e = ρ e = (cid:20)
00 0 (cid:21) , ˜ ρ e = ρ e = (cid:20) (cid:21) , ˜ ρ e = ρ e = (cid:20) (cid:21) , ˜ ρ e = ρ e = (cid:20) (cid:21) . (15)In this case, from (14), it is easy to recognize that the densitymatrix ρ t of the teleported qubit coincides with the densitymatrix ρ ψ of the unknown pure quantum state | ψ (cid:105) for everypossible outcome of the measurement. Conversely, wheneverthe entanglement generation and distribution process is imper-fect, (14) continues to hold but the sub-matrices { ˜ ρ e ij } i , j = , deviate from their ideal expressions as a consequence of thenoise.In the following, stemming from the result derived inLemma 2, we evaluate in Proposition 1 and in the subsequentCorollary 3 the performance gain – in terms of reduction ofthe imperfections affecting the teleported qubit – achievablethrough the superposition of casual orders via the quantumswitch. For this, we resort to the fundamental figure of meritknown as quantum fidelity F . In a nutshell, the fidelity F of animperfect quantum state with density matrix ρ , with respect toa certain pure state | ψ (cid:105) , is a measure – with values between0 and 1 – of the distinguishability of the two quantum states,and it is generally defined as F = (cid:104) ψ | ρ | ψ (cid:105) [34], [35]. Proposition 1.
The average fidelity F QS of the teleportedquantum state at Bob’s side when the EPR pair is distributedvia a quantum switch is given by:F QS = (cid:40) F QS |−(cid:105) = , with probability pq , F QS | + (cid:105) = − p − q + pq ( − pq ) , otherwise . (16) where p and q are the error probabilities of the two considerednoisy channels D and E given in (1) and (2) , and F QS |−(cid:105) and (a) Average fidelity F QS | + (cid:105) given that the control qubit | ϕ c (cid:105) is measuredinto state | + (cid:105) . (b) Average fidelity F QS = pq F QS |−(cid:105) + ( − pq ) F QS | + (cid:105) . Fig. 6: Average fidelity of the teleported qubit when the EPR pair member | Φ + (cid:105) B is distributed at Bob’s side via a quantumswitch as a function of the error probabilities p and q of the two considered noisy channels D and E given in (1) and (2). F QS | + (cid:105) denote the average fidelity when the measurement of thecontrol qubit correspond to the state |−(cid:105) and | + (cid:105) , respectively.Proof: See Appendix E Remark.
From (16) it is easy to recognize that – withprobability pq heralded by a measurement of the control qubitequal to |−(cid:105) – the quantum trajectory corresponding to a evensuperposition of the two alternative noisy evolutions D → E and
E → D constitutes a noise-free channel. In fact, a fidelityequal to one – which corresponds to the case of a teleportedqubit at Bob identical to the original qubit to be teleported atAlice – is obtained whenever the measurement of the controlqubit returns state |−(cid:105) . Remark. (16) can be equivalently written in a compact formas: F QS = pq F QS |−(cid:105) + ( − pq ) F QS | + (cid:105) = − p − q + pq . (17)Stemming from Proposition 1, in Fig. 6 we report theaverage fidelity achievable with a quantum switch, as a func-tion of the error probabilities p and q of the two considerednoisy channels – i.e., the bit flip channel and the phase flipchannel given in (1) and (2), respectively. More in detail,in Fig. 6a we show the density plot of the average fidelity F QS | + (cid:105) obtained when the control qubit | ϕ c (cid:105) is measured intostate | + (cid:105) as a function of p and q . As discussed within theremark following Corollary 1, whenever | ϕ c (cid:105) is measuredinto state | + (cid:105) , the noise on the quantum channels cause anunavoidable and irreversible degradation of the entanglement,which maps into a degradation of the teleported quantuminformation. This is evident in Fig. 6a: for any p , q > theaverage fidelity F QS | + (cid:105) < , with values lower than . for thehighest values of the error probabilities p and q . However,as it will be shown in the following with Fig. 7, when a quantum switch is utilized for the entanglement distribution,the degradation of the teleported quantum state is always lowerthan the degradation introduced by the classical trajectory forany value of p (cid:44) and q (cid:44) . As regards to the averagefidelity F QS |−(cid:105) obtained whenever | ϕ c (cid:105) is measured into state |−(cid:105) , given that F QS |−(cid:105) = for any value of p and q a graphicalplot is not necessary. Finally, in Fig. 6b we report the densityplot of the average fidelity F QS = pqF |−(cid:105) + ( − p ) F QS | + (cid:105) as afunction of p and q . It is worthwhile to note that the quantumswitch guarantees an average fidelity exceeding the threshold – that is the maximum fidelity achievable by distributingentanglement through classical channels [36] – for most of thevalues spanned by p and q . An exception arises whenever p isclose to zero and q is close to (and vice versa). The rationalefor this performance is that, in this case, the superposition ofalternative orders collapses into a classical trajectory giventhat one of the two quantum channels behaves as an identicalchannel. Corollary 3.
