From LOCC to continuous LOCC

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Master Physics & Astronomy

Theoretical physics

Master’s thesis

From LOCC to continuous LOCC

A study of continuous classical-quantum interactions

by

Marten Folkertsma

10428666

Supervised by Maris Ozols and Michael Walter

December 2020,

60 EC,

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Abstract

In this thesis we study a subset of quantum channels known as local operations and classical communication (LOCC), which are central to the theory of entanglement. LOCC channels have a natural discrete structure – they can be understood as two parties taking turns to manipulate their local quantum states while exchanging classical messages. We suggest a new form of LOCC – timeless continuous LOCC (TCLOCC). We first define a timeless LOCC (TLOCC) channel and then use it in combination with the Lindblad formalism to create a timeless continuous version of LOCC. Our formalism uses quantum-classical channels (qc-channels) together with qc-states to implement TCLOCC channels. We use our formalism to derive a very natural continuous version of a protocol by Fortescue and Lo for distilling entanglement between two parties from a three-party W state.

Acknowledgements

I want to express my sincerest appreciation of supervisors Maris Ozols and Michael Walter , for assisting me during this thesis. Every meeting we had was a lot of fun. We had enthusiastic discussions about the subject matter at hand and afterwards I always felt excited about working on. I will remember that it is always good to reward yourself with a piece of chocolate after doing difficult mathematics. I’m grateful for their support during my covid-19 infection, it gave me the rest I needed at the time.

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List of Symbols

H Hilbert space 9

U(H) Set of unitary operators on H 9 L(H) Set of linear operators on H 9

D(H) Set of positive semidefinite operators with trace one on H 10 ρ Density matrix 10

Pos(H) Set of positive semidefinite operators on H 10 ρqc Quantum-classical state 11

C(X , Y) Set of channels from X to Y 13

CP(X , Y) Set of completely positive superoperators from X to Y 13 TP(X , Y) Set of trace-preserving superoperators from X to Y 14 vec(ρ) Vectorization of matrix ρ 14

D Dephasing channel 16

K Manipulation of the classical register 17 L Lindbladian 31

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1 Introduction

Quantum computers see an ever increasing rate of development. The quantum supremacy paper by Arute et al. (2019) shows a calculation done in minutes on a 53 qubit quantum computer, where a classical computer would take thousands of years. Companies like Google, IBM and smaller ones like IonQ now host small quantum computers online to be used for experimentation. Even though real problems have yet to be solved on quantum computers, there are a lot of very promising steps in their development. A natural question to be asked is how these quantum computers will interact with each other. The idea of a quantum network has been studied by many parties, but an experimental realization outside of the lab has yet to be built.

This gives rise to a “distant“ lab paradigm in which a multipartite quantum system is distributed between various parties. These parties share a quantum system but are only allowed to act on local qubits. All parties have a local quantum computer with their own qubits on which they can apply operations. These local operations can be seen as applying gates and doing measurements to their local subsystems that are available. Even though all the parties only have a local system with which they can interact, the entire system is in a global multipartite quantum state, a state shared by multiple parties. This multipartite state can involve correlations, classical as well as quantum, between the different subsystems. To get use out of these correlations, local operations are accompanied by classical communication. This can be used to optimize operations and measurement strategies. In quantum information theory the class of channels describing these kind of protocols is called LOCC, or local operations and classical communication. This class can be understood as everything one can do with two distant quantum computers and a telephone line.

Every physical process involving local quantum operations and classical communication can be de-scribed by an LOCC protocol. As LOCC is a restricted subset of global protocols it serves as a good tool for studying quantum correlations and other non-local quantum effects as well as channel capacities (Chitambar et al. (2014)).

A classic example of an LOCC protocol is that of quantum teleportation, first shown by Bennett et al. (1993). In this protocol two parties, Alice and Bob, share one ”e-bit”, a maximally entangled qubit pair, and use this, as well as two bits of classical communication, to send a qubit in a possibly unknown state from Alice to Bob. The use of this e-bit and hence entanglement is essential for this protocol to succeed. Notice that if Alice and Bob share infinite e-bit pairs using this protocol that LOCC allows Alice and Bob to do any shared physical evolution as if having a quantum network.

Example 1.1. Teleportation involves two parties, say Alice and Bob, who share an e-bit |Φ+_{i =} _√1 2(|00i+

The protocol consists of two steps. First Alice does a local measurement on her qubits, the message system |ψi and her qubit of the entangled pair, in the Bell basis, explained below, and sends the classical outcome to Bob. Then Bob does a unitary rotation depending on this measurement outcome to receive the final state. A schematic representation can be found in Figure 1.

This unitary rotation is a Pauli rotation as given by UP auli= {I, X, Z, XZ}, where XZ = −iY . The

Bell measurement consists of a measurement in basis {Pi = |φii hφi| : |φii = I ⊗ Ui|Φ+i , with Ui ∈

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The state shared by Alice and Bob at the start of the protocol is as follows: |Ψi_A0_AB = |ψi_A0⊗|Φ+i_AB.

First Alice measure the two local qubits A0A of her system with the Bell measurement. Assume that the measurement outcome is i. Note that i runs from 1 to 4 and hence it can be sent over using only two classical bits. The post-measurement state is given by:

(Pi⊗ I) |Ψi_A0_AB = (|φii hφi| ⊗ I) |Ψi_A0_AB

= (|φii ⊗ I)(hΦ+| ⊗ I)(I ⊗ Ui†⊗ I)(|ψiA0⊗ |Φ+i_AB)

We can rewrite this state by using the unitary on entangled state trick from appendix A.2. By rewriting we find a more familiar form of the measurement outcome

(Pi⊗ I |Ψi) = 1 2(|φii ⊗ IB)( X i∈{0,1} hii| ⊗ IB)(IA⊗ IA⊗ Ui)(|ψA0i X j∈{0,1} |jji_AB) =(|φiiA0_A⊗ I)(I ⊗ I ⊗ Ui) X i,j∈{0,1}

hi|ψi hi|ji |ji_B

= |φii_A0_A⊗ Ui|ψi_B

Note that Ui is real and hence Ui = Ui. This is the desired state up to a local rotation on Bob’s

system. Note that this rotation depends on Alice’s measurement outcome, hence Bob needs to know the measurement outcome to remedy this.

To solve this Alice sends her measurement outcome i to Bob. Bob now uses this message to apply the necessary unitary U_i† to his local system. After applying U_i† he will be left with state |ψi exactly.

A schematic representation of this channel is given by Figure 1.

Figure 1: A schematic illustration of how Alice and Bob perform quantum teleportation. Note that the only line crossing from Alice to Bob is a line containing classical information. This fact illustrates that this is an LOCC protocol.

The use of entanglement in teleportation turns out to be essential. Without it Alice and Bob wouldn’t be able to send any qubits exactly, however many classical bits they would use. Teleportation shows how an LOCC protocol can manipulate and use entanglement to achieve classically impossible outcomes. This is indeed the case and as shown by Werner (1989). He shows that any LOCC protocol maps a seperable state to a seperable state, hence it can not create any entanglement. Separable states do not contain any

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entanglement, hence LOCC can’t create any entanglement. This runs even deeper as almost all notions of entanglement measures have as crucial property that under action of an LOCC the entanglement cannot increase. Hence the study of entanglement is deeply entwined with the study of LOCC.

As seen in Example 1.1 an LOCC channel seems to be naturally discrete. One party does a mea-surement and shares his or her outcome with another party who reacts on that message. However, there is a new interest in a continuous version of LOCC. Bennett et al. (1999) describe how by doing weak measurements any LOCC channel can be made continuous. They set up a problem in which two parties share non-local orthogonal states which can be perfectly distinguished by separable measurements, but as it turns out, cannot be perfectly distinguished by LOCC. The key ingredient of this proof is that all LOCC channels can be made continuous, by using many extremely weak measurements, but not all separable channels can.

Another study in which a continuous version of LOCC seems to be necessary is the study of classical gravity. Although it is generally believed that gravity is a quantum force, Di´osi (2011) and Hu (2014) are studying what it would mean if gravity was a classical force. Classical gravity entails a continuous classical interaction between quantum systems, hence a continuous LOCC channel.

What all these versions of continuous LOCC are missing is a general framework in which one can study continuous LOCC. This thesis focuses on this problem, the structure of continuous LOCC. As a sneak peak we describe a continuous version of the teleportation protocol as shown in Example 1.1.

Example 1.2. For continuous communication we need to extend the state given to the two parties with a classical register. This register will be shared between both parties and represent a classical shared message. It is hence vital that this register remains classical throughout the time evolution. The state is hence modified in comparison to the original state and is given by |Ψi_A0_ABM = |ψi_A0⊗ |Φ+i_AB⊗ |0i_M.

The evolution is driven by a Linblad equation, discussed thoroughly in chapter 4.1.

d dtρ = X i LiρL†i− 1 2{L † iLi, ρ} (1)

with the Li’s being linear operators. For the Li’s there are two sets, Alice’s set and Bob’s set. Alice first

needs to do a measurement and doing so load a message. For this she uses the following set of operators:

LA_i = |φii hφi|A0_A⊗ IB⊗ |ii h0|M (2)

With |φii being the Bell basis and i running from 1 to 4 as before. These operators perform a continuous

Bell measurement and store measurement outcome i in the classical register. Note that these operators do not interact with Bob’s local system. Bob needs to apply the correct rotation to get back the state |ψi depending on what Alice has measured. For this Bob has his own set of operators:

LBi = IA0_A⊗ U_i,B† ⊗ |5i hi|_M (3)

With Ui being a Pauli matrix rotation as given in Example 1.1. Note here that Bob reads out message

i and changes the classical register to state 5. The latter ensures that the corrections U_i† is only applied once. At this point the process stops as there are no operators acting on the classical register in state |5i h5|. By solving the differential equation with the given operators one finds a solution for ρ(t), which

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shows the desired behaviour that with increasing time the local state of Bob changes to |ψi as can be seen in Figure 2.

Figure 2: The fidelity of ρB with |ψi hψ| when the running continuous teleportation protocol as described

above. Note that the fidelity at t = 0 is √1

2 as this is the fidelity between the maximally mixed state and

a pure state. Asymptotically the fidelity changes to 1.

Example 1.2 gives an example of a continuous LOCC channel. From this example we can already get some insights on properties of these continuous LOCC channels that will be discussed throughout the thesis.

Firstly, a classical register M is shared between the two parties, both can read it out and write into it. The register has to remain classical throughout the entire evolution. This is achieved by the structure of the Lindbladian.

Secondly, the Lindbladian itself is static i.e. it does not depend on time. In later chapters we will call an LOCC channel having this property timeless LOCC, or TLOCC.

Lastly, by embracing the Linblad equation this describes a continuous channel. For any possible t we obtain a valid LOCC channel, which will be shown in later chapters. This continuity does however come with a price. In general it takes an infinite time to implement a measurement using a Lindbladian evolution. This fact causes the continuous teleportation protocol to succeed only as t → ∞.

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will be done in several steps. In chapter two we will first discuss a general basis of quantum information theory needed to understand these protocols. This is to clarify the material and give a clear set of definitions as we use them throughout the thesis.

Secondly in chapter three, we will discuss the notion of timeless LOCC. This will be the first restriction to the general LOCC protocol and gives some extra insights into how the continuous version will work. We will give a definition of timeless LOCC channels as a concatenation of qc-channels.

Then in chapter four we will focus on the Lindblad equations and their effect on qc-states. Here we will also discuss Markovian channels, as they are very closely related to the study of the Lindblad equations.

In chapter five we will combine the notion of creating LOCC out of qc-channels and the Markovian description of channels to give a clear definition of timeless continuous LOCC. We will also discuss the relation between timeless continuous LOCC and general LOCC and show that up to closure they are the same.

In chapter six we will show the new found framework in action by showing a more natural form of an entanglement distilation protocol. This entanglement distillation protocol involves a three-party entangled state, from which the parties want to distill an e-bit.

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2 Preliminaries

In this section we will discuss the basics of quantum information theory. The idea is to give the reader some intuition of the mathematical formulation of quantum information theory. Next to that we will clearly define the mathematical structure needed to discuss the rest of the thesis. We start out by discussing what a quantum state is and how it is mathematically defined. After that, we describe how entanglement arises from this formulation. Then we will take a closer look at quantum channels to finally discuss LOCC channels.

The idea is that after this chapter the reader is comfortable with the basic notions of quantum infor-mation theory and understands how Example 1.1 can be mathematically written as an LOCC channel.

2.1 Quantum states

In physics there are many systems that display quantum mechanical properties, such as the spin of an electron, the binding of atoms in molecules or even larger materials such as semi-conductors. In quantum information theory we associate such a system to a Hilbert space, H. A Hilbert space is a vector space equipped with an inner product. For the purpose of this thesis one can always think of H = Cd _{with the}

standard inner product: hφ|ψi =Pd

i φiψi, where |φi = hφ| †

the complex transpose. Quantum states of a system are the vectors in this Hilbert space, with the particular property that they have norm 1, |ψi ∈ H with hψ|ψi = 1.

As an example one can think of a one qubit system. A one qubit system is associated with a Hilbert space H = C2_{. All possible norm 1 vectors in H are of the form |ψi = α |0i + β |1i with |α|}2

+ |β|2= 1. This set describes all possible legitimate physical states that a qubit can be in.

Physical interactions are described mappings between these physical states. On pure states, vectors with norm one, all these mappings can be understood as a multiplication by a unitary matrix, U ∈

U(H) ⊂ L(H) with the property U U†= I.

Besides unitary mappings another important operation is a quantum measurement. In physics this corresponds to the measurement of some classical attribute of a quantum system like the spin of an electron. Given a finite set of possible outcomes Ω, a measurement is described by the function µ : Ω → Pos(H) with P

ω∈Ωµ(ω) = I. Here Pos(H) ⊂ L(H) is the set of positive semidefinite operators. With

the probability of measuring a property state being ω is given by:

p(ω) = hψ| µ(ω) |ψi . (4)

From this definition it is clear that the total probability is one,P

ωp(ω) =

P

ωhψ| µ(ω) |ψi = hψ|ψi =

1 and that any p(ω) ≥ 0. Thereby {p(ω)}ω∈Ω forms a probability distribution. After a measurement the

state collapses into a post-measurement state. If µ is a projective measurement, µ(ω)2= µ(ω) we can find a post-measurement stat. Given measurement outcome ω this post-measurement state is given by

|ψωi =

µ(ω) |ψi

hψ| µ(ω) |ψi. (5)

This definition of measurement has some intrinsic degree of freedom to manipulate states without changing any possible measurement. One can always add an arbitrary phase to a state |ψi → eiθ_{|ψi and}

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because the probability is defined as the absolute value square, one will never be able to measure this added phase. This gives rise to another formulation of quantum states, a density matrix.

Given a pure state |ψi ∈ H, its accompanying density matrix ρ is given by ρ = |ψi hψ| ∈ Pos(H). The global phase of eiθ|ψi is lost when computing its density matrix: eiθ_{|ψi hψ| e}−iθ _{= |ψi hψ|. Density}

matrices are positive semidefinite operators of unit trace. The set of all density matrices for a given Hilbert space H is given by

D(H) = {ρ ∈ Pos(H) | Tr(ρ) = 1}. (6)

When we are discussing a quantum state in this thesis, we talk about its accompanying density matrix. The measurement of a density matrix is also defined by a map µ : Ω → Pos(H), but outcome probabilities are given by

p(ω) = Tr[µ(ω)ρ]. (7)

Note that this also forms a probability distribution.

As µ(ω) ∈ Pos(H), µ(ω) can always be written as µ(ω) = A†_ωAω with Aω∈ L(H) being some linear

operator. We can use this fact to define the post-measurement state as

ρω=

AωρA†ω

Tr[µ(ω)ρ]. (8)

A density matrix as given before, ρ = |ψi hψ|, is what one calls a pure state. Besides pure states density matrices can also describe mixed states. Given any set of pure states {|ψii} and a probability

distribution pi withPipi= 1 and pi ≥ 0 for all i, the corresponding mixed state is given by

ρmixed=

X

i

pi|ψii hψi| . (9)

They correspond to probability distributions over pure quantum states. To understand these states one must first get familiar with the notion of bipartite states.

2.1.1 Bipartite states

One can imagine a situation where there are two parties, Alice and Bob, and they both have a quantum computer. The quantum systems associated with these quantum computers are HAfor Alice and HBfor

Bob. These two systems can also be considered as one whole system, a global system, given by Hilbert space HAB = HA⊗ HB. Any one of Alice’s states ρA and any one of Bob’s states ρB can be included as

a bipartite product state:

ρAB = ρA⊗ ρB. (10)

The construction of HAB also leaves room for states that cannot be written as a tensor product. These

states are called entangled. We will discus entanglement more thoroughly in the next section.

The inclusion of any of Alice’s local states ρA into the larger system is done by the tensor product.

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state ρAB and let Alice perform a measurement on her local system. This corresponds mathematically

to applying measurement map µ(ω) on Alice’s system and doing nothing with Bob’s system:

p(ω) = Tr[(µ(ω)A⊗ IB)ρAB]. (11)

This is a global measurement of the entire system, but it does not matter what is measured in Bob’s local system. In fact we measure Bob’s system and discard the outcome directly. This intuition can be translated to a mathematical mapping HAB → HA, namely the partial trace:

ρA=

X

i

(I ⊗ hi|)ρAB(I ⊗ |ii) = TrB[ρAB], (12)

where {|ii} forms a orthonormal basis of HB. Note that TrA[TrB(ρABD)] = TrB[TrA(ρABD)] = ρD.

If ρAB is a pure state with some entanglement, then Alice’s local state is a mixed state.

In physics this situation can happen when studying a system in a laboratory. If the state one is studying is actually part of a larger global system, then the local state is actually mixed instead of pure. This mixedness can usually be interpreted as the state being noisy. This noise is the reason why it is so hard to build a stable quantum computer.

2.1.2 Classical and quantum-classical states

Density matrices allow for a construction containing both quantum and classical information. While pure states contain only quantum information, one can embed any classical probability distribution into a density matrix. Given a probability distribution {pi}, withPipi= 1 and pi≥ 0, one can embed this

into the following diagonal density matrix:

ρc=

X

i

pi|ii hi| , (13)

with |ii denoting the i-th standard basis vector. A density matrix of this form is called a classical state. A hybrid of quantum and classical states are called qc-states (quantum-classical states). Given a set of density operators {σi} and a probability distribution {pi} one can define a qc-state as follows:

ρqc=

X

i

piσi⊗ |ii hi| . (14)

We call this first register the quantum system and the second register the classical system. One can relate this to an experimental setup where a quantum system is connected to some measurement device. Given a classical measurement outcome i the quantum state collapses to σi, this happens with probability pi.

2.2 A notion of entanglement

As shown in the introduction and briefly discussed above, entanglement is a very interesting property of multiparty states to study, especially when looking at LOCC. In physics, entanglement can be understood as a property of a multi-particle system. A multi-particles pure state is entangled if it is in a state that cannot be described as a product of the individual particle states. A simple example of this is the

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maximally entangled Bell state used in Example (1.1): |Φ+_i AB= 1 √ 2(|00i + |11i). (15)

Assume Alice and Bob share this state and Alice does a measurement and gets outcome 0. This means the entire state collapses to |00i which leaves Bob’s local state to be |0i.

In quantum information theory entanglement arises from the structure of multipartite systems. A shared Hilbert space between two parties can be written as the tensor product of the individual Hilbert spaces: HAB= HA⊗ HB. This gives rise to two different sets of states, those that we call entangled and

those that are separable. A pure state |Ψi_AB∈ HAB is called separable if it can be written in the form

|Ψi_AB= |ψi_A⊗ |ψi_B (16)

for any |ψi_A∈ HAand |ψiB∈ HB. If a state is not separable it is entangled.

This notion can be extended to mixed states as well. A mixed state ρAB ∈ D(HAB) is said to be

separable if it can be written as

ρAB=

X

i

piρA,i⊗ σB,i, (17)

whereP

ipi= 1 and pi≥ 0 for all i, and ρA,i∈ D(HA) and σB,i∈ D(HB). Any mixed state that is not

separable is said to be entangled. As example of a separable state one can think of the qc-state in 14. Entanglement can cause noise in experiments, as mentioned before, but it can also be used to build interesting protocols, e.g. teleportation by Bennett et al. (1993), superdense coding by Bennett and Wiesner (1992) and many other quantum protocols. Entanglement is considered the primary feature of quantum mechanics that differentiates it from classical mechanics.

It is useful to quantify the amount of entanglement in a given bipartite state so that we can compare how entangled different states are. For a pure state there is a essentially unique quantity we can use for measuring entanglement. As described earlier, whenever a pure state is entangled, the local states become mixed. When a state is more entangled the local states are more mixed and vice versa. There is a clear quantity that indicates the mixedness of a state which we call the entropy of a state.

In quantum information theory the entropy of a state is given by the von Neuman entropy:

S(ρ) = − Tr[ρ log₂(ρ)]. (18)

which can be calculated in terms of the eigenvalues {λi} of ρ:

S(ρ) = −X

i

λilog2(λi). (19)

Note that this measure of entropy is non-negative for any quantum state, as the eigenvalues λi of any

quantum state have properties λi≥ 0 andP_iλi= 1. A pure state has no entropy: H(|ψi hψ|) = 0. A fully

mixed state in dimension d on the other hand has maximal entropy: S(I/d) =Pd

i=1 1

dlog2(d) = log2(d).

Hence the von Neumann entropy fully captures this notion of mixedness of a state.

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ρA= TrA[|Ψi hΨ|AB], the entanglement entropy is thus given by

E(|Ψi hΨ|_AB) = S(ρA). (20)

d

Pd

i=1|iii the measure gives

E(|Ψi hΨ|_AB,ent) = S(I_d) = log₂(d).

This measure of entanglement cannot be used to find out if a mixed state is entangled or not. This entanglement measure for mixed states measures both entanglement and classical correlation at the same time. One can see this by looking at he following two states

E

d

X

i=1

1

(Christandl, 2006) wrote a great overview of possible entanglement measures for mixed states. This subject will however not be necessary for the rest of the thesis.

2.3 Quantum channels

In the picture of quantum density operators, mappings between different states become a bit more involved than unitary mappings. As density operators are linear operators on a Hilbert space themselves, a simple matrix multiplication will not be enough. Instead we use a class of mappings called superoperators. A superoperator

Φ : L(X ) → L(Y) (23)

is a linear map that maps linear operators on one linear space to linear operators on another linear space. Physically allowed mappings, or mappings that restrict density matrices to density matrices are called quantum channels. For a superoperator to be a channel it has to satisfy two additional constraints:

1. Φ is completely positive,

2. Φ is trace-preserving.

All such linear mappings Φ form the set of quantum channels, denoted C(X , Y). A superoperator Φ is completely positive if for any complex Euclidian space Z and positive semidefinite operator P ∈ Pos(X ⊗ Z),

[Φ ⊗ IZ](P ) ∈ Pos(Y ⊗ Z). (24)

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any P ∈ Pos(X ) it holds that

Tr[P ] = Tr[Φ(P )]. (25)

We denote the set of all such superoperators by TP(X , Y).

2.3.1 Representations of quantum channels

The action of a quantum channel on a quantum state can be represented in several ways. In this section we will discuss three of them. We will also show how to relate them to one another.

The first and most used representation is known as the Kraus representation. For any completely positive superoperator Φ ∈ CP(L(X ), L(Y)) there exists an alphabet Σ and a set of linear operators

{Aa ∈ L(X , Y) | a ∈ Σ} (26)

such that for any ρ ∈ D(X ),

Φ(ρ) =X

a∈Σ

AaρA†a. (27)

Φ is trace preserving if it also holds that

X

a∈Σ

A†_aAa = IX. (28)

This makes Φ a channel. It is important to note that the Kraus representation is in general not unique. However if a superoperator has a Kraus representation of this form it is a channel. The converse also holds. Any set of linear operators {Aa ∈ L(X , Y) | a ∈ Σ} with the property Pa∈ΣA†aAa = IX forms

a quantum channel Φ(ρ) = P

a∈ΣAaρA†a. Note that the Kraus representation and post-measurement

states are very closely related to one another.

The second representation is called the natural representation. Any matrix can be represented as a vector of a larger linear space. For instance a density matrix ρ ∈ Cd×d _{can be represented as a vector}

in Cd2 _{by stacking the columns. This procedure is called vectorization and for a basis vector |ii hj| is}

defined as follows

vec(|ii hj|) = |ii |ji . (29)

This action can be extended by linearity to vectorize any arbitrary matrix. The vectorization of a density matrix in its diagonal eigenbasis is given by

vec(ρ) =X i vec(pi|ψii hψi|) = X i pi|ψii |ψii. (30)

With this it is clear that

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is a linear mapping. Therefore there is a linear operator K(Φ) ∈ L(X ⊗ X , Y ⊗ Y) such that

K(Φ)vec(ρ) = vec(Φ(ρ)). (32)

This K(Φ) is called the natural representation and is given by

K(Φ) =X

ijlk

h

hi| Φ(|li hk|) |jii|ji hl| ⊗ |ii hk| . (33)

The natural representation is unique but depends on a choice of linear basis. It is however the least used representation as it does not have a nice mathematical structure. In this thesis it will be used for calculating the determinant of a channel.

The third representation is the Choi representation. The Choi representation requires a specific mapping from channels to linear operators, J : L(L(X ), L(Y)) → L(Y ⊗ X ) given by

J (Φ) =X

i,j

Φ(|ii hj|) ⊗ |ii hj| . (34)

The operator J (Φ) is called the Choi representation of Φ. The action of channel in terms of the Choi representation is given by

Φ(ρ) = TrX[J (Φ)(IX⊗ ρT)]. (35)

A superoperator Φ ∈ L(L(X ), L(Y)) is completely positive if and only if J (Φ) ≥ 0 is positive semidefinite, and Φ is trace-preserving if TrY[J (Φ)] = IX.

This representation is unique for any channel, and therefore studying Choi representations can be considered the same as studying channels. The Choi representation will mostly be used to prove bounds on norms.

These representations can all be related to one another. Given a channel Φ ∈ C(X , Y) with a Kraus representation Φ(ρ) =P

aAaρA †

a. Its natural representation is given by

K(Φ) =X

a

Aa⊗ Aa (36)

and its Choi matrix is

J (Φ) =X

a

vec(Aa)vec(Aa)†. (37)

2.3.2 Particular useful channels

In this section we will discuss some examples of channels to get some intuition for them. After that we will show some simple structures of channels involving classical and qc-states.

As hinted at before, the post-measurement state has a very close resemblance to the Kraus represen-tation. Given a measurement µ : Ω → Pos(H) with µ(ω) = A†_ωAω the measurement channel Φµ is given

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by.

Φµ(ρ) =

X

ω

AωρA†ω. (38)

It describes the average post-measurement state of the measurement µ. When µ(ω) is a projective measurement in basis {|ψωi}, this channel can be understood as the projection of the state on that basis

Φ{|ψωi}(ρ) =

X

ω

|ψωi hψω| ρ |ψωi hψω| (39)

A special example of this is the measurement channel that measures in standard basis. This channel is called the dephasing channel :

D(ρ) =X

i

|ii hi| ρ |ii hi| . (40)

Note that this channel destroys all quantum information of a system. Any quantum state is projected onto a classical state, hence it is also known as a quantum-to-classical channel.

Channels are often used to study information protocols involving multiple parties. A simple example of this is a channel that shares a classical message from party A to party B. For this channel we need to assume an initial state between parties A and B to be

ρAB= |mi hm|A⊗ |0i h0|B, (41)

where m is an integer (classical message) that A wants to send to B. The channel that achieves this is

Φcm(ρAB) =

X

i,j

(|ii hi| ⊗ |ii hj|)ρAB(|ii hi| ⊗ |ji hi|) (42)

=X

i

|iii hi| ρA|ii hii| . (43)

We see this by directly looking at the action of Φmc on ρAB:

Φcm(ρAB) =

X

i,j

=X

j

|mi hm|_A⊗ |mi δj,0hm|B= |mi hm|A⊗ |mi hm|B.

Note that this channels shares a message from A to B, but A also retains the message. We can combine the measurement channel with the classical message channel, to better understand a quantum measurement in the lab. In the lab one has a quantum system to measure and a classical device to read out the measurement outcome. We can assign the combined system a state of the following form:

ρlab= σs⊗ |0i h0|_c (44)

where σsis the quantum system and |0i h0|_cthe classical measurement monitor. Assume we do a

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channel, we can construct the lab measurement channel :

Φlab(ρlab) =

X

ω,j

AωσsA†ω⊗ |ωi hj|0i h0|ji hω|c. (45)

Note that this channel is more general than a direct combination of measurement and messaging. If we apply this channel on ρlab, we get

Φlab(ρlab) = X ω p(ω) AωσsA † ω p(ω) ⊗ |ωi hω|_c (46)

which is exactly of the form of a qc-state as in Equation (14). The quantum density operators are given by the post-measurement states and the classical state is the classical outcome on the monitor.

Figure 3: Image sketching a classical measurement protocol with a monitor.

2.3.3 Creation and trace operations

In this thesis we will often be working with an explicit message register. These message registers will always be classical as we are discussing LOCC. There are a few operations which we will use very often, such as creation of a message register, projection onto a certain state in the message register, jump operations in the message register, conditioning on the message register, and the removal of the message register.

For ease of notation we define a set of operators which will take care of these operations. The construction of a classical register is done by

K_∅→i(ρ) = ρ ⊗ |ii hi| . (47)

Note that this is a channel for any i. Projection is the opposite of constructing:

K_i→∅(ρ) = (I ⊗ hi|)ρ(I ⊗ |ii) = ρi. (48)

Note that this is a completely positive superoperator but not a channel. This operation will mainly be used to study what happens at different states on the message register. A jump operation changes the message register from some state i to a state j:

Ki→j(ρ) = |ji hi| ρ |ii hj| . (49)

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of the jump operation is used to condition on the message register. Here we introduce some shorthand notation:

Ki→i(ρ) = Ki(ρ) = |ii hi| ρ |ii hi| . (50)

The removal of a message register is done by projecting on all possible states of the message register, which is the same as taking a partial trace (12):

X i K_i→∅(ρ) =X i (I ⊗ hi|)ρ(I ⊗ |ii) = Tr2(ρ) (51)

2.4 LOCC channel

Quantum channels provide a very general notion for transmission of information, so they are the natural mathematical objects for describing quantum protocols between multiple interacting parties. In commu-nication, we can distinguish different protocols by the types of resources that the parties involved are allowed to use. In this thesis we will be looking at protocols in which parties have their own quantum systems but are only allowed to communicate classically. The set of channels describing such protocols is called local operations and classical communication or LOCC.

Before we can define LOCC we need one more mathematical object, a quantum instrument. A quantum instrument can be understood as a generalized measurement that produces a post-measurement state with some of the outcomes potentially grouped together, or as a measurement combined with a channel. It is defined as follows:

Definition 2.1. Given an alphabet Σ, a quantum instrument is a collection of completely positive super-operators {Φa: a ∈ Σ} ⊂ CP(X , Y) such that P_a∈ΣΦa∈ C(X , Y).

With this we can define bipartite LOCC. Before giving a definition we want to give a bit of an intuition of these channels. For this we introduce two parties, Alice and Bob, who perform a one-way LOCC channel. The first step is for Alice to perform a local channel combined with a measurement. Mathematically this is the same as applying an instrument. Alice then transmits the classical outcome message to Bob. Depending on this message, Bob applies a local channel. In this manner they are allowed to do any local operation but only communicate classically.

The definition we use in the thesis is from Watrous (2018). He starts out defining one-way LOCC from Alice to Bob. General LOCC than consists of applying multiple one-way LOCC operations after one another.

Definition 2.2. Let X , Y, Z, and W be complex Euclidean spaces and let Ξ ∈ C(X ⊗ Y, Z ⊗ W) be a channel. Here we assume that Alice’s input space is X and output space is Z and Bob’s input space is Y and output space is W. The channel Ξ is an LOCC channel under these conditions:

1. If there exists an alphabet Σ and a collection

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of completely positive maps satisfying

X

a∈Σ

Φa ∈ C(X , Z)

along with a collection

{Ψa: a ∈ Σ} ⊆ C(Y, W)

of channels such that

ΞA→B=

X

a∈Σ

Φa⊗ Ψa,

then ΞA→B is a one-way right LOCC channel.

2. If there exists an alphabet Σ and a collection

{Ψa: a ∈ Σ} ⊂ CP(Y, W)

of completely positive maps satisfying

X

a∈Σ

Ψa ