Quantum Communication via Spin Networks

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FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea Magistrale in Fisica della Materia

TESI DILAUREAMAGISTRALE

Quantum Communication

via

Spin Networks

Candidato: Stefano Chessa Matricola 504508 Relatore: Vittorio Giovannetti Anno Accademico 2017–2018

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Chapter 1 Intro

1.1 A brief introduction to Quantum Information

The concept of information is one those to which almost everyone of us is exposed since the very early age of our lives but goes often unnoticed, hid-den and confused among the multiple connotations it takes through daily tasks, no matter we’re at school or at work, reading a book or watching TV. The fact is that the ability to produce, store and transmit information has played a huge role in modern human civilisation: starting from the inven-tion of writing, the capacity to preserve knowledge has boosted the progress in agriculture, commerce and culture, reshaping the social, politic and eco-nomic structures that characterised human kind till about 5000 years ago.1

Recently even life itself has been seen as an emerging effect of information processing and replication, offering an alternative approach to the question of origin of life.2,3

Information appears to be more interesting than what it seemed at first sight.

But for what concerns this piece of work, the aspects of information that we most care about are those involving science, Physics in particular. It wouldn’t be accurate and maybe not even possible to set a specific day in which the idea of information appeared in Physics and Mathematics. We could cite the introduction by John Von Neumann in 1927 of the homonym entropy, but it was not contextualised in an informational frame as we use it today. Moreover we could object that if we consider the Von Neumann entropy we should do the same with the Boltzmann entropy, introduced in 1872. Said that the history game isn’t so funny after all, we can still name some of the corner stones that steered the wheel of that branch of Mathe-matics and Physics that later would have been called Information Theory. Chronologically, the first step toward modern formalization of computation was an article published in 1936 by Alan Turing entitled “On Computable Numbers, with an Application to the Entscheidungsproblem”.4 In this

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mark Turing solved one of 23 Hilbert’s problems, the Decision Problem, and incidentally introduced the Turing Machine, a formal tool aimed to emulate a generic process of computation given an input, an alphabet and a set of rules. The second milestone we must cite is a paper by Claude Shannon published in 1948: “ A Mathematical Theory of Communication”,5 _{which is}

considered the foundation of Information Theory. Here, to be extremely con-cise, is presented a formalisation of the process of communication between two parties together with the introduction of information entropy.

To be fair we should spend way more space to portrait all the decisive contributors who dedicate their work to the development of the subject but, again, the exposition of a detailed timeline isn’t our purpose. t But where is Physics?

More or less in parallel to the first steps of Information Theory, Physics was living a period of fervent evolution. The advent of Quantum Mechanics opened the path to a huge and brand new framework that shed light upon unknown aspects of Nature at the microscopic scale and that turned out to be incredibly powerful in terms of predictions and precision. Still, how infor-mation and Quantum Mechanics are even related? Long story short, one of the main reason humans were pushed to improve their computational capac-ity was the desire to understand Nature, and in order to do it, to simulate it. The point is that quantum mechanical systems appeared to be really not manageable from this point of view. If the reader had ever had the opportu-nity to attend an introductory talk or seminar about Quantum Information or Computation, he/she will probably be familiar with the following (by now super mainstream) example: the simulation of a N spin-1₂ particles state, in the relative Hilbert space, needs at least 2N _{parameters to be stored.}

Assuming to dedicate d bits per parameterwe’ll need d 2N _{bits. The most}

optimistic (but never realistic) case would take d 1. In this situation, for N 40 it makes 3.2 1013_{bits (}_{4TB), for N 80 makes 10}12_{TB, about}

104 _{times the entire capacity of information storage of humankind in 2007.}6

As we said: not manageable. The first description of the issue was given by Richard Feynman in 1982 in an article, “Simulating Physics with comput-ers”,7 _{where he hypothesised the convenience of the use of quantum systems}

to simulate in “reasonable” time other quantum physical systems instead of standard computers. This was the first rationalisation of the chance to taking advantage of quantum properties of Nature to perform calculations not accessible to classical computation. And there’s more. The eventual-ity of a “Quantum Computer” able to outperform classical ones exposed a breach in the theoretical construction of computation. Going back for a moment to the work of Turing, together with the Turing Machine came the

_{Since parameters are complex number we necessarily have to do a truncation, but}

we can hypothesize that the computation can be carried on within reasonable margin of errors.

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1.1. A BRIEF INTRODUCTION TO QUANTUM INFORMATION 7 Church-Turing Thesis, a hypothesis about the computability of a given func-tion, stating roughly that a function can be said computable if it’s Turing computable, namely if there exists a Turing Machine able to compute it. So, assuming Turing Machines to be classical, the existence of a different machine capable to solve a problem inaccessible to them would violate the Church-Turing Thesis. The Thesis was then modified by David Deutsch in 19858 introducing the Quantum Turing Machine and leading to a new statement: every physical process can be simulated by a Quantum Turing Machine. Consequently the horizon of computability was enlarged, but to what extent? Is it possible to take advantage of “quantumness” of Nature to grow our computational power? These are questions still unanswered. Start-ing from the early ‘90s the first quantum algorithms emerged, showStart-ing that quantum mechanics was useful to easily solve problems hard to be solved classically. Most famous the one by Peter Shor, who proposed a quantum algorithm to solve the factorization problem.9 _{Unfortunately there is no}

di-rect proof at the moment for the so called “Quantum Supremacy”, that is a substantial gap in computational performance between classical and quan-tum computers guaranteeing the latter ones to be able to solve efficiently “hard” problems. Very recently however it has been shown10 that there’s at least one example of a problem not solvable by classical computers and accessible to quantum computers. In this sense, from the theoretical point of view, there are no certainties but moderate clues that let us hope for a useful speed up, at least for certain classes of problems, provided by a quan-tum computer. The other side of the coin we haven’t considered yet is the practical one. We’ve being talking about quantum computers for decades, where are they? Nowhere, at the moment, and this is one of the reasons that has fuelled some scepticism among detractors about the effective feasibility of such kind of devices. The correct and reliable manipulation of information in quantum systems in a stable and coherent way needs extremely precise control and efficient isolation from environment, features often not simply available at small scales. Indeed this facet is probably the main focus of current research in the field, also because results are coming. If we cannot say that we have at disposal a functional quantum computer, we still can

_{We’re being voluntarily vague and inaccurate here about the whole topic of}

com-putability because it’s not our matter of interest. There are actually two versions of the Church-Turing Thesis: the first one (Weak ) is the one we reported in the text, the second one (Strong ), which would be violated by a quantum computer, states that a function is Turing-computable if there exists a Turing machine able to compute it efficiently. Here we would need to talk about how the efficiency is defined but we settle for the following: a Turing machine is efficient if the resources needed for the computation (usually time or space) scale up polynomially with the size of the input. Anyway, a more satisfying cover-ing of the subject can be found on any introductory textbook about Complexity Theory and, for what concerns Quantum Computing aspects, Quantum Complexity Theory.

_{Actually they proved the separation between complexity classes relative to a specific}

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strongly affirm that substantial steps toward the goal have been made: in less than two decades we passed from a proof of principle experiment with a 7-qubits computer11_{to Google’s 72-qubits device this year}_{. Hence there are}

reason to be optimistic and expect functional devices, maybe over a limited set of problems, in the near future.

The purpose of this compressed introduction was to set a wider context to the work developed in this thesis and, again, it was not intended to be deep or particularly accurate. Our interest will be focused on the aspect of communication and in particular how specific physical systems can be used as communication lines in the landscape of Quantum Computation and Communication. Keeping this in mind we’ll briefly cite David Di Vincenzo’s work and then switch to the notions about communication that we need.

_{The threshold estimated by Google to exhibit the Quantum Supremacy on their}

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1.2. REALIZATION OF QC AND DI VINCENZO’S CRITERIA 9

1.2 Realization of Quantum Computers and

Di Vincenzo’s criteria

We’ve seen how Quantum Computers could reveal themselves really useful and powerful tools in the realm of computation. We did not mention though how physicist intended to engage with the issue of their realization. In general we could ask: how such device should be built to function? What kind of features should characterise it?

In 2000 David Di Vincenzo addressed these questions in a paper entitled “The Physical Implementation of Quantum Computation”,12_{where exposed}

some basic requirements a quantum computer should fulfil, later named Di Vincenzo’s criteria. We report them here.

1. A scalable physical system with well characterized qubits

A qubit is a two level system, so we have to ensure the physical system we use, whose parameters need to be known, to be two-level. If it’s not, we have to ask to suppress possible jumping to undesired states. To perform non trivial computations a set of several qubits is necessary. 2. The ability to initialize the state of the qubits to a simple fiducial state,

such as S000...e

Typically to execute any computation it’s necessary to know the initial state of the registers hosting the data processing. To do so we have to be able to initialize those registers at least to one known state. 3. Long relevant decoherence times, much longer than the gate operation

time.

Just like in classical computation, if during the data processing the register undergoes corruption mechanisms the final result will be use-less. To avoid this issue it’s absolutely necessary for the state of the register to be stable for a time sufficiently long to be reasonably sure that no errors occurred during the stages of computation. The quan-tum state should then remain coherent long enough to be processed, therefore longer than the gate operation time.

4. A “universal” set of quantum gates

If we want a device capable to perform universal computation we’ll need a set of universal gates, i.e. a finite number of feasible and func-tioning gates whose combination can reproduce or approximate within a certain threshold any possible transformation.

5. A qubit-specific measurement capability

To retrieve the result of the computation we’ll need to read out the output registers. In the quantum mechanical case this translates to a measurement of some of the qubits or all of them.

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6. The ability to interconvert stationary and flying qubits

We would expect that a more or less complex calculation would need several steps to be carried out, so an input qubit would pass through a number of gates before it could be measured. If we imagine gates like stationary like in a circuit, then the qubit will have to move (flying qubit) from gate to gate and somehow interact with them in order to transmit the information they transport and let that information be processed.

7. The ability faithfully to transmit flying qubits between specified loca-tions

Clearly the displacement of those qubits has to take place without loss or corruption of that information they are supposed to carry.

All these prescriptions are totally general: they compose a intuitive and reasonable guide toward the construction of an operational quantum device but tell us quite little about how effectively we should choose some physical systems to be our qubits or engineer others to be quantum gates. But again it’s not the topic we’ll care about, also considering its huge extension in research and literature.

As we mentioned before, our interest will be focused on the aspects of communication via quantum systems. From this perspective, criteria num-ber 6 and 7 appear to be the most stimulating among the ones Di Vincenzo enumerated. When he wrote his paper the main applications of quantum communications were those aiming to prove that photons could be success-fully employed to implement long distance quantum communication, primar-ily adopted to achieve quantum key distribution tasks. It’s not surprising then that when thinking of communication lines inside a potential quantum computer he came up with “flying qubits” and did not consider different so-lutions. But if we think about it, keeping in mind that generally a physical realisation of such a device would be preferred to be as compact as possible because of isolation from external environment, the delegation of the com-munication between various components of the computers to photons could not be the most practical solution. One could think, going on with the parallelism with classical computation, to implement stationary communi-cation lines through which send quantum states exploiting some appropriate protocol. These kind of lines could be made of the same kind of systems composing the quantum gates, facilitating the connection and the transfer of states and, above all, they could be fixed once for all at the moment of fab-rication and properly calibrated, limiting in this way the resources needed to control and monitor the entire system; resources that could then be spent elsewhere. One of the proposed applications of this idea is the solid state

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1.2. REALIZATION OF QC AND DI VINCENZO’S CRITERIA 11 platform, which presents a number of similarities to what is today’s electron-ics. In many of these models, qubits are represented by spins while gates by proper interactions to reproduce the requested Hamiltonian. Qubits are fixed and quantum states are transferred through “quantum wires”, consist-ing themselves in chains or similar structures of spins. The state is somehow encoded into one of the ends of the chain, it’s transported by the evolution of the whole system and in some way is recovered at the other end.

With this discussion we are getting closer to what will be the main topic of the thesis. It would be interesting to give a deeper look into the topic of Communication and Information Theory, together with the quantum equivalent version but, since we won’t make use of it and for the sake of brevity, we are not going to. Nevertheless the interested reader can find a streamlined introduction abou the topic for instance on the popular Nielsen and Chuang’s book.15

We are ready then to get started to the main subject of the thesis, i.e. why and how we can take advantage of spin systems to perform quantum communication.

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1.3 Spin Quantum Networks

In the previous sections we introduced the broad subject of Quantum Infor-mation, trying to convey an approximative but faceted prospective, covering the essential history and thoughts that shaped and brought it to us. In section 1.2, talking about Di Vincenzo’s criteria, we anticipated that the ability to prepare, transmit and receive faithfully quantum states is an unconditional necessity to be able to perform those tasks of information processing and communication that physicists in the last decades have been aiming at. As we said Quantum Computing is the primary target towards which most efforts are put into, but other applications such as long-distance quantum communication have raised considerable interest in recent years. In this regard, experiments conducted via optical fiber17 or satellite trans-mission18,19 _{have shown that photon-based quantum communication can be}

already reliably accomplished and scaled. Because of that it’s not unrea-sonable to expect these configurations to find shortly extensive application in the domain of quantum cryptography, that makes key distribution safer, virtually airtight.

But when debating about quantum computing, photons don’t appear to be the most immediate and handy option. As a matter of fact, even if the codification of information on photons isn’t difficult (as an example in po-larization states), a quantum processor would need these photons to interact somehow, for instance in a 2-qubits gate. Unfortunately, since photons don’t interact in standard conditions, their employment in a performing quantum computer becomes less viable, even if in this this sense some proof of prin-ciple has been proposed.20 _{The other aspect relevant to such evaluation is}

the miniaturization: photons are exceptional means to perform log-distance quantum communication, but in computation, due to resources, manage-ability and isolation requirements, the processing elements will be preferred in the most compact configuration affordable. Photons lose in this way their principal asset inside a quantum computer, although they remain probably the best medium to connect different quantum computers.

So if we want to depict how a quantum computer could be realised, keeping in mind Di Vincenzo’s prescription and the practical necessities that emerge in every experimental makeup, we’d probably imagine a setting where the input is coded on qubits of some kind, which are put in contact with gates forming the processing stage after which the output is read. Now, the actual nature and disposition of these gates obviously will depend on the physical means we choose to take advantage of. The approaches most discussed in the literature can be summarized in the following sub-areas:

Atoms or ions in optical lattices or traps. Solid State systems.

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1.3. SPIN QUANTUM NETWORKS 13 Others: NMR, photonic systems, hybrids.

The most promising one, at least for useful implementations in the near term is arguably the one involving the use of solid state systems, which com-prehend among the others superconducting qubits, electron spins in quan-tum dots and defects in structures like diamonds, silicon-based matrices and others. We are not going to detail the specifics of these configurations, what we need to know is that the structures assigned to the information process-ing are posed on/in a bulk of a convenient material and there they are fixed. Then, if we imagine the computational apparatus as an ordered combina-tion of gates (a circuit), we’ll have to be able to connect them in a way that let us transfer quantum states in a reliable manner to perform properly all the transformations expected. Consequently the immediate solution to the issue appears to be, like in standard electronics, the direct fabrication of lines of communication between quantum processors, registers and other elements. These “quantum wires” will be made in their turn of opportune quantum systems e.g. atoms or spins that somehow will be implanted in the compound and let interact. A configuration of this kind can suffer some limitations, the principal one is that in a realistic setting experimentalist will neither have enough resources and likely nor the capacity to control every element of such a wire: on a micrometric scale, depending on the interaction set, the realisation of a single wire could require from tens to thousands of spins or atoms. This is the reason why it is worth studying this type of system: if we managed to characterize a composite structure of many elements, able to transfer faithfully quantum states without the need of constant supervision and manipulation by the experimenter, we could, if in possession of a suitable fabrication procedure for it, focus the resources to the computational components.

In this thesis we’ll concentrate on spin systems employed as communica-tion lines. Specifically we’ll see whether and how well many body systems like networks of spins (spin networks) can transport quantum states from one end to another and how the simplest models can be enriched or modified to better the performances. In the next chapter we present an overview of the specific subject, dealing with general schemes that has been proposed in the last decade to address the issue of quantum communication through spin systems. We’ll make use of these concepts as a basement for our successive work.

_{To my knowledge all those institutions claiming to be able to perform non trivial}

quantum computation in the next future (Google, IBM, D-Wave, Microsoft, Intel) use quantum computer based on superconducting qubits.

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1.4 Preliminary conclusions and outlook

Now that we have presented the landscape in which we’re going to move, we give a brief outlook of what will follow.

We’ll study some of the implementations thought in the last years for quantum communication on spin networks, mainly spin chain models be-cause of their simplicity, a couple of very simple protocols and we will try to understand how to evaluate their quality, acknowledging limitations and relative solutions.

Then we’ll start the first manipulations: we want to understand how our interaction with these system can influence the communication, in positive or negative. We’ll see for instance that an excessive monitoring can be bad while some amount of control over the system can help . We’ll also try in a basic characterisation of a typical time of propagation in a next neighbour chain. This last aspect in particular will lead to interesting questions that will shift the topic to Chapter 3 subject: Lieb-Robinson bounds in spin systems.

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Chapter 2 Quantum Communication &

Spin Systems

2.1 Introduction

We said that spin networks and spin system in general can be employed as communication facilities to perform transfer of quantum states and quantum information. As one can imagine and as we anticipated in the previous chapter the communication features will depend on the physical properties of the system at disposal and on the procedures put in place by users. In this regard we want to examine what kind of model are more suited to carry out quantum communication and understand how to manipulate it. The chapter is structured as follows:

in Sec. 2.2, following [21], we’ll introduce the spin chain, simplest model imaginable to communicate, and the main typologies of Hamil-tonian to describe them. We’ll present some protocols and introduce the fidelity as a figure of merit to test the transfer quality.

In Sec.2.3we’ll analyze a protocol where the receiver owns a memory and we’ll discuss how his operations performed affect the effective state transfer.

In Sec. 2.4, in a system with controlled and time dependent interac-tions, we’ll consider whether and how the right management of the system parameters can improve the communication performance.

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2.2 Quantum Communication through spin chains

We introduced in Sec.1.3the issue of quantum communication, i.e. the pro-cess of transport of a quantum state between different regions of space. We also suggested a relevant application in the realm of quantum computation and we’ll stick to that to facilitate the exposition letting us exploit examples and figures when we’ll need some practical backing.

This said, we can start going into more detail. We are in possession of two quantum registers A and B composed by a certain number of qubits. We aim to make them communicate: we could think to take another qubit, put it in contact with the first register, then move it close the other one and put the qubit in contact with it (Fig. 2.1).

Figure 2.1: Flying qubit (red) connecting registers.

But this would be a “flying qubit” (see Sec. 1.2, point 7) and we want a fixed intermediary. So we take several qubits, dispose them in line and connect the two registers. In parallelism with digital electronics, we could say to have built a “quantum bus” (Fig. 2.2).

A

B

1 2 3 4 5 6

Figure 2.2: Quantum bus connecting distant registers.

Assuming at the moment the qubits not to be interacting, we haven’t specified yet how the state is transferred from qubit to qubit. The most ingenuous idea would be to equip the system with a controllable interaction, for instance a field, which is switched on and off in order to make “talk” only two qubits at a time. In the system pictured in Fig. 2.2 we’d have: (A,1)¨(1,2)¨(2,3)¨(3,4)¨(4,5)¨(5,6),¨(6,B). But in Sec. 1.3 we already

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2.2. QUANTUM COMMUNICATION THROUGH SPIN CHAINS 17 discussed the fact that such control over a large system wouldn’t be too much realistic and expensive in terms of resources. So it would be better if we could reach a different solution requiring lower control capacity, with permanent coupling between qubits: spin chains can represent one of these solutions.

2.2.1 Spin Chains

A spin chain is a monodimensional collection of spins. Since we’re pushing forward a description in terms of qubits, i.e. 2 levels systems, when talk-ing about spins we’ll consider only spin-1₂ systems and from now on words “qubit” and “spin” will be considered completely interchangeable, unless differently specified.

Clarified what the system is, we have to characterise its physical behaviour, namely specify the Hamiltonian governing it. One of the most general model for spin system is represented by the exchange interaction Hamiltonian:

Hij JijSi Sj, (2.1)

where the summation over indices is assumed. Si, Sj represent the spin

operator for i-th and j-th particle, Jij the coupling strength between them

and Si Sj SxiSjx S y iS

y

j SizSjz. Since we deal with spin-1

2 we have that

Sα

i σiα, with σiα Pauli matrices:

σx 0 1 1 0 σ y 0 i i 0 σ z 1 0 0 1. (2.2) Therefore we can see that with this Hamiltonian coupling coefficients are fixed, not necessarily uniform and in principle every particle can interact with each other, like in Fig. 2.3.

A simplification of this model that we’ll take advantage of is expressed by the XY interaction, in which for each particle one component, z by convention, of the spin vector is null:

HXY_ij JijSixS x j S y iS y j. (2.3)

A further simplification we’ll adopt later is the next-neighbour approxima-tion, in which each spin interacts only with adjacent spins:

Hnn Q

i

Ji,i1Si Si1. (2.4)

It’s not unusual in the literature to face situations in which a magnetic field is present, maybe to force the spins’ orientation like we’ll see in sub-section 2.2.2. In that case assuming a static field Bi A 0 for each site along

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Figure 2.3: Interactions between spins are indicated by arrows, only first and second neighbours interactions are portrayed.

HB Q

ij

JijSi Sj Q i

Biσiz. (2.5)

It worth noticing that all the Hamiltonians listed above could be also applied to more complicated structures other than spin chains. If we imagine a network where at each node (or site) lies a spin, we can attach an index i to identify the site and safely generalize Hamiltonians in Eq. 2.1, 2.3, 2.4

and2.5.

2.2.2 A first protocol

We established in the previous subsection some of the options to describe the dynamics of our spin system. We’re interested in understanding how this dynamics helps with the transfer of a quantum state. We imagine now to have our chain of N spins and we assume to be able to initialise it in a known state. Assuming for the sake of simplicity to be dealing with a ferromagnetic chain, i.e. Jij @ 0, the easiest configuration would be where

all the spins are aligned in a definite direction, condition achievable for instance with the application of a magnetic field. The chain is then in its ground state. Each spin is oriented towards the same direction, selected by the field, that we assume without loss of generality to be downward. Hence every spin is in the pure state Se, which in the computational slang we

_{Technically Hamiltonian}_2.4_{should be rewritten in terms of couples}_{`ije such that the}

distance between sites i and j is 1: Hnn P

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2.2. QUANTUM COMMUNICATION THROUGH SPIN CHAINS 19 express as S0e while for Se we’ll have S1e. The overall chain state will then be: S0e a S0e a a S0e S000e (Fig. 2.4).

Figure 2.4: Chain in the ground state, qubits aligned in the same direction.

Two parties, Alice (A) and Bob (B) to be original, decide to use the chain to perform quantum communication. A, wants to send B a generic stateSψe α S0eβ S1e she prepared on her qubit. To do so she places it near the first end of the chain (Fig. 2.5) and lets the composite system interact.

A

B

Sψe

Figure 2.5: Alice places her qubit in a generic state near the chain.

The chain will evolve unitarilyaccording to the Hamiltonian governing the system. Bob has to wait for a certain time t0, depending on the

Hamil-tonian and the size of the system, until the qubit lying at the end at his disposal is in a state Sψe equal or sufficiently close to Sψe (Fig. 2.6): now, on the last qubit, he has access to the state sent by Alice.

It’s clear then that for this protocol to succeed Bob will need: – A figure of merit to quantify the “closeness” of states. – The ability to compute it’s evolution in time.

_{Here and later, if not specified, we assume that no noise affects the system so that}

the evolution is exclusively specified by the “closed” Hamiltonian.

_{It has to be taken account of the Hamiltonian term linking Alice’s qubit to the chain.}

Alternatively we can assume that Alice encodes the desired state directly on the first spin of the chain.

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A

Sψ

_B

e

Figure 2.6: The chain evolves and Bob collects the state after an opportune amount of time.

2.2.3 Fidelity

We learned in the previous section a basic protocol of quantum state transfer where Alice “inject” her state Sψine α S0e β S1e in the first qubit of the

chain and Bob can collect an output state Sψoute waiting for the chain to

evolve. Bob needs an instrument to evaluate whether there’s resemblance between his state and the one sent by Alice: such instrument is fidelity. We can think of a figure of merit that owns its maximum forSψine Sψoute and

its minimum forSψine Ù Sψoute. A solution is found if we think to Bob’s qubit

density matrix ρout. As we said, the last qubit, like other ones, is initialised

in the state S0e so ρout0 S0e `0S. After a time t0 and unitary evolution

we’ll have ρoutt0. To verify how much this density matrix “overlaps” with

the stateSψine we can simply compute the expectation value of ρoutt0 over

Sψine. Hence we define the fidelity F :15

F `ψinS ρoutt0 Sψine . (2.6)

Since any density matrix is positive semi-definite we have that 0B F B 1. It follows from the definition that if Sψoutt0e Sψine then ρoutt0

Sψine `ψinS and F 1, while if the states are orthogonal F 0. This definition

satisfies the requirements we wished for above.

If it’s clear that higher the fidelity, the better the communication, still from this definition it’s not evident how high it has to be for a quantum channel to be said good. There is no unambiguous answer to this question, in the sense that the possible application determines an acceptable value. A threshold is given by the maximum fidelity achievable with classical com-munication, which is f 1

2 without sharing of entanglement between the

parties an F 2₃ with entanglement, also known as the no-cloning limit22 from the no-cloning theorem:23 _{it goes without saying that a quantum}

chan-nel outperformed by a classical one is of little use.

Now we can show an explicit expression for the fidelity for Hamiltonian in Eq.2.5 the most generic we listed.

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2.2. QUANTUM COMMUNICATION THROUGH SPIN CHAINS 21 Recalling the protocol in subsection 2.2.2, we have the chain with N spins initialised in the overall state S000e, that for the sake of concision we call hereS0e, and A has access to the first qubit while B to the last one, in order to perform their operations. A “writes” her state Sψine on the first qubit.

Exploiting the Bloch sphere parametrization for qubit states we can write Sψine cos

θ 2S0e e

iφ_sinθ

2S1e . (2.7)

The overall state will then be written as: SΨeC cos

θ 2S0e e

iφ_sinθ

2SÑ1e , (2.8)

where SÑ1e represents the overall state with S1e on the first spin and S0e elsewhere (S100e), and generalizing, SÑje will be the state with S1e on the j-th qubit andS0e on others. This representation becomes helpful when we consider that Hamiltonian HB _{in Eq.} _2.5 _{leads to total spin conservation}

along the z direction, i.e.:

HB,

N

Q

i

σz_{i 1} 0. (2.9)

So, considering that except for Alice’s qubit that is in a superposition of S0e and S1e the chain is in the S0e state, at every time of evolution at most only one among all spins will be in the stateS1e, accordingly the total state will be a superposition ofS0e and SÑje states. From the total spin conservation follows also that the stateS0e does not evolve: in some way we can say that the properties of communications are related only to the capacity of the system to propagate the “excitation”S1e.

Now, being eiHBt_{the unitary evolution operator} _{from time 0 to t, we can}

write SΨte_C: SΨteC eiH B_t SΨeC cos θ 2S0e e iφ_sinθ 2 N Q j 1 `ÑjS eiHB_t SÑ1e SÑje cosθ 2S0e e iφ_sinθ 2 N Q j 1 fjt SÑje . (2.10)

To extrapolate the density matrix for B’s qubit we have to perform the partial trace over the remaining qubits of the chain with respect to the overall pure state SΨte_C, explicitly:

_{The proof of this statement is straightforward applying Pauli matrices commutation}

relations.

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ρoutt Tr1,2N1 SΨteC`ΨtS . (2.11)

Using the fact that `0SÑje 0 and `ÑjSÑke δjk we obtain:

ρoutt P t Sψoutte `ψouttS 1 P t S0e `0S , (2.12)

where we defined: Pt cos2 θ 2 SfNtS 2_sin2 θ 2 Sψoutte 1 » Ptcos θ 2S0e e iφ_f Nt sin θ 2S1e . (2.13)

Looking in Eq.2.13at the shape of Pt and Sψoutte is trivial to see in

Eq. 2.12that for fNt 1 we have P t 1 and ρout Sψine `ψinS. So for a

specific time ¯t and certain Bi in the Hamiltonian, F 1 could be achievable.

In general we have:

Ft `ψinS ρoutt Sψine

Pt S`ψinSψoutteS2 1 P t S`ψinS0eS2,

(2.14) and averaging over all the possible initial statesSψine on the Bloch sphere

(pure states): `F te 1 4π S Fθ, φ, tdΩ RefNt 3 SfNtS2 6 1 2. (2.15) Again, we note that adjusting the magnetic fields Bi we in principle

could obtain fNt 1 (or such that argfNt is a multiple of 2π) and

consequently an average fidelity of 1.

2.2.4 Limitations and proposed solutions

In subsection 2.2.3 we introduced the “instrument” represented by fidelity and we computed it to quantify the transfer efficiency for the quantum com-munication protocol exposed in2.2.2. From Eq.2.15, as we could expect, a dependence from the Hamiltonian arises in the form of the transition am-plitude fNt. What naturally comes next is to ask if any system or model

owns equal capability of good transfer, in other words what is the maximum fidelity attainable for each configuration within a definite amount of time. Again, as we could imagine, different systems lead to different fidelities. We want to show here that the protocol we used till now isn’t satisfactory in many cases and we’ll mention some improvements that allow a better (if not perfect) state transfer.

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2.2. QUANTUM COMMUNICATION THROUGH SPIN CHAINS 23 Following the exposition in [28], we consider an XY next-neighbour in-teraction model of an unmodulated chain, i.e. with uniform couplings:

J 2 N Q i σ_ixσ_ix₁ σy_iσ_iy₁, (2.16) where J represents the unmodulated couplings.

Without entering the details, authors in [28] find a condition for symmetric systems (and an unmodulated chain described by Eq. 2.16 is one of those) to be satisfied to permit perfect state transfer. Then in the paper they show that for this kind of spin chain perfect state transfer is achievable only for a number of spins N B 3.

So for NC 4 and unmodulated chain we can’t have F 1. Anyway, as we’ve already discussed at the beginning of2.2.3, it’s matter of interest at least to know how far from 1 the transmission could hit and also whether it exceeds the classical limit of f 2

3. From Eq.2.15we see that through the numerical

estimation of the maximum of SfNtS it’s possible to find the maximum for

the fidelity. In [21] author shows the numerical results for the highest fidelity value within a reasonably long time (4000~J units) evaluated in chains with N varying from 2 to 80 (Fig.2.7).

We can see that the behaviour is not totally predictable but there’s a general trend that shows a decrease of the maximum fidelity with the growth of N . For big N the quantum channel moves towards a classical channel’s performance in terms of fidelity.

Configurations for perfect transfer

The results showed in fig. 2.7 aren’t encouraging: for long chains fidelity drops quite far from the unity value but also from those thresholds that often define near-perfect transfer (see2.2.3, 95% or 99%). Luckily the issue has been addressed and some solutions have been found. We just enumer-ate some of them because their technical discussion isn’t our purpose, if interested the reader will find references, what is relevant is to know that modifications of the protocols or the system under exam can bring consid-erable improvements.

Engineered chains.

The modulation of coupling constants in an appropriate fashion can improve the communication features. For instance in [28] a method is presented to achieve perfect state transfer in a next-neighbour XY N spins chain imposing Jl,l1

»

lN l. Wave-packet encoding

If sender and receiver have access to more than one qubit, the state can

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Figure 2.7: Maximum fidelity numerically evaluated for different N within an evolution period of 4000

J , with J energy unit of the coupling constant. The straight

line indicates the value of the classical limit F 2₃. Taken from [21].

be encoded into a “Gaussian-like” wave-packet that suffers no or little dispersion along the chain. In [29] authors suggest to encode the S1e state into a truncated Gaussian stateSGj0, ke P

j

e

jj02

N23 eikjSÑje.

Dual rail protocol

Instead of using a single chain we can employ two of them at the same time enhancing the transfer. In [25] is presented the first dual rail protocol: two chain are posed in parallel, Alice and Bob have access to respectively the first and last qubits of each chain. Alice encodes her state in a state of the kind αS01eβ S10e, while Bob waiting enough time and doing the right operations can retrieve the state with fidelity 1.

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2.2. QUANTUM COMMUNICATION THROUGH SPIN CHAINS 25 If the receiver has the chance to store information can use it for a better state reconstruction. In [26] authors show that if Bob periodically swaps the last qubit of the chain with another one initialised in S0e, doing unitary operations over the set of qubits he cut from the chain he can recover the state sent by Alice.

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2.3 Quantum Zeno Decoupling

Now that we have explored some of the principal aspects regarding quantum communication using spin systems we can try to get into more detail.

In this section we will focus on a specific scheme, already and quickly mentioned in the previous section, the receiver with memory one. In partic-ular we’ll assume the receiver to be able to act on his qubit and we’ll analyze the possible consequences of a high frequency rate of interaction receiver-network on the spin receiver-network dynamics and on the transfer efficiency. 2.3.1 High frequency and decoupling

We are a in the situation in which Alice and Bob have the usual chain com-posed of N spin-1₂ particles that we assume interacting through one of the Hamiltonians listed in Subsec.2.2.1that will be identified simply by H. The choice of Hamiltonian here could be more general, we could even not require H to conserve the overall spin.

As already described in2.2.2, A encodes her message on the first qubit and B measures or operates on the last qubit. Regarding communication by means of spin chains there exist several protocols taking multiple operations to be done on the qubit before being able to extract the necessary informa-tion.21,25–27 _{We assume to be in the situation described in [}₂₆_] _{i.e. B has}

at disposal a quantum register employed as a memory, consisting in a large number of qubits initialized in theS0e state which neither interact with each other nor with the chain (when they aren’t swapped). He periodically swaps the last spin of the chain with one of the memory, in order to be able to reconstruct A’s state with some information processing technique later (Fig.

2.8).

Figure 2.8: Schematisation of the swapping process.

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2.3. QUANTUM ZENO DECOUPLING 27 Assuming Bob to be in possess of an arbitrarily large memory, i.e. num-ber of spins to swap, and wondering which strategy could allow him to minimize the transfer time, we could intuitively think that a fast regime of swaps would be the best option: he shrinks the time of interaction, but collecting enough “samples” he’ll be finally able to reconstruct the target state. This comes to be false.

We don’t care about the reconstruction problem here, but we want to show that if the period of the swap, which from the point of view of the chain system acts like a projection on S0e, is too short, then B can’t retrieve any information about the overall state and consequently about Alice’s message. So A uploads her stateSψe_Aon the first qubit, others qubits are initial-ized toS0e: we express the overall state as Sψ00e_ACB Sψe_AS0...0e_CS0e_B (the subscripts A and B indicate evidently Alice’s and Bob’s spins respectively, C represents the remaining spins of the chain between A’s and B’s ones). The chain is let evolve under Ut eiHt and, with period τ , B swaps his qubit. Then we have that the state evolves as:

ρtACB Ut Sψ00eACB`ψ00S Ut, (2.17)

and the swap produces:

ρts_ACB trBUtSψ00eACB`ψ00SUt a S0eB`0S, (2.18)

where we called ρts

ACB the overall state after the swap.

If we define τH as an “effective” time of signal propagation related to the

evolution induced by H, in the high frequency swapping regime we have τ P τH. It follows that we can write Uτ eiHτ 1 iHτ oτ2, and

substituting above:

ρts_ACB TrB1 iHτ oτ2Sψ00eACB`ψ00S1 iHτ oτ2 a S0eB`0S.

Keeping only the linear order in τ :

ρts_ACBTrBSψ00eACB`ψ00S iτHSψ00eACB`ψ00S

iτSψ00eACB`ψ00SH a S0eB`0S.

(2.19) Let’s take the trace over B: in the first term inside the trace in Eq. 2.19

looking at how Sψ00e_ACB is built, only the projection overS0e_B is not 0. In particular we have that:

TrB Sψ00eACB`ψ00S Sψ0eAC`ψ0S . (2.20)

_{Here neither we’re interested in the specific technique showed in [}₂₆_{] nor how in general}

it could be done.

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Furthermore, considering the remaining two terms, we can write HSψ00eACB Q Ñj α_ÑjSÑj, 0eAC,B Q Ñj β_ÑjSÑj, 1eAC,B, (2.21)

where we’ve introduced the notation: SÑj, 0eAC,B Sj1...jN1eACS0eB with

ji 0, 1. Then, projecting on S0eB:

B`0SHSψ00eACB Q

Ñj

α_ÑjSÑjeAC. (2.22)

This vector within the restricted system AC can be written as H_Sψ0e

AC:

His a Hamiltonian acting over the Hilbert spaceHAa HC spanned by the

two subsistems A C obtained taking the coefficients α_Ñj from H.

H

01

H

10

H

11 £¨¨ §¨¨ ¥SÑj, 0eAC,B £¨¨ §¨¨ ¥SÑj, 1eAC,B (2.23)

Inserting in the previous expression we obtain:

ρts_ACB Sψ0eAC`ψ0S iτHSψ0eAC`ψ0S iτSψ0eAC`ψ0SH a S0eB`0S (2.24)

But this is equivalent, neglecting quadratic terms in τ , to: ρts

ACB 1 iHτSψ0eAC`ψ0S1 iHτ a S0eB`0S

UτSψ0eAC`ψ0SUτ a S0eB`0S

Sψ_e

AC`ψS a S0eB`0S.

(2.25)

The final state appears to be factorized between A C and B and the iteration of the swapping protocol always produces a state of this kind, substantially the dynamics remains confined outside B, the only relevant condition is B’s qubit to be initialised toS0e_B.

In particular after the n-th protocol iteration, it’s easy to verify that the projection onS1e_B, and consequently the probability for Bob’s qubit to be excited is always 0:

B`1S ρts,ACBnS1eB B`1SUτSψn1eAC`ψn1S a S0eB`0SUτS1eB

B`1S1 iHτSψn1eAC`ψn1S a S0eB`0S1 iHτS1eB B`1Sψn10eACB`ψn10S1eB iτ_B`1SHSψn10eACB`ψn10S1eB

B`1Sψn10eACB`ψn10SHS1eB.

(2.26)

_{If we rearrange the basis vectors} _{SÑj, 0~1e}

AC,B in order to have SÑj, 0eAC,B before and

SÑj, 1eAC,B afterwards, H coefficients result to be the H’s top-left quadrant ones,

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2.3. QUANTUM ZENO DECOUPLING 29 all these terms, because of the form of Sψn10eACB, are null.

Therefore the last qubit doesn’t participate to the system evolution, it’s decoupled. So, if we are interested in a protocol for the recovery of information, we cannot sample the system at arbitrary frequency.

2.3.2 Time of propagation

In the previous subsection, while discussing a high-frequency swapping regime, we introduced an effective time τH in relation to which we could estimate a

time scale sufficiently short to let us approximate the unitary evolution at the linear order. How can we determine it?

To simplify the exposition we choose among the Hamiltonians showed in

2.2.1a model of next-neighbour XY Hamiltonian:

H g 2Q_i σ x iσxi1 σ y iσ y i1, (2.27)

being g the coupling constant quantifying the strength of interaction. As we’ve already mentioned before, it’s easy to verify that the total spin along z direction is conserved and that H acts on the stateSψ00eACB (with

SψeA αS0eA+βS1eA) producing a new one in the form αS000eACB+βS010eACB.

So, because of its structure, an excitation on the first qubit can propagate only spin-by-spin, then the matrix will be in the form:

0 g g 0 g g 0 g g 0 g (2.28)

How much time will take the excitation in A to reach the opposite site B? Often in the literature this is accomplished by evaluating the fidelity, as we did before, and finding the t for which it’s 1.28 _{As we already observed}

in 2.2.4 not for every system though a fidelity Ft 1 is achievable, and as we’ve already mentioned in [28] it’s shown not to be in chain with more than 3 particles, except for a proper choosing of the couplings. We will settle for its first maximum: we will look for the tmax corresponding to the first

peak of the fidelity, not caring whether it’s actually the absolute maximum value in general it will not). Following the procedure in [28] we find the Hamiltonian’s spectral decomposition (eigenvectorsSke and eigenvalues Ek)

in order to express the unitary evolution and then calculating the fidelity.

_{We choose a uniform coupling constant to further simplify, but that doesn’t}

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We have: Ek 2g cos kπ N 1, Ske N Q n 1 sin kπn N 1Sne (2.29) and consequently the fidelity:

Ft W 2 N 1 N Q k 1 sin πk N 1 sin πkN N 1e iEktW _(2.30)

being Sne the state with excitation on the n-th site and N the number of particles composing the chain.

Thus we have an analytical expression for Ft but it wouldn’t seem immediate finding when and whether it assumes a determinate value for a generic N . We will compute it numerically: we expect the time propagation to grow with the number of particles N and decrease when g, the interaction strength, grows. The calculations agree with expectations, and we show below the plots (Fig. 2.9): left column constant g and growing N , right column growing g and constant N .

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2.3. QUANTUM ZENO DECOUPLING 31

Figure 2.9: On the left (blue) fidelities for growing N . On the right (red) fidelities for growing g. Time in units of 1

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Even if numerically from the plots we can extrapolate a value for τH for

various combinations of g and N , in this regard we report the qualitative behaviours in Fig. 2.10 and Fig. 2.11, we’d rather look for a more formal and comprehensive tool regarding signal propagation in spin systems. We’ll see in chapter3how the Lieb-Robinson bound can help us.

Figure 2.10: Values of τH in unit of 1_g for different N and g 1. The plot

shows a linear trend, but the fidelity of the first maximum (which not necessarily corresponds to the absolute maximum) decreases.

Figure 2.11: Values of τH in unit of 1_g for different g (in unit of g) and N 20.

The plot shows a hyperbolic trend 1

g. Also here the fidelity of the first maximum

(which not necessarily corresponds to the absolute maximum) more or less decreases with larger N .

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2.3. QUANTUM ZENO DECOUPLING 33 2.3.3 Quantum Zeno Effect

In 2.3.1 we encountered a first sophistication of the simple transmission protocol, where other than the chain Bob had at disposal a set of additional qubits employable as a memory. He had to swap those qubits with the last spin on the chain and collect somehow information about the state sent by Alice on the used qubits. We asked then how quickly he could execute this swap to fasten the protocol and we arrived at the conclusion that an excessive frequency of swapping spoiled the communication.

This effect of a dynamic that gets “frozen” by a continuous action by an external subject can be led back to what is called Quantum Zeno Effect (QZE).

QZE was first described in [30] to illustrate how a continuous process of measurement could freeze the evolution of a quantum system. The effect’s peculiar name is due to one of well known paradoxes elaborated by Zeno from Elea, specifically the one dealing with motion, in which the philosopher stated that since we can decompose the trajectory of a body in motion in an infinite number of static frames, motion itself emerges as paradoxical.

The quantum version of this “paradox” goes like this.

Let’s imagine to have a quantum system with Hilbert spaceHS and

Hamil-tonian H prepared at t 0 in the pure state Sψ0e. It will evolve unitarily

under Ut eiHt_{. We define:}

Survival amplitude At `ψ0Sψte `ψ0S eiHtSψ0e .

Survival probability pt SAtS2 T`ψ0S eiHtSψ0eT 2

. (2.31) These quantities serve us to compute the probability for the system to stay in the initial stateSψ0e after an evolution time t.

In the short time regime we can expand the unitary propagator Ut to get the expression for pt:

pt 1 t2 τ2 Z ot3_, _(2.32) where _τ12 Z `ψ0S H 2_Sψ 0e `ψ0S H Sψ0e2.

We imagine then to perform a series of N measurements with a time interval of τ _NT to check out whether or not the system has remained in the initial state. So we have that the survival probability after N measurements pNt is:

pNt pτN p t N

N

. (2.33)

And in the limit of large N :

pNt 1 t 2 NτZ2 N e t2 N τ 2 Z ÐÐÐ 1.N ª (2.34)

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The survival probability tends to 1 then, the state is kept where it was. In our case, the swapping protocol, Bob didn’t simply measured the qubit but exchanged it with another in the stateS0e. The procedure acted then as a projective measurement of the spin in the state on the S0e state and, in the limit of high frequency swapping, actually froze it preventing it to evolve and acquire information about the remaining part of the chain.

Before concluding the section it is worth mentioning that a different as-pect of QZE which could be interesting for our purposes is the so called Quantum Zeno Dynamics. If instead of dealing with measurement project-ing on sproject-ingle states (1 dimensional subspace) we had to handle incomplete measurements, i.e. projection on multidimensional subspaces, we could ac-complish a result similar to the monodimensional case. Specifically we would find that a given system prepared in a stateSψ0e in a subspace HP delimited

by the projector (incomplete measurement) P remains in that subspace in the high frequency limit measurement.31

This feature shows compelling properties for possible application to the topic of communication through spin chains or networks in general. An example in this regard is given in [32]: authors apply a Zeno-regime measurement with the additional option given by the control of the subspace in which the incomplete measurement acts, namely the projector P evolves with time so P Pt. In short, the measurement basis is rotated. They show that the appropriate dynamics for Pt is able to induce a non trivial evolution of the system comparable to the action of an adiabatically evolving Hamiltonian Ht. This means that without direct control over the system (H fixed) it’s still possible the manipulation of its dynamics, and consequently in the case of spin chains or networks of the information propagation, “passively” only by controlling the measuring process.

2.4 Quantum communication in presence of

partial control along the spin network

Hitherto we analysed quantum communication protocols almost exclusively from the point of view of sender-receiver, in the sense that we focused on those actions that Alice and Bob could execute on the respective ends of the chain and assuming them to be completely out of control for what concerned the body of the chain itself. In the introduction we gave in1.3to motivate our focus on spin system as quantum channels, this was justified by the fact that handling efficiently a large amount of degree of freedom as a hundreds or thousands spins chain could be would require an excessive quantity of resources. In the following we’ll relax this premise, we’ll hypothesize for instance to work with more compact systems, e.g. chains made of few tens of qubits or maybe we’ll assume the operations to be done over the chain

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2.4. QUANTUM COMMUNICATION AND PARTIAL CONTROL 35 to be sufficiently simple to be scaled. However this doesn’t mean to be an experimentally accurate or too reliable discussion, we just want to expose a proof of principle to state that even elemental procedure adjustments can improve substantially the transfer.

One of the main reasons to worry about alternative solutions for the advancement in terms of performances of such systems is suggested by the arguments in 2.2.4 and summarized in Fig. 2.7: for a uniformly coupled and static chain an acceptable value for maximum fidelity is not attainable already in the case of only 10 spins. Hence if we deal with systems of this kind the assumption of control over them isn’t totally unjustified.

Ballistic regime with couples

As usual we start with the most basic scenario. Let’s imagine to have our ordinary chain of spins but this time couplings aren’t static, they are tun-able. In particular we assume to be able to switch them on and off at will. Again we could ask how reasonable effectively such an assumption would be in a real experimental setup: it has been shown that for example in superconducting qubit arrays it is already an achievable feature,33,34 sub-stantially the simple switching operation can be considered a weak form of control, probably one of the less demanding in terms of resources and system management.

Our first goal is to get a communication line able to implement perfect state transfer (F 1) regardless of the length of the chain. Since we know that for a 2 qubits spin chain F 1 is feasible an immediate configuration comes up: we can switch on only one coupling at a time starting from Alice’s one to Bob’s one, keeping it on for the amount of time needed to reach F 1 and then switching it off and starting the same procedure with the next spin. In this way the state is transported from spin to spin with fidelity 1, in a time that scales linearly with the dimension of the chain. This type of transfer is also known as ballistic regime.

To give a practical instantiation we consider the XY next-neighbour model described by:

Ht N Q i 0 gitσxiσix1 σ y iσ y i1 N Q i 0 gitAi,i+1, (2.35) where git γχit and χit ¢¨¨¦¨¨ ¤ 1 i τR@ t @ i 1 τR 0 e.w.

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with τRthe time needed to transport the state from one qubit to the next

one, γ the coupling strengthand the χifunctions so that χitχjt 0 ¦t

when ix j.

As we saw in2.2.3this Hamiltonian conserves the total spin, so if every spin except A’s one are initialised in S0e we have that the overall state S0e doesn’t evolve and in term of communication we can focus only on how A’s S1e is transferred. Then looking at the first two qubits we have that the Hamiltonian in the subspace formed by vectorsSÑje defined in 2.2.3 acts on the stateS1e S0e as:

HS10e 0 γχ1t

γχ1t 0 10 (2.36)

We have then to calculate the time τR needed by the system to go from

S1e S0e to S0e S1e. It’s straightforward to show (it can be seen as an example of Rabi flop) that:

τR

π

2γ. (2.37)

The total ttottransfer time for a N spins chain in the ballistic regime results

to be: ttot N_2γπ. We have then a linear trend comparable to the one already

seen in section 2.3.2, Fig. 2.10, but in this case with perfect state transfer achievable.

Ballistic regime with triplets

In 2.2.4 we reported results stating how for unmodulated chains perfect transfer is achievable only in the cases in which NB 3. So, other than N 2 we can analyse the employment of triplets of spins activated in sequence to deliver quantum states with fidelity F 1. With respect to the 2-spin-case this kind of configuration offers a wider set of combinations of working parameters. As a matter of fact, within a triplet (spin 1,2,3 and couplings g12t, g23t), we could fix equal or different values for Sgi,i1S and times in

which the couplings act: in sequence, together or a mix. Even if it diverges from the unmodulated coupling condition, we’ll explore if the balancing between the coupling strength and operation times can offer improvement in term of total time of transfer, also in the situation of near-perfect transfer (f @ 1).

In short here we’ll switch on and off 2 couplings and we’ll deal with a Hamiltonian of the type described in Eq.2.35where, if previously we stated git γiχit such that χitχjt 0 ¦t when i x j, we now release this

condition admitting χitχi1t x 0 for some values of t. In particular we

assume that the first interaction g1t γ1χ1t, linking spin 1 to spin 2, will

be active since the initial time t0 0 to time t while the second interaction

will be tuned on at time τ , being t0B τ B t. Graphically:

_{We choose an uniform coupling but it’s not strictly necessary.}

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2.4. QUANTUM COMMUNICATION AND PARTIAL CONTROL 37

Figure 2.12: Two interactions: configuration γ1 γ1γ2.

Here also holds 0 B Sγ1S, Sγ2S B Γ. Since we know that the Hamiltonian

conserves the total spin, if the initial state for our 3 spin system is S100e we’ll have non zero couplings only towards the two states S010e and S001e. Explicitly we can write :

HS100e 0 γ1t 0 γ1t 0 γ2t 0 γ2t 0 1 0 0

To reproduce the ballistic propagation we’d like to see if we can achieve a perfect (or at least good) state transfer from spin 1 to 3, being then able to reiterate from spin 3 to 5 and so on. Other than this we want investigate if the modulation of γi and τ parameters lets us shorten the transfer time

with respect to the case of a uniform coupled chain (γi γ ¦i): fixed

γ1 and γ2 we find an expression for the fidelity Fγ1, γ2, τ, t and we look

for which values of τ it’s maximised on a t @ _max_γπ

1,γ2, condition that

ensures us a gain in transfer time compared to a uniform chain with couplings γi maxγ1, γ2.

The fidelity is expressed as Fγ1, γ2, τ, t

»

S `001S Ut S100e S2_{. But not}

to deal with complications related to the Hamiltonian’s time dependence we split the evolution in two steps each driven by a constant Hamiltonian:

H1 0 γ1 0 γ1 0 0 0 0 0 , H2 0 γ1 0 γ1 0 γ2 0 γ2 0 and consequently the fidelity:

Fγ1, γ2, τ, t 3

Q

j 1

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To evaluate it we have to diagonalise the Hamiltonians and find the eigen-vectors. With some calculations (reported in AppendixA) we get:

Fγ1, γ2, τ, t γ1γ2 γ2 1 γ22 cosγ1τ 2γ1 Dγ2 cos¼γ2 1 γ22t τ cosγ1τ 2 » γ2 1 γ22 Dγ2 sin¼γ2 1 γ22t τ sinγ1τ , with D 1 γ2 1γ22 γ2 2 γ12 γ2 2 (2.39) We look for for maxima, hence imposing ∂tFγ1, γ2, τ, t 0 follows

tmax 1 » γ2 1 γ22 arctan » γ2 1 γ22 γ1 tanγ1τ τ. (2.40)

Since tmax has to be greater than τ and τ A 0, a gain in transfer time is

possible only under these conditions: ¢¨¨¨ ¨¨¨¨ ¦¨¨ ¨¨¨¨¨ ¤ 1 ¼ γ2 1γ22 arctan ¼ γ2 1γ 2 2 γ1 tanγ1τ τ B π maxγ1,γ2 tanγ1τ B 0 τ A 0 (2.41)

Via numerical analysis we search for t satisfying the requirements listed above and we evaluate the fidelity to understand the quality of the transfer. If perfect transfer (Fγ1, γ2, τ, tmax 1) can’t be achieved we can still set a

threshold Fmin and consider values of F beyond it still acceptable.

From Eq.2.40 we define ∆ tmax_max_γπ₁_,γ₂. To find a gain in transfer

speed we must have ∆@ 0 for some of the parameters values, so we look for them (Fig.2.13).

So there are some points in which the condition is satisfied. We compute for them the fidelity F and we estimate the ratio:

r tmax_π

maxγ1,γ2

. (2.42)

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2.4. QUANTUM COMMUNICATION AND PARTIAL CONTROL 39

Figure 2.13: Value of ∆ in function of γ2 (in unit of Γ) and τ (in unit of _Γ1), for

fixed γ1 0.5.

Figure 2.14: Value of F in function of γ1 and γ2 (in unit of Γ). The colorbar

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The majority of these points represents though a small value for the fidelity, consequently we select only those such that F A 0.75 (Fig. 2.15).

Figure 2.15: Value of F in function of γ1 and γ2 (in unit of Γ), only F A 0.75

selected. The colorbar shows r values.

Therefore we are able to find some tmax allowing a gain in time transfer

but fidelity, which does not overcomes 0.8, does not completely fulfil the typical “good transfer” threshold.

To attain a higher magnitude fidelity we try a slightly different config-uration: instead of turning on lately the second interaction γ2 we turn off

early the first one γ1. Graphically in Fig.2.16.

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2.4. QUANTUM COMMUNICATION AND PARTIAL CONTROL 41 The procedure goes on as before (see Appendix A), except that under these circumstances H1 0 γ1 0 γ1 0 γ2 0 γ2 0 , H2 0 0 0 0 0 γ2 0 γ2 0 Here the expression for the fidelity comes to be: Fγ1, γ2, τ, t γ1γ2 γ2 1 γ22 cosγ2t τ 2γ1 » γ2 1 γ22 Dγ2 2 sin ¼ γ2 1 γ22τ sinγ2t τ 2γ1 Dγ2 cos ¼ γ2 1 γ22τ cosγ2t τ , with D 1 γ₁2γ2₂ γ2 2 γ12 γ2 2 (2.43)

Then we derive the equation for the maxima: tmax 1 γ2 arctan 2γ 2 1 γ22 3 2 sin » γ2 1 γ22τ γ2 Dγ22 2γ12 γ22 cos » γ2 1 γ22τ τ (2.44) And taking account of the additional constraints we have a transfer time gain when: ¢¨¨¨ ¨¨¨¨¨ ¨¦ ¨¨¨¨¨ ¨¨¨¨ ¤ 1 γ2arctan 2γ₁2γ₂232sin¼γ2 1γ22τ γ2 Dγ₂22γ₁2γ₂2 cos ¼ γ2 1γ 2 2τ τ B π maxγ1,γ2 sin¼γ2 1γ22τ γ2 Dγ222γ12γ22 cos ¼ γ2 1γ22τ C 0 τ A 0

Anew we numerically compute the fidelity for possible tmaxobeying the

above conditions and we look for perfect or quasi-perfect state transfer. Like before we check whether there are points for which the parameter ∆ defined above is negative (Fig.2.17).

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Therefore there are points that satisfy the necessary condition. So we compute the fidelity F in dependence of γ1and γ2(and τ ) and we show it in

Fig. 2.18where we report also the value of the parameter r defined before.

Figure 2.18: Value of F in function of γ1 and γ2 (in unit of Γ). The colorbar

shows r values.

Immediately we can notice how high values of F and low values of r (blue points) are achieved. Low r means faster transfer, but we have to consider only those points with a high fidelity. To better visualize them we operate some selections. We start looking for points with F A 0.99 (Fig.2.19).

Figure 2.19: Value of F in function of γ1 and γ2 (in unit of Γ), points with

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2.4. QUANTUM COMMUNICATION AND PARTIAL CONTROL 43 Now we add a filter to r we restrict to those points such that r @ 0.75 (Fig. 2.20).

Figure 2.20: Value of F in function of γ1 and γ2 (in unit of Γ), points with

FA 0.99 and r @ 0.75 are selected. The colorbar shows r values.

We see then that there are points for which perfect state transfer sub-stantially stands (F A 0.99) with a reduction of the transfer time of more than 25% (r@ 0.75).

Finally we look for the minimum r with F A 0.99 and we find it to be 0.7. In conclusion we can say that through the capacity to switch on and off couplings in the chain, we can find sets of values for γ1, γ2 and τ allowing

us to perform near-perfect state transfer in a shorter time with respect to the simple Rabi-Flop process.

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Quantum Communication via Spin Networks