The average fidelity F QS of the teleported quan-tum state at Bob’s side when the quantum switch is adopted isalways greater than the average Fidelity F CT of the teleportedquantum state when a classical trajectory is adopted, for everyp , q (cid:44) : F QS > F CT , (18) where F CT = − p − q + pq .Proof: See Appendix F Stemming from Corollary 3, in Fig. 7 we compare theaverage fidelities achievable with either a quantum switch ora classical trajectory, as a function of the error probabilities p and q of the bit flip and the phase flip channel, respectively.More in detail, in Fig. 7a we report the density plot of the ratio (a) Ratio F QS / F CT between the average fidelity of the teleportedqubit when the EPR pair member | Φ + (cid:105) B is distributed to Bob: i) viaa quantum switch, and and ii) through a classical trajectory. (b) Average fidelity of the teleported qubit as a function of p when q = p : i) F CT : average fidelity of the teleported qubit when the EPR pairmember | Φ + (cid:105) B is distributed to Bob through a classical trajectory; ii) F QS | + (cid:105) : average fidelity when the EPR pair member | Φ + (cid:105) B is distributedto Bob via a quantum switch, given that the control qubit | ϕ c (cid:105) ismeasured into state | + (cid:105) ; iii) F QS |−(cid:105) : average fidelity when the EPR pairmember | Φ + (cid:105) B is distributed to Bob via a quantum switch, given thatthe control qubit | ϕ c (cid:105) is measured into state |−(cid:105) ; iv) F QS : averagefidelity when the EPR pair member | Φ + (cid:105) B is distributed to Bob viaa quantum switch; Fig. 7: Performance comparison between the quantum and classical trajectories as a function of the error probabilities p and q of the two considered noisy channels D and E given in (1) and (2). F QS / F CT between F QS – the average fidelity of the teleportedqubit when the EPR pair member | Φ + (cid:105) B is distributed to Bobvia a quantum switch – and F CT – the average fidelity ofthe teleported qubit when the EPR pair member | Φ + (cid:105) B is dis-tributed to Bob through a classical trajectory. Indeed, Fig. 7aclearly shows the performance gain achievable by distributingentanglement via a quantum switch. To better visualize theperformance gain in terms of fidelity, in Fig. 6b we plot theaverage fidelity as a function of p when q = p . Remarkably,when p = q = – i.e., when no quantum information can besent through any classical trajectory traversing the channels D and E – it results that F QS = , whereas F CT = .Furthermore, greater is the noise affecting the communicationschannels D and E , higher is the performance gain in termsof fidelity provided by the quantum trajectory implementedvia quantum switch. Differently, F CT decreases by increasing p and q . In the limit case of having p = q → , F CT → whereas F QS → . Remark.
In a nutshell, distributing the entanglement througha quantum switch provides a significant performance gain –in terms of fidelity of the teleported qubit at Bob’s side – foreach level of the noise affecting the quantum communicationchannels. More remarkably, by retaining at Bob’s side theentangled particles heralded by a |−(cid:105) -measurement of thecontrol qubit | ϕ c (cid:105) and by discarding the particles heralded bya | + (cid:105) -measurement, the quantum switch realizes a noiselessentanglement distribution through noisy channels .VI. C ONCLUSIONS AND F UTURE P ERSPECTIVES
In this paper, we investigated the utilization of the quantumswitch to face with the noise degradation introduced by the entanglement distribution within the quantum teleportationprocess.The theoretical analysis revealed that exploiting the possi-bility for a quantum particle to experience a set of evolutionsin a superposition of alternative orders is key to enhance thefidelity of the teleported qubit. Specifically, by utilizing thequantum switch, the teleportation is heralded as a noiselesscommunication process with a probability that, remarkably andcounter-intuitively, increases with the noise levels affectingthe communication channels considered in the indefinite-ordertime combination.These preliminary results are encouraging. Nevertheless, asubstantial amount of conceptual and experimental work hasto be developed in order to tackle the challenges and openproblems associated with the utilization of the quantum switchin the Quantum Internet. In the following, we outline some ofthese issues.
Quantum Switch vs Entanglement Distillation:
A wellknown technique to counteract the noise impairments affectingthe entanglement generation/distribution process is the entan-glement distillation (or entanglement purification) [37]. Ac-cording to this technique, if the contamination of the entangledqubits is below a certain threshold, it is possible to purifymultiple imperfectly entangled pairs into a single ”almost-maximally entangled” pair, albeit at the price of additionalprocessing. Hence, the entanglement purification exploits mul-tiple transmissions of imperfect entangled pairs to obtain asingle more entangled pair [22]. By comparing entanglementpurification and quantum switch from a communication net-work perspective, we can argue that the communication delayinduced by the former seems to be higher that the one inducedby the latter. However, further research is needed to quantifythis possible delay-advantage. Finally, it is worthwhile to note that the benefits provided by the quantum switch and theentanglement purification can be mutually combined – withthe quantum switch enhancing the fidelity of each imperfectentangled pair, hence reducing the number of imperfect pairsrequired at the destination to distill a maximally entangled pair– rather than constituting mutually-exclusive alternatives. Network Design Issues:
The possible photonic implemen-tation of a quantum switch sketched in Sec. III-B requiresthe availability of two communication links between Aliceand Bob, interconnected through a swapping device. Althoughmultiple physical implementations of the quantum switch havebeen proposed in literature [12]–[14], [20], [21] as discussedin Sec. I, further research is needed to face with the challengesarising with the quantum network design. As instance, whenthe quantum communication links are implemented throughoptical fiber links, the interconnection through the quantumswap requires a spatial proximity between the fibers, which inturns poses additional constraints on the network topology.
Channel Noise:
The assumption of channel D being the bit-flip channel and channel E being the phase-flip channel isnot restrictive, since other types of depolarizing channels areunitarily equivalent to a bit flip and a phase flip channel. Hencethe analysis can be easily extended by considering suitablepre-processing and post-processing operations, as noted in[1]. Nevertheless, further research is needed to quantify theperformance gain achievable when both the entangled qubitsare distributed via quantum switches through noisy channels.Finally, the question whether the quantum switch can beintegrated within the framework of quantum error correctiontechniques [38] is an open and interesting problem.A PPENDIX AP ROOF OF L EMMA | Φ + (cid:105) A ,is alreadyat Alice’s side, thus it does not need to go throughout anycommunication channel. Differently, the second qubit of theEPR pair | Φ + (cid:105) B needs to be distributed to Bob.By distributing | Φ + (cid:105) B through a quantum switch, the stateof the global system constituted by the entangled pair ρ e , thecontrol qubit ρ c = | + (cid:105) (cid:104) + | and the communication channels D and E can be described through the Kraus operators W ij given by: W ij = ( I ⊗ D i ) (cid:0) I ⊗ E j (cid:1) ⊗ | (cid:105) (cid:104) | + (cid:0) I ⊗ E j (cid:1) ( I ⊗ D i ) ⊗ | (cid:105) (cid:104) | , (19)being the first qubit of the entangled pair (virtually) travel-ing throughout an ideal channel represented by the unitarytransformation I given in Table I. By exploiting the tensorproduct properties, such as A ⊗ C + B ⊗ C = ( A + B ) ⊗ C and ( A ⊗ B )( A ⊗ B ) = A A ⊗ B B , (19) can be rewrittenequivalently as: W ij = I ⊗ (cid:0) D i E j ⊗ | (cid:105) (cid:104) | + E j D i ⊗ | (cid:105) (cid:104) | (cid:1) . (20) Since D and E denotes the bit flip channel and the phase flipchannel, respectively, their Kraus operators are given by [33]: D = ( − p ) I , D = pXE = ( − q ) I , E = pZ . (21)By substituting (20) and (21) in (5), and by exploiting againthe tensor product properties, after some algebraic manipula-tions it results: P(D , E , ρ c )( ρ e ) = ( − p )( − q )( ρ e ⊗ | + (cid:105) (cid:104) + |) + (22) + ( − p ) q ( I ⊗ Z ) ρ e ( I ⊗ Z ) ⊗ | + (cid:105) (cid:104) + | ++ p ( − q )( I ⊗ X ) ρ e ( I ⊗ X ) ⊗ | + (cid:105) (cid:104) + | ++ pq ( I ⊗ X Z ) ρ e ( I ⊗ X Z ) † ⊗ ( Z | + (cid:105) (cid:104) + | Z ) From (22), the proof follows by recognizing that Z | + (cid:105) (cid:104) + | Z = |−(cid:105) (cid:104)−| and that: I ⊗ Z = (cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) I ⊗ X = (cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| I ⊗ X Z = (cid:213) i , j (− ) j ⊕ | i j (cid:105) (cid:104) i ( j ⊕ )| (23)A PPENDIX BP ROOF OF C OROLLARY
P(D , E , ρ c )( ρ e ) at the output of the quantum switch is amixture of pure states {| + (cid:105) , |−(cid:105)} of the control qubit. As aconsequence – by measuring the control qubit in the Hadamardbasis – whenever the measurement outcome is equal to |−(cid:105) , theglobal state collapses into the state ρ QSe , |−(cid:105) reported in equation(24) at the top of the next page. In this case – happening withprobability pq – Bob receives the particle | Φ + (cid:105) B of the EPRpair without any error. In fact, ρ e can be recovered perfectlyfrom ρ QS e , |−(cid:105) by simply applying on ρ QS e , |−(cid:105) the unitary correctiveoperation ( I ⊗ X Z ) , defined in (23).Differently, when the measurement outcome of the controlqubit is the one corresponding to the state | + (cid:105) , the globalstate collapses into the state ρ QSe , | + (cid:105) reported in (25) at the topof the next page. In this case – happening with probability ( − pq ) – Bob cannot receives the particle | Φ + B (cid:105) withouterror. Nevertheless, also in this case as it will be shown inProposition 1, a considerable gain is assured with respectto the standard channel composition arising with classicaltrajectories. A PPENDIX CP ROOF OF C OROLLARY
D → E - the density matrix ofthe entangled pair ρ CT e at Bob’s side is given by: ρ CT e = E [D ( ρ e )] = E (cid:34) (cid:213) i = , D i ρ e D † i (cid:35) = (26) = (cid:213) j = , E j (cid:34) (cid:213) i = , D i ρ e D † i (cid:35) E † j . ρ QSe , |−(cid:105) = (cid:34)(cid:213) i , j (− ) ( j ⊕ ) | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j (− ) ( j ⊕ ) | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † (24) ρ QSe , | + (cid:105) = ( − p )( − q ) ρ e + p ( − q ) (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) ρ e (cid:34)(cid:213) i , j | i j (cid:105) (cid:104) i ( j ⊕ )| (cid:35) † − pq ++ ( − p ) q (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) ρ e (cid:34)(cid:213) i (| i (cid:105) (cid:104) i | − |( i ⊕ ) (cid:105) (cid:104)( i ⊕ ) |) (cid:35) † − pq (25) ρ ψ H ˜ ρ e X Z ρ t AliceBob ρ ρ ρ { ρ ij } i , j = , Fig. 8: Pictorial Representation of the Quantum TeleportationProcess in terms of density matrices.By substituting (21) in (26) and by accounting for (23), theproof follows after some algebraic manipulations.A
PPENDIX DP ROOF OF L EMMA ρ ∈ C × = ρ ψ ⊗ ˜ ρ e is an × matrixgiven by : ρ = ρ ψ ⊗ ˜ ρ e = (cid:20) ρ ψ ˜ ρ e ρ ψ ˜ ρ e ρ ψ ˜ ρ e ρ ψ ˜ ρ e (cid:21) . (27)As indicated in the main text, we denoted with ˜ ρ e theentanglement density matrix, since we do not formulate anyassumption on the scheme employed for the entanglementgeneration and distribution process as well as for the noiseaffecting the process. Specifically, ˜ ρ e can be either given by(7) in absence of noise or can be in some way affected by thenoise.The teleportation process starts with Alice applying theCNOT-gate of Table I to the pair of qubits at her side. In termsof density matrix, this is equivalent to consider an unitaryoperator U = C NOT ⊗ I × acting on the global state ρ , so thatthe Bob’s qubit is left unchanged: ρ = U ρ U † = ( C NOT ⊗ I × ) ρ ( C NOT ⊗ I × ) . (28) By accounting for the expression of the CNOT gate and bysubstituting (27) in (28), after some algebraic manipulationsone obtains: ρ = (cid:20) ρ ψ ˜ ρ e ρ ψ ˜ ρ e χρ ψ χ ˜ ρ e ρ ψ χ ˜ ρ e χ (cid:21) , (29)with χ ∈ R × equal to: χ (cid:52) = (cid:20) × I × I × × (cid:21) = χ † . (30)Then, as shown in Fig. 8, Alice applies the H-gate of Table Ito the state to be teleported. Hence the global state ρ afterthe H gate is: ρ = ( H ⊗ I × ⊗ I × ) ρ ( H ⊗ I × ⊗ I × ) † . (31)By accounting for the expression of the H gate, it results: H ⊗ I × ⊗ I × = √ (cid:20) I × I × I × − I × (cid:21) . (32)By substituting (32) and (29) in (31), after some algebraicmanipulations, one obtains equation (33) reported at the topof the next page, , with Γ ∈ C × and Λ ∈ C × defines as: Γ = (cid:20) ρ ψ I × ρ ψ I × ρ ψ I × ρ ψ I × (cid:21) , (34) Λ = (cid:20) ρ ψ I × ρ ψ I × ρ ψ I × ρ ψ I × (cid:21) . (35)Finally, as shown in Fig. 8, Alice jointly measures thepair of quantum states at her side, with chance offinding each of the four combinations , , , . Alice’smeasurement operation instantaneously fixes Bob’s quantumstate – regardless of the distance between Alice and Bob –as a consequence of the entanglement. However, Bob canonly recover the original state after he correctly receives thepair of classical bits conveying the specific results of Alice’smeasurement. This further step projects ρ on the subspacesdescribed by the operators Π ij ∈ R × = | i j (cid:105) (cid:104) i j | ⊗ I × , with i , j ∈ { , } .More in detail, let us suppose that the measurement outcome is ρ = Γ ˜ ρ e + Λ ˜ ρ e χ Γ ˜ ρ e − Λ ˜ ρ e χ ( I ⊗ Z ) Γ ( I ⊗ Z ) ˜ ρ e + ( I ⊗ Z ) Λ ( I ⊗ Z ) ˜ ρ e χ ( I ⊗ Z ) Γ ( I ⊗ Z ) ˜ ρ e − ( I ⊗ Z ) Λ ( I ⊗ Z ) ˜ ρ e χ . (33)the one corresponding to the state | (cid:105) . After the measurement,the global quantum state collapse into the state: ρ = Π ρ Π † Tr [ Π ρ Π † ] . (36)As a consequence of its definition, Π is equal to: Π = | (cid:105) (cid:104) | + | (cid:105) (cid:104) | (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:20) I × × × × (cid:21) × × o × (37)By substituting (37) and (33) in (36), and by exploiting theexpressions of Γ and Λ given in (34) and (35), after somealgebraic manipulations, it can be recognized that (36) isequivalent to: ρ = | (cid:105) (cid:104) | ⊗ (cid:16) ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) , (38)where we utilized the block-structure of the matrix ˜ ρ e in termsof the × sub-blocks { ˜ ρ e ij } i , j = , .From (38) – by tracing out the composite Alice’s state andby recalling that Tr C (C ⊗ D) = D Tr (C) – it results that thedensity matrix ρ t of the teleported qubit at Bob’s side, whenthe measurement outcome at Alice is equal to | (cid:105) , is givenby: ρ t = (cid:16) ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) . (39)Hence by accounting for Definition 2, the proof follows.With the same reasoning, the lemma can be proved fordifferent outcomes of the measurement process at Alice’s side.As instance, let us suppose that the measurement outcome isthe state | (cid:105) . By reasoning as above, it results: ρ = Π ρ Π † Tr [ Π ρ Π † ] = (40) = | (cid:105) (cid:104) | ⊗ (cid:16) ρ ψ ˜ ρ e − ρ ψ ˜ ρ e − ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) . From (40) – by tracing out the composite Alice’s state – oneobtains that the density matrix ρ t of the teleported qubit atBob’s side, after having applied the Z gate – is given by: ρ t = Z (cid:16) ρ ψ ˜ ρ e − ρ ψ ˜ ρ e − ρ ψ ˜ ρ e + ρ ψ ˜ ρ e (cid:17) Z . (41)A PPENDIX EP ROOF OF P ROPOSITION F QS of the teleported qubit at Bob’sside can be evaluated by averaging the conditional fidelity F QS ( θ, φ ) (cid:52) = (cid:104) ψ | ρ t | ψ (cid:105) on all the possible values of the qubit | ψ (cid:105) , i.e.: F QS = π ∫ π d θ ∫ π F QS ( θ, φ ) sin ( θ ) d φ == π ∫ π d θ ∫ π (cid:104) ψ | ρ t | ψ (cid:105) sin ( θ ) d θ d φ (42)By adopting a quantum switch for the entanglement dis-tribution scheme, from Corollary 1 the density matrix ofthe entangled pair ρ QS e at the output of the quantum switchcoincides with ρ e whenever the measurement of the controlqubit | ϕ c (cid:105) provides as outcome the one corresponding to thestate |−(cid:105) . And this outcome is obtained with probability pq . Asa consequence, by substituting ρ QS e = ρ e in (14) of Lemma 2,it results ρ t = ρ ψ . Hence, from (42), the average fidelity F QS |−(cid:105) given that the control qubit | ϕ c (cid:105) is measured into state |−(cid:105) isequal to .Conversely, whenever the measurement of the control qubit | ϕ c (cid:105) provides as outcome the one corresponding to the state | + (cid:105) , from Corollary 1 the density matrix of the entangled pair ρ QS e at the output of the quantum switch is given by (9).And this outcome is obtained with probability ( − pq ) . Asa consequence, by supposing without any loss of generalitythat A = , and by substituting the expression (9) of ρ QS e in(14), after some algebraic manipulations it results: F QS | + (cid:105) = π ∫ π d θ ∫ π (cid:104) ψ | ρ t | ψ (cid:105) sin ( θ ) d θ d φ == π ( − pq ) ∫ π d θ ∫ π (cid:2) ( − p ) − q ( − p ) sin ( θ ) ++ p ( − q ) sin ( θ ) cos ( φ ) (cid:3) sin ( θ ) d θ d φ. (43)and the proof follows by solving (43).A PPENDIX FP ROOF OF C OROLLARY ρ CT e when no quantum switch is adoptedis given by (10). As a consequence, by assuming without anyloss of generality that A = , and by substituting ρ CT e in (14) Whenever the indicator function of the measurement process at Alice isdifferent from A = , all the above analysis continues to hold, since it issufficient to single out the corresponding value of ρ t in (14). When the indicator function of the measurement process at Alice isdifferent from A = , all the above analysis continues to hold, since itis sufficient to single out the corresponding value of ρ t in (14). of Lemma 2, it results that the average Fidelity F CT when noquantum switch is adopted is given by: F CT = π ∫ π d θ ∫ π F ( θ, φ ) sin ( θ ) d θ d φ == π ( − pq ) ∫ π d θ ∫ π (cid:2) ( − p ) − q ( − p ) sin ( θ ) ++ p ( − q ) sin ( θ ) cos ( φ ) (cid:3) sin ( θ ) d θ d φ == − p − q + pq . (44)The proof easily follows by considering that (16) can beequivalently written in a compact form as: F QS = pqF QS |−(cid:105) + ( − pq ) F QS | + (cid:105) = − p − q + pq . 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IEEE CommunicationsSurveys Tutorials , vol. 21, no. 1, pp. 970–1010, 2019. Marcello Caleffi (M’12, SM’16) received the M.S.degree (summa cum laude) in computer scienceengineering from the University of Lecce, Lecce,Italy, in 2005, and the Ph.D. degree in electronicand telecommunications engineering from the Uni-versity of Naples Federico II, Naples, Italy, in 2009.Currently, he is with the DIETI Department, Uni-versity of Naples Federico II, and with the Na-tional Laboratory of Multimedia Communications,National Inter-University Consortium for Telecom-munications (CNIT). From 2010 to 2011, he waswith the Broadband Wireless Networking Laboratory at Georgia Institute ofTechnology, Atlanta, as visiting researcher. In 2011, he was also with theNaNoNetworking Center in Catalunya (N3Cat) at the Universitat Politecnicade Catalunya (UPC), Barcelona, as visiting researcher. Since July 2018, heheld the Italian national habilitation as
Full Professor in TelecommunicationsEngineering. His work appeared in several premier IEEE Transactions andJournals, and he received multiple awards, including best strategy award, mostdownloaded article awards and most cited article awards. Currently, he servesas associate technical editor for IEEE Communications Magazine and as editorfor IEEE Communications Letters and. He has served as Chair, TPC Chair,Session Chair, and TPC Member for several premier IEEE conferences. In2016, he was elevated to IEEE Senior Member and in 2017 he has beenappointed as Distinguished Lecturer from the
IEEE Computer Society . InDecember 2017, he has been elected Treasurer of the Joint
IEEE VT/ComSocChapter Italy Section . In December 2018, he has been appointed member ofthe IEEE
New Initiatives Committee . Angela Sara Cacciapuoti (M’10, SM’16) receivedthe Ph.D. degree in Electronic and Telecommunica-tions Engineering in 2009, and the