Tools and Methods for the Distillation of Entanglement in Continuous Variable Quantum Optics

Contents

1 Introduction
1.1 Entanglement: an Introduction
1.2 Monogamy, Decoherence and Other Challenges
1.3 Practical Quantum Optics CV Distillation

2 Continuous Variable Distillation of Entanglement with Linear Optics
2.1 Motivation
2.2 The (ρ (i) , ρ (i)) Protocol
2.3 Probabilities
2.4 Non-Gaussian Resource States
2.5 The (ρ (i) , ρ (⁰ )) Protocol
2.6 Work in Progress: Homodyne Distillation Pumping
2.7 Open Problems and Conclusion

3 First Steps Towards Experimental Procrustean Distillation of Entangle- ment
3.1 Introduction
3.2 Squeezed Vacuum States, How useful are they?
3.3 Ideal Photon Subtraction
3.4 Chosen Subtraction Setup
3.5 Entanglement Quantification
3.6 Characterization Strategy

4 Detector Tomography
4.1 Introduction
4.2 The Detectors
4.3 Reconstruction
4.4 Ill Conditioning and Regularisation
4.5 Detailed Assumptions
4.6 Conclusion

5 Conclusion
5.1 Summary of Thesis Achievements
5.2 Future Work

A Tomography Appendix

B Homodyne Measurement Appendix
B.1 Explicit Calculation for the Gaussian Projective Measurement of a Gaus- sian State

C Quantavo Appendix
C.1 The Quantavo Maple Toolbox
C.2 Getting Started
C.3 Toolbox
C.4 Practical Example
.1 Dictionary of Procedures:
.2 Copyright and Disclaimer
.3 Maple Code

Bibliography

Publications

- A. Cabello, A. Feito, A.L. Linares

Bell’s inequalities with realistic noise for polarization-entangled photons Phys. Rev. A 72, 052112 (2005)

- J. Eisert, M. B. Plenio, D. E. Browne, S. Scheel and A. Feito

On the experimental feasibility of continuous-variable optical entanglement distillation, Optics and Spectroscopy, Volume 103, Number 2 p 173-177 (2007)

- H. B. Coldenstrodt-Ronge, J. S. Lundeen, K. L. Pregnell, A. Feito, B. J. Smith,

C. Silberhorn, J. Eisert, M. B. Plenio and I. A. Walmsley. A proposed testbed for detector tomography, Journal of Modern Optics (2008)

- A. Feito

A Maple toolbox for quantum optics in Fock space, arXiv.org: quant-ph/0806.2171 (2008),

- J.S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, T.C. Ralph, K.L.Pregnell, Ch. Silberhorn, J. Eisert, M.B. Plenio, and I.A. Walmsley, Measuring measurement, Nature Physics (2008)

Abbreviations

Abbildung in dieser Leseprobe nicht enthalten

Introduction

The thesis is organised as follows: This first chapter is a brief introduction to the literature and the context. Subsequently I introduce the motivation behind this thesis and my contributions to the field. The next chapter (2) deals with improvements to current entanglement distillation protocols. It can be read independently from the other chapters and constitutes the first block of the thesis. Chapter 3 is a more detailed analysis of the experimental and theoretical challenges involved in the actual implementation and can be read independently too. Chapter 4 deals with the concept of detector tomography. Even though this tomography is used to characterise the tools from chapter 3 it can also be read independently. Finally some software tools I developed are included for reference in the appendix but are not needed to understand the rest.

1.1 Entanglement: an Introduction

Since the publication of the famous paper in 1935 by Einstein Podolsky and Rosen [EPR35] the way we picture entanglement and how we talk about it have evolved greatly. It was described as “spooky action at a distance” and later as “statistical correlations revealing non-locality” [Pop95] to have widespread acceptance from the mid1990’s on as a possible resource for communication and computation having its own unit, the “ebit”. I will briefly review its origins, uses and why distillation of entanglement appears as a very natural question in the field of quantum information.

1.1.1 What is Entanglement?

Entanglement is a property which is only encountered in the realm of quantum mechanics and exhibits some striking features. One way to reveal its effects involves two distant particles having previously interacted in a specific way. When these particles are measured, the formalism seems to suggest that measuring one particle affects the properties of the distant particle instantaneously. It is worth noting that these effects cannot be seen in a single run experiment but become apparent in the statistics relating the results of many measurements. In that regard the exact interpretation of this result requires a careful examination of the many assumptions involved in the formalism [Bel87, HR07]. However, irrespective of the interpretation, the coexistence of concepts such as locality, realism, logic or probability sets is brought into question or may even have to be abandoned [EPR35, Har93, Pop95, Ish97].

To give a more formal definition of entanglement we must note that two key elements give rise to the structure of entanglement:

- First the mathematical object that describes the properties of an isolated physical system is a ray embedded in a Hilbert space. Therefore linear combinations of states also describe physical systems.
- Secondly the way to describe two or more particles or systems makes use of the direct product. The states defined by linear combinations of direct products of rays can give rise to the correlations defining entanglement.

To be more precise and general about the definition of entanglement in the bipartite case we can refer to [Wer89]) :

Definition 1. The state described by the density matrix ρ on HA ⊗ HB is said to be separable iff it can be written as the convex combination:

Abbildung in dieser Leseprobe nicht enthalten

or can be approximated in trace norm by the states of that form. Otherwise the state is said to be entangled.

(Note that the multipartite generalisation of the definition is more or less straightforward although the concept of N-separability adds to its complexity [HHHH07]).

1.1.2 Why Use Entanglement?

The existence or not of these correlations between distant particles has been the subject of heated debate for most of the 20 th century [EPR35, Boh35, Sch35, Ein53, Bel64, Bel87]. However, in the 1980’s a series of experiments in quantum optics confirmed many of the properties entanglement was expected to display [CHSH69, APR81, AGR82, ADR82, SA88]. Giving a practical use to this correlations has sparked the imagination of numerous scientists in computation, information science and physics.

Among the notable applications of entanglement for communication we find the so called ‘teleportation’ [BBC+93a], secret key distribution for quantum cryptography [Eke91] or quantum dense coding [BW92]. In the field of computation various algorithms have been found for which entangled states allow some computational problems to be solved far faster than is possible using classical resources alone [Ste98, Sho97, Sho96]. However, the exact role entanglement plays (or not) in this speed-up is still a subject of active investigation.

Additionally the epistemological and philosophical questions involved in the transition from classical physics to quantum physics can be elucidated as we enhance our control of quantum systems and refine our means of interacting with them.

1.2 Monogamy, Decoherence and Other Challenges

It must be noted that the very structure of entanglement, while enabling useful correlations between some particles, can also introduce unwanted ones. Entanglement between any number of particles is generated by means of pairwise or multiple-particle interactions. It should therefore be obvious that when unwanted interactions (for instance with the environment) occur, entanglement beyond the reach of our measurement devices can be generated. Two key concepts help us grasp the problem behind this phenomenon: Monogamy of entanglement and decoherence.

Loosely speaking monogamy of entanglement [CKW00] makes reference to the following: If two particles A and B share certain amount of entanglement and one becomes entangled with a third one the pairwise entanglement between A and B will weaken. As a limiting case when two particles are maximally entangled, they cannot be entangled with a third one in that same degree of freedom. More specifically we can write the state of these three particles and look at the entanglement between different parts of the system. For instance, the entanglement between A and the pair BC can be quantified by a certain entanglement measure E (A (BC)). This quantity then limits the entanglement A can have with B and C taken individually. For qubits this relation can be expressed by the inequality E (AB) + E (AC) ≤ E (A (BC)), where E () is some entanglement measure (or originally the square of the concurrence)[CKW00].

It is often impossible to control all the interactions between a system and the environment (air, optical fibres, atoms, radiation, etc). Therefore if our system becomes entangled with third particles the entanglement between the two (or more) particles that interest us will eventually decrease. Similar results hold for more particles, higher dimensions and for certain infinite dimensional systems making it a general problem [KW04, AI06, AI07, AI08] .

1.2 Monogamy, Decoherence and Other Challenges

It is however not always possible to describe or have access to every particle that interacts with our system and we often adopt a more coarse grained point of view. For our quantum communication purposes, particles will need to be sent through a channel (optical fibre or air for instance). The inevitable interaction with the particles of the channel cannot be described in full detail but will be effectively described as a decoherence process. Tracing out the channel will also reveal a state whose correlations have ‘leaked’ into the environment [NC00, Pre98]. Put differently, part of the entanglement that the original system contained is now shared with inaccessible particles.

On top of the decoherence noisy channels introduce, the creation of entanglement can be a noisy process in the first place. Entangled states are often the outcome of probabilistic events and need to be described with mixed states similar to those having experienced decoherence. These two considerations (partial accessibility and uncertainty in the state generation) imply that usually the available states are less entangled than the pure state description we often encounter. We can loosely say that entanglement is either difficult to produce or is lost in inaccessible parts of the system.

To summarise, the mathematical structure of quantum mechanics reveals the interesting and useful property of entanglement. However the correct description of experimentally available states shows that entanglement is hard to produce, manipulate and preserve.

1.2.1 Fighting Decoherence with Distillation

As we have seen above, if we want to exploit the power pure entangled states have to offer, we need to fight decoherence and mixedness (or lack of purity in the quantum sense). In other words we ultimately want to have highly entangled pure states as a resource. Many areas in Quantum Information can help achieve this goal. Notably better sources of particles will contribute to the solution (be it photons [MLSW08], trapped ions [LDM+03], Nuclear Magnetic Resonance [LKC+02] or any other implementation). Isolating our qubits from the environment is obviously another interesting path.

Abbildung in dieser Leseprobe nicht enthalten

Figure 1.1: LOCC: Local Operations and Classical Communications are the operations Alice and Bob are allowed to use in their quantum communication tasks. LOCC precludes the coherent exchange of quantum particles between the two or more parties.

This idea has lead to clever arrangements of particles which protect the coherent information [Kit02, FKLW01].

However, we must remember that if these particles are to travel long distances even highly entangled pure states will degrade through a noisy channel. We may nevertheless choose to tame the effects of this noise with accurate and controlled local quantum processes in the distant labs. These are commonly referred to as local quantum operations (LO). Local operations are aimed at avoiding the typical long-distance decoherence. Additionally, standard telecom technologies allow perfect classical communication which can help coordinate the quantum operations in each lab (see fig. 1.1). The use of these resources is commonly described under the acronym LOCC: Local Operations and Classical Communication. Another crucial aspect of LOCC-operations is its fundamental relationship to entanglement. LOCC-operations between two nonentangled particles cannot create entanglement. In fact one can define classical correlations between two quantum systems as those arising from LOCC-operations alone [EJPP00, CLP01].

The compelling question is therefore: can LOCC fight decoherence or even increase the entanglement if it is present? There is a partial affirmative answer to this question. Distillation can achieve this but with certain restrictions. This will not be a deterministic process and not all states will be distillable [Ken98a, Rai99, PV07]. Purity and entanglement-increasing LOCC-operations are generally classified depending on the resource states and the target states. Increasing the entanglement of pure states is called entanglement concentration. When this operation encompasses the use of mixed states too it is called entanglement distillation. Also, purification refers to the process of increasing the purity which can be quantified with measures such as Tr {ρ ² } or the Von Neumann entropy.

Entanglement Concentration A simple example introduced in 1995 will help us grasp the spirit of distillation. The first insight leading towards distillation was that LOCC operations on a state not violating any bell inequality could turn it, probabilistically, into a state revealing non-locality [Pop95]. To understand this, consider a state of the form:

Abbildung in dieser Leseprobe nicht enthalten

Without loss of generality we will restrict ourselves to the case in which α > β. Let Alice add an ancillary qubit in the state | 0 〉 forming the state:

Abbildung in dieser Leseprobe nicht enthalten

and let her perform a unitary operation on her two qubits that will map:

Abbildung in dieser Leseprobe nicht enthalten

Now with probability β ² measuring the ancillary state will give the result “0”. If this result is obtained the other two particles will be in a maximally entangled state whereas obtaining “1” will return the product state | 00 〉. Using LO (ancilla + local unitary) and CC (i.e. communicating the success or failure of the measurement) to increase entanglement was introduced by Bennett et al. [BBPS96] as well as Gisin [Gis96] who described it as “hidden quantum non-locality revealed by local filters”. A single pure state probabilistic distillation of entanglement following that scheme was done in 2001 [KBLSG01] . The last 12 years have of course seen impressive advances beyond the pure, single copy, bipartite and probabilistic distillation that was just presented. Dealing with mixed states, qudits, infinite dimensions or multipartite settings have been some of the issues addressed. A recurring issue concerns the speed at which one can distill these pure entangled states, the resources involved and the trade-off between probabilities and yield (be it maximally entangled states or other target states).

1.2.2 Distillation in Finite Dimensional Hilbert Spaces

Although the first appearance of entanglement [EPR35] was in continuous variables, the Bohm-experiment version of the EPR paradox did get more attention due to its simplicity. Two level quantum systems indeed provide a simple system to work with. Photon polarization, two level atoms or spin-¹² particlesareidealsystemsforthestudyofentanglement. The collaboration of information scientists, computer scientists and physicists has also brought a lot of attention towards systems containing ‘qubits’ due to the analogies with the classical ‘bits’. Such systems also suffer the effects of decoherence which hinder the realisation of many quantum information tasks. Distillation of entanglement is therefore a crucial question in these systems.

Bipartite Entanglement Distillation: This example is one of the most frequently studied due to its obvious consequences for quantum information theory. Sharing, encoding or sending a message using ‘qubits’ between two distant parties A and B are basic communication tasks. Furthermore some discoveries about its mathematical structure under

LOCC make it a fertile ground for investigation (for example its relation to majorization [LP01]) . Let us then study some of the generalisations beyond the simple example of concentration presented above. Bennett et al.

The BBPSSW recurrence protocol [BBP+96] is a first well defined contribution to distillation aimed at facilitating teleportation. It works for 2 ⊗ 2 states with fidelity F = 〈φ + |ρ|φ + 〉 > 1 / 2. The resource is N copies of a state ρ. Both parties perform U ⊗ U∗ twirling to get N copies of a 2 ⊗ 2 isotropic state ρF and then locally Alice and Bob perform two pairs of XOR operations (also known as CNOT gates):

UXOR|x〉|y〉 = |x〉|x + y〉 (1.1)

Where the sum in (1.1) is performed modulo 2. The first particle will be called source and the second target. To complete the distillation they take pairs of ρF states. Source particles are taken from the first of the two pairs and target particles from the second pair. This leads to many copies of

ρ′ = UXORA ⊗UXORB (ρF ⊗ ρF) U† XORA ⊗UXORB

For each of these four qubit states Alice and bob measure target qubits locally in the computational basis. If the results agree they keep the remaining pair of source particles and “twirl” it. Otherwise they discard it. The surviving pairs will have a new fidelity of:

Abbildung in dieser Leseprobe nicht enthalten

Efficiency

The problem is that the success probability goes to zero in the limit F → 1 with the above protocol. In fact it can be shown that this is the case for almost all mixed states [Ken98b]. Nevertheless if F is high enough to ensure that S < 1 where S is the Von Neumann entropy, then the hashing protocol [BDSW96] gives asymptotically nonzero distillation rate providing (1 − S) N maximally entangled pairs. Following these same ideas another protocol was presented and applied to Quantum Privacy Amplification.

Quantum Privacy Amplification (QPA)

This method aimed at increasing the security of quantum cryptography over noisy channels (in the entanglement based scheme) appears in [DEJ+96]. Based on the ideas from the distillation protocols above it improves the rates and applies it to key distribution over noisy channels.

Requirements

Let us assume that pairs are generated in the state |φ + 〉 and then become mixed when distributed over a noisy channel. The basis to describe the state of our pairs will be the Bell state basis {|φ + 〉, |ψ−〉, |ψ + 〉, |φ−〉}. In it, the density operator will by assumption have diagonal elements {a, b, c, d} following the notation in [BEE00]. Therefore, the first diagonal element will be the fidelity: a = 〈φ + |ρ|φ + 〉. The purpose of QPA will be to achieve a = 1 and therefore b = c = d = 0. The off diagonal elements do not contribute on average to the QPA algorithm so one does not need to specify them. The details of the procedure are similar to the one presented above by Bennett et al. Pairs of states are considered, UA and UB rotations are applied on both pairs at each respective side followed by a Controlled-NOT operation on both copies. Afterwards the target pair is measured and coinciding outcomes are kept.

Efficiency

The QPA procedure looses at least one half of the particles (the ones used as targets) at each iteration. In spite of this, it is about 1000 times more efficient than the proposal in [BBP+96] when a is close to 1/2. It has also been proved [C.98] analytically that the target point a = 1 , b = 0 , c = 0 , d = 0 is a global attractor for a > ¹

.Neverthelessit does not guarantee the security of the cryptographic protocol because of finite detection efficiencies.

Some general statements

When it is possible for Alice and Bob to transform one or more copies of the resource state ρ into at least one copy of |φ + 〉 with high accuracy using LOCC, ρ is loosely said to be distillable. The question “are all entangled states distillable?” had, as was mentioned, a negative answer. Indeed one can see [HHH97] that for example entanglement of n ⊗ m positive partial transpose (PPT) states cannot be distilled and they are nevertheless entangled. This leads to the concept of ‘free’ and ‘bound’ entanglement. The former being distillable and the latter not. All currently known examples of bound entangled states have a positive semi definite partial transpose of the density operator. Every PPT state is known to be undistillable. The converse is a central open question.

Another concept arises when trying to optimise this process. The concept of distillable entanglement of a state ρ is intuitively the maximum over all allowable protocols of the expected rate at which “good” EPR pairs can be obtained from a sequence of identical states. A rigorous formulation of it was given by E.M. Rains [Rai99]. To ease the formulation of a rigorous and computable definition it is useful to consider a more powerful set of operations than LOCC, namely operations preserving the positivity of the partial transpose (PPT operations). This set is easier to describe but PPT operations allow more general operations. For instance, to map a product state onto a bound en-

tangled state and ensure the distillability of any NPT (negative partial transposed) state [EVWW01]. Nevertheless It is still an open question if NPT bound entangled states exist with respect to LOCC. The distillable entanglement of a bipartite state ρ under LOCC can be expressed as [APE03]:

Definition 2. The optimal rate of maximally entangled states that can be distilled from ρ, by LOCC, in the asymptotic limit is:

Abbildung in dieser Leseprobe nicht enthalten

mensions, and the supremum is taken over all possible sequences of integers {Kn}.

The distillable entanglement for example provides a bound to the optimal rate any protocol of the BBPSSW kind may achieve [BBP+96]. However it remains a definition of limited practical applications due to the difficult optimisations it entails.

Now the next obvious question is which states are distillable. If ρ is a pure entangled state, distillation is always possible [BBPS96]. If ρ has a small amount of entanglement, sufficiently many copies of it allow copies of |φ + 〉 to be distilled with high accuracy. Furthermore, if ρ is a mixed state of exactly two qubits, if it is entangled it is distillable [BBP+96, HHH97]. In a more general fashion a necessary and sufficient condition for a state to be distillable can be expressed as [HH01]:

Proposition 1. A bipartite state ρ on HAB = HA ⊗ HB is distillable iff for some twodimensional projectors P, Q and for some number N, the “two-qubit-like” state

Abbildung in dieser Leseprobe nicht enthalten

In spite of the above proposition there is no effective known procedure to determine whether a given state is distillable or not.

A further subtlety was introduced when enquiring precisely how many copies are required for a given distillation [Wat04]. The concept of n-distillable state was introduced. A state is said to be n -distillable if there exists an LOCC protocol that allows Alice and Bob to convert n copies of ρ to a shared pair of qubits that is entangled. Note that n-distillability does not require the n copies of ρ to become a maximally entangled pair, but only the conversion to an entangled pair. Therefore ρ is distillable iff ρ is n -distillable for some n. Indeed once we have grouped our states in groups of n and distilled entangled pairs, we can use these in a usual protocol like [HHH97] to distill singlets. Actually for pure and mixed states on a single shared pair of qubits distillability and 1-distillability are equivalent. An interesting result from [Wat04] is that

Proposition 2. For any choice of integers d ≥ 3 and n ≥ 1 , there exists a d ² ⊗ d ² bipartite mixed quantum state that is distillable but not n-distillable.

This means that entanglement distillation is nonlinear with respect to the number of copies used in the distillation process. There are instances of states ρ where 10⁶ copies do not suffice for a single shared pair of non-separable qubits to be created. That distillability is in general not equivalent to n -distillability has therefore important consequences.

For a rigorous treatment of distillability and bound entanglement see [Rai99]. For a review containing the first ideas see [HH01, VP98]. For a thorough and rigorous treatment see [DW05].

Multipartite entanglement distillation

Various proposals have been made to distill multipartite entanglement [MPP+98, ADB05, DAB03]. This can lead to the distillation of Greenberger-Horne-Zeilinger (GHZ) states

[GHSZ90], that is states of the form:

Abbildung in dieser Leseprobe nicht enthalten

These and other truly entangled multipartite states are a very interesting resource for quantum communication in networks. Indeed communication networks usually involve more than two parties so this will be essential for scalable quantum information processing. Multipartite entanglement is more difficult to quantify in high dimensions but the creation of multipartite entangled particles is already an experimental reality for photons [BPD+99, LZG+07] molecules [LKZ+98], spins in diamonds [NMR+08] or ions [HHR+05].

Pair assisted distillation

Another multipartite distillation approach involves purifying entangled pairs first and building multipartite entanglement afterwards using the methods from teleportation [BBC+93b, ZHWidZ97, BVK98]. Since we know how to purify two particles we can do that first and then entangle them successively. Let us study the case for tripartite distillation.

Description

The procedure [ZHWidZ97, ZHWidZ97] consists of four main steps:

1. Divide the original ensemble in two equal sub-ensembles.
2. Bob and Claire perform projections of particles onto:

Abbildung in dieser Leseprobe nicht enthalten

Bob does it with particles from one sub-ensemble and Claire with particles of the other. When they obtain a successful projection onto |−〉 Alice performs a σz on her particles, otherwise she does nothing.

3. A-B on one side and A-C on the other then perform a standard two particle pu rification process. This results in two maximally entangled ensembles of pairs of particles, shared between Alice and Bob and between Alice and Claire.

4. To obtain a single GHZ state out of two maximally entangled pairs shared be tween A-B and A-C she chooses one entangled pair from each sub-ensemble. She performs a CNOT operation on her two particles and projects the target parti cle onto | 0 〉 or | 1 〉. A successful projection onto | 1 〉 is followed by a σz operation on Claire’s particle, and otherwise nothing is done.

Requirements

Since two particle entanglement distillation requires f > 1 / 2 if we do not know the initial state this will set a limit here too. Otherwise, if we have additional information, a state not fulfilling the above condition could be purified [HHH97]. For more than three particles the criteria are more difficult and we have to turn our attention to schemes that directly distill multipartite entanglement.

Direct distillation

For two particles the singlet state is invariant under any bilateral rotation and this plays an important role in the aforementioned purification schemes. For three or more particles there is not always a known maximally entangled state which is invariant under multi-lateral rotations. This makes it difficult to convert an arbitrary state into a Werner state. In the absence of a maximally entangled state invariant under random bilateral rotations we may introduce a Werner-type state [BEE00]:

Abbildung in dieser Leseprobe nicht enthalten

This state could describe the attempt to distribute a state |φ + 〉 to many parties through a noisy channel. The fidelity of the transmitted state would be evaluated as:

Abbildung in dieser Leseprobe nicht enthalten

resulting therefore in the expression [Abbildung in dieser Leseprobe nicht enthalten] forWerner-typestates.

A protocol going beyond the pairwise distillation was proposed in¹⁹⁹⁸ [MPP+98] and was called P1+P2. It can purify a Werner-type state of any number of particles, provided the fidelity of the initial mixed state is above a certain threshold.

Description

The protocol consists of Alice and Bob performing each on their side iterations of the operations P1 followed by P2. P1 is a local CNOT and a measurement M1. M1 keeps the control qubits if an even number of target qubits are measured in the state | 1 〉. Otherwise the control qubits are discarded.

P2 is a local CNOT operation and a measurement M2 in which the control qubits are kept if all target qubits are found to be in the same state (otherwise they are discarded). Therefore when purifying 3 particles only | 000 〉 and | 111 〉 are kept.

For instance 4 states can be taken by Bob. P1 is done on one pair and P1 is done on another pair. Two states come out of each P1 operation. Two states are afterwards fed to the P2 operation.

Achievements

This purification is not restricted to Werner-states. Other states can be purified by P1 or P2 alone. For example if the initial state has no weight on |φ−〉 and the other states have

equal weights (or even if other states have zero weight) then P2 alone can purify to |φ + 〉

1.2.3 Distillation in Infinite Dimensional Hilbert Spaces

A lot of progress has been made studying qubits or low dimensional systems. Qubits, due to the low dimensionality and symmetries in two dimension offer many opportunities to solve quantum information problems. The maintained efforts in the experimental community to isolate and control two level systems have also contributed to the progress in the area. However both technical and mathematical limitations exist which make it worthwhile exploring beyond qubits or qutrits.

To begin with, the infinite dimensional stage offers new and as yet unexplored possibilities. The rich structure of the infinite Hilbert space makes its analysis more complex but also could unveil new insights, protocols and technologies. For instance one can find highly entangled states very close (in trace norm) to non-entangled ones [ESP02b]. Another characteristic is that very few states are non-distillable [HCL01] or that one can in principle achieve arbitrarily high entanglement. All these features make it an exciting arena. However, one of the main aspects that encourages the use of infinite dimensional systems is the available expertise in quantum optics. The quadrature amplitudes of the quantised electromagnetic field provide the continuous variables, which observe commutation relations analogous to those of position (X) and momentum (P) in the quantum harmonic oscillator.

Most quantum communication protocols require some form of preparation, unitary manipulation and efficient measurement. It turns out that standard optical tools such as non-linear crystals, beam splitters, phase shifters or phase-quadrature measurements fulfill all these requirements. Moreover the breadth of possible continuous variables

(CV) implementations makes its exploration very encouraging. Among the physical systems where CV are studied we can count phonons, photons, polarisation of intense beams, cold atoms, Josephson Junction Circuits, Bose-Einstein condensates or nano- mechanical resonators. In many of these systems standard techniques such as Quantum Key Distribution [GG02, Ral03], teleportation [Vai94, DBL+03], quantum erasing [AGL+04] or Universal Quantum Computing [SBd02, BL05] have been ported from the discrete variables setting. Decoherence obviously affects continuous variables too [SPID05] and so the distillation concepts must be adapted to this setting.

Bipartite entanglement distillation

In the distant lab paradigm, the case of two parties is the simplest. The infinite dimensional Hilbert space however introduces many difficulties. For instance on a bi-partite infinite-dimensional Hilbert space one can find arbitrarily close states (in trace-norm) whose difference in entropy of entanglement is infinite [ESP02b]. In that sense, many of the definitions from the finite dimensional setup need to be carefully revised or redefined. Since many problems and open questions remain unsolved in finite dimensional spaces it would seem that little can be said about the complex infinite dimensional case. Nevertheless some subsets of states defined in the continuous variable setting prove to be easily described. A notable example are Gaussian states and their manipulation through Gaussian operations. Gaussian states have various advantages since they have a simple mathematical description, are easily generated and standard optical tools can apply Gaussian operations to them.

Gaussian States in CV

Since Gaussian states can be described by a small number of parameters (as opposed to an infinite number of parameters for a general CV state), and due to their importance in linear optics they play a special role in the field of distillation in infinite dimensions. Let us review some of their properties.

We can define some position and momentum operators (for instance representing the position and momentum of a harmonic oscillator or the quadratures of an electromagnetic field) as a linear combination of creation and annihilation operators: X = 1 (â + â†),

Abbildung in dieser Leseprobe nicht enthalten

where X, P are real valued variables. For entangled states, and therefore for states with

with more than one mode, R =(

be generalised to:

where Σ is the symplectic matrix,

Abbildung in dieser Leseprobe nicht enthalten

where we have employed the symplectic product, ξT ΣR between the vector of real valued variables ξ and the vector of operators R.

We will say that the state ρ is Gaussian when its characteristic function

Abbildung in dieser Leseprobe nicht enthalten

is Gaussian in the variables ξ [SSM87, AMS97]. This means that the characteristic function χρ (ξ) can be cast in the form:

Abbildung in dieser Leseprobe nicht enthalten

The 2 n × 2 n matrix Γ is called the covariance matrix and d is the displacement vector. The displacement vector gives the coordinates of the centre of the Gaussian in phasespace, and the covariance matrix contains the variances and co-variances. The first and second moments { d , Γ } fully characterize Gaussian states and therein lies the simplicity of their description.

To make things more interesting, many states currently produced in standard quantum optics labs are Gaussian. For instance thermal states, coherent (Glauber) states, or squeezed states [LK87]. More interesting yet is the fact that Gaussian operations (those mapping Gaussian states onto Gaussian states) can be just as easily described [EP03, ADMS95]. These operations can be implemented using phase shifters, beam splitters, squeezers and homodyne detection; again standard tools in the linear optics experimental scene.

What Gaussian states will not do An optimistic hope before 2002 was that distillation could be done using Gaussian states and Gaussian operations. However it was shown [ESP02a, Fcv02, GIC02] not to be true. More precisely distilling Gaussian states with Gaussian local operations and classical communication (GLOCC) is impossible. It was later shown that the more general set of non-Gaussian operations allows for distillation of Gaussian states [GDCZ01]. A number of protocols that use this alternative have been put forward. All of them require some non-Gaussian element. Either using non-Gaussian states, non-linear interactions or non-Gaussian measurements. I will outline different proposals presenting the ideas and methods involved.

CV Entanglement swapping and entanglement distillation

An early proposal from 1999 [PBP00] noted that purification is always possible if the CV entangled states are projected onto the two levels of the Schmidt basis with the largest Schmidt coefficients. Afterwards one can perform a standard discrete distillation towards two level states with possibly higher entanglement than the original ones. However, an approach producing CV entangled states was also presented beyond the discrete case. A procedure to distill superpositions of coherent states (called cat-states)

was introduced in [PBP00] inspired by entanglement swapping. The key idea was to substitute the Bell state measurement by a reverse entangling operation accompanied by a projective measurement on the two particles. For certain parameters of the initial cat-states distillation was proved to be possible.

These continuous variable macroscopic systems are very interesting systems for distillation. Systems like Bose-Einstein condensates (BEC) or coherent light states are optimal candidates. Further research in the linear optics domain using cat states was developed in 2001 [JK02b]. In this context quasi Bell-states are:

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An interesting property is that quasi-Bell states can be unambiguously discriminated using only linear elements like beam splitters and homodyning. The purification aims at purifying states like:

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where the fidelity F is defined as [Abbildung in dieser Leseprobe nicht enthalten]. In the protocol two copies of ρab are taken and modes a, a′ are mixed in a beam splitter on Alice’s side and b, b′ on Bob’s side. After the beam splitter, each party performs measurements on the out-coming branches testing if a and a′ (b and b′) are in the same state by means of a BS and two detectors. In principle a photon parity measurement can reveal which quasi-Bell state was obtained. Practical implementations however suffer a high sensitivity to photon loss given that a single photon lost will change the parity [JK02a]. However, assuming this

is overcome, with certain probability the states are kept and the new fidelity becomes:

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giving an increase as long as F > 1 / 2. The probabilities involved can be quite high and for |α| ≫ 1 they can be 1 / 8 ≤ Psucc ≤ 1 / 4. The use of linear optics and the high probabilities make it a promising approach.

Entanglement distillation in continuous variables using non-linearities

To overcome the no-go theorems found in [ESP02a, Fcv02, GIC02] another idea is to make the states interact with non-linear media. The problem is often that the size of the non-linearities reduce the probability of the distillation to impractical levels.

Protocols

Two protocols were published in [FMF03] in 2003. They attempt to distill two-mode squeezed vacuum states of the form:

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The first scheme is based on the dispersive interaction of a two level atom with the microwave cavity field together with atomic state detection. The second scheme makes use of a cross Kerr interaction, coherent states, homodyne measurements and linear optics.

Description

The mechanism behind these schemes involves an ancillary system. The ancilla experiences a phase shift dependent on the number of photons on one mode of the shared state. This phase modulation is then converted into amplitude modulation via interference which allows to control the amplitude of the Schmidt coefficients. This is a proba- bilistic method which relies on the result of the measurement on the ancillary state. This result will tell us whether the distillation succeeded or not. Recent developments have refined this techniques but non-linearities are still too small for practical applications [MK06, MK07].

Entanglement distillation in continuous variables by means of linear optics and light measurements.

This idea was introduced in [OKW00] as a method to increase teleportation fidelity through a photon-number measurement. The entanglement increase was however limited by the detector inefficiencies and could not improve much beyond the subtraction. A more general scheme introduced the non-Gaussian character of the procedure in the resource states [BESP03, EBSP04]. This last protocol was iterative and therefore one could increase the entanglement beyond the photon subtraction. It was then the first feasible protocol for distillation of entanglement in continuous variables that used exclusively linear optics and non-number resolving photo-detection.

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Figure 1.2: borrowed from [BESP03], Diagram showing one iteration of the linear optics distillation protocol for continuous variables. The tensor product ρ⊗ρ distinguishes the upper and lower branch, and the product |n〉 ⊙ |m〉 the left and right modes.

One step of the iterative procedure from the [BESP03, EBSP04] distillation protocol is depicted above. Two copies ρ ⊗ ρ (upper and lower copies) of two mode states are mixed at the beam splitters (BSs). This is followed by an avalanche photo-diode (APD) measurement on the two upper modes. A successful step of the protocol occurs when zero photons are detected, and the out-coming modes are then kept. One such step of the protocol can be described as taking ρ (i) to ρ (i +¹ ) by means of:

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where the unitary operations U describe the action of the BS. Another version of it [EPB+07] uses the more efficient Homodyne Detection instead of vacuum projections. This setup can be equivalent but is more complex to analyse in its full generality as we will see in section 2.6. It has been shown that each starting state ρ (⁰ ) needs to be non-Gaussian if ρ (N) is to have an entanglement greater than the one of the original state. These non-Gaussian resource states ρ (⁰ ) can be obtained for example using APD detectors and linear optics in a photon subtraction scheme.

It has been shown in [EBSP04] that non-Gaussian ρ (⁰ )’s lead to enhanced entanglement and purity after an arbitrary number of iterations of the protocol. Note that that mixed input states can converge to pure Gaussian ones. This has been shown to be possible for mixed and pure states and for current APD detection efficiencies. Furthermore the family of states leading to this increase has been characterised. These states converge towards a Gaussian state after a few iterations.

Therefore, once the necessary non-Gaussian states are obtained the distillation procedure mixes them to obtain a single state with higher purity and entanglement. The lowest probabilities involved in the problem are found in the ‘Degaussification’, or preparation of the resource states [BESP03]. These low probabilities make it hard to scale the procedure to distill many copies. Additionally in that protocol the resources scale exponentially with the number of steps of distillation. I will present and study these problems in chapter 2.

Multipartite entanglement distillation

This idea is promising since the CV states can be easy to manipulate with linear optics and easy to distribute. Monogamy of entanglement in this case introduces different limitations since maximally entangled states are not limited to have 1 unit of entangle-

ment but can achieve arbitrarily high values both in the bipartite and multipartite case [AEI07, AI07].

Distillation in a CV multipartite network

This idea has barely been developed. Nevertheless a distillation in a CV quantum teleportation network has been proposed [vB00]. The basic idea is that a single mode squeezed state is sufficient to allow quantum teleportation between any two of N parties with the help of all other parties. The assistance of the other N-2 parties relies only on LOCC. Because of these N-2 measurements, bipartite entangled states are distilled from the initial N-partite entanglement.

1.2.4 Alternatives

We could also ask if there are alternatives to quantum entanglement distillation that overcome the difficulty of not having maximally entangled states. Indeed, in continuous variables, it has been shown that it is possible to distill a secure secret key with Gaussian operations on Gaussian states. For example [IVAC04] shows that the transmission of Gaussian-modulated coherent states and homodyne detection is equivalent to an entanglement purification protocol using CSS error correcting codes [CS96, Ste96] followed by key extraction. Also in [NBC+04] it is shown that it is possible to distill a secure key (under certain assumptions) from sufficiently entangled Gaussian states with non-positive partial transposition. This process does not require distillation and makes use of Gaussian states and Gaussian operations alone. Other novel ideas include using the effect of the environment to enhance entanglement [GMN06], or converting stabilizer codes to distillation protocols [Mat02].

As we have seen, many new ideas are being developed with the Gaussian state CV formalism trying to improve probabilities, use of resources and scalability of the protocols. As it stands it is a challenging area full of theoretical and experimental open questions.

1.3 Practical Quantum Optics CV Distillation

Our focus will be on implementations of CV distillation that try to use linear optics and photo-detection as the main resources. Part of the reason is that these resources are readily available today. Photons with CV entanglement were generated 20 years ago using squeezed states [WXK87]. However, distillation from these sources remains up to this day an experimental challenge. Some of the reasons involve not having pure enough states or not having them at sufficiently high rates. Other reasons have to do with low probabilities or with the complexity of the setups. On a different arena, revealing entanglement increase requires one to measure the entanglement before and after the distillation procedure. Finding simple and practical ways to determine this CV entanglement rigorously has been part of the problem too. I will explain how to use some mathematical tools to evaluate this entanglement precisely in chapter 3.

1.3.1 Photon Subtraction and Procrustean Distillation

Figure 1.3: Diagram of a photon subtraction setup. A quantum states goes through a beam splitter (meeting the vacuum at the other port). A successful subtraction occurs when a photon is measured in one of the ports. This non-Gaussian operation can create non-Gaussian states.

As was discussed earlier, distillation with the standard linear optics (and thus with

Gaussian operations) requires non-Gaussian states. Generating non-classical non-Gaussian states is an active and challenging field in quantum optics with promising applications to quantum information. One simple way to generate non-Gaussian states involves subtracting a photon from a Gaussian state as shown in Fig C.1. Great progress has been made generating cat states, kittens and photon subtracted states [OP05b, NNNH+06, AOBG07, OTBLG06, ODTBG07]. Other non-Gaussian states are NooN states (of the form |φ〉 ∼ |N 0 〉 + | 0 N 〉) [SOG06, EHKB04, MLS04, WPA+03], close approximations to Fock states [BAS+06, LO05, qGlFzX03, ROW+07, OTBG06], or photon-added states [AZ04].

In spite of all this progress one crucial proof of principle experiment remains elusive: demonstrating an increase of CV entanglement solely with LOCC. My calculations in chapter 3 elucidate the challenges and opportunities in this area.

Another crucial aspect that requires careful examination is the adequate characterisation of entanglement. Experimental data can reveal strong correlations in quantum states and not imply any entanglement [AP06, GRW07]. The purity of the states and the relationships between the measured observables must be studied carefully in these cases. Of course, a full tomographic reconstruction of the density matrix can unveil the available entanglement. However, even avoiding the arduous task of complete tomography, one can construct rigorous statements about the entanglement present from partial measurement on the state. The use of quantitative entanglement witnesses [EBA07, AP06, GRW08] is a tool very well suited for such situations and I will show how it is applied to specific examples.

1.3.2 CV Experimental and Theoretical Tools in Quantum Optics

The complexity in the recent quantum optics experiments generating non-Gaussian states is all too often ignored or approximated. However studying some of its elements in greater detail can lead to improved performance and understanding. The full description of the down-converted states in all their degrees of freedom (frequency, polarization, photon-number, angular-momentum, etc) is one example. Another exam- ple relates to the detectors and processes used to manipulate, measure and prepare the states. Photo-detection has been acquiring increasing complexity. In the last decade apparatuses such as single-carbon-nanotube detectors [FMM+03], charge integration photon detectors (CIPD) [SWM+04], Visible Light Photon Counters (VLPC) [KTYH99], quantum dot arrays [SOF+00], superconducting edge or picosecond sensors[MNMS03, GOC+01] or time multiplexing detectors [ASS+03a, ASS+03b] have made their appearance. Understanding in full detail the physical processes that occur in such detectors is of course out of our reach and we must resort to partial calibrations. Yet we rely on these detectors for state preparation and state tomography. I will present the first results attempting a more rigorous characterisation of detectors through detector tomography [LSS99, LFCR+08].

Continuous Variable Distillation of Entanglement with Linear Optics This second chapter builds upon the ideas presented by Browne et al. between 2003 and 2005 [BESP03, EBSP04, Bro05]. It analyses the inherent limitations of those optical CV distillation protocols and introduces modifications geared towards making the protocols less resource consuming and more efficient.

2.1 Motivation

The practical implementations of entanglement distillation in discrete finite Hilbert spaces often run into crucial limitations. Proof-of-principle experiments have been performed in optics [KBLSG01, ZYC+03, YKOI03] but they require single photons or photon number resolving counters which are either difficult to produce or expensive. Furthermore current experiments require the destruction of the state in order to prove the distillation was successful (making any iteration impossible). Proposals without post-selection exist (for 2-dimensional systems) [XbH03, HK07] but are yet to be implemented. For more details concerning the problems detector efficiency, mode matching, post-selection or bandwidth impose on distillation see, for example, Rohde et. al, 2.2 The (ρ (i) , ρ (i) ) Protocol [RRM06].

It is in this context that we turn our attention to entanglement distillation in continuous degrees of freedom by means of linear optics (Gaussian operations). This proposal has seen an increasing experimental interest [EPB+07, FMF03] due to major experimental improvements in both linear optics and the detection of light. After presenting the scheme from [BESP03, EBSP04] I will identify the problems involved in a realistic implementation. I will evaluate the success probabilities for distillation and the resources needed. In an attempt to improve it, I will introduce my variations on the protocol to obtain a deeper insight. This will raise different questions answered in further chapters.

We will call the protocol from Browne et al. [BESP03] the (ρ (i) , ρ (i)) protocol. The motivation behind this notation will be clarified later. We will also refer to it as the ‘Gaussifier’ or ‘Gaussification protocol’ since states converge towards Gaussian states when fed into the protocol.

2.2.1 Brief Description

To picture the whole distillation process we can first imagine that the two parties, say Alice and Bob, have 2 N -copies of a bipartite state. They also have lots of beam-splitters

(BS) and avalanche photo-diode (APD) detectors. Now they group those bipartite states in pairs, say ρAB - ρAB. They will use BS and APD-s to operate on each of the 2 N− ¹ pairs. If the procedure is successful (and classical communication will inform Alice and Bob of it) each pair will generate a single bipartite state ρAB - ρAB → σAB. The 2 N− ¹ states left will be grouped in 2 N− ² pairs. Each pair will be acted on with the same configuration of BS, APD and classical communication (CC) as before. If they are all successful, we will now have 2 N− ³ pairs and so on until we have a single state left.

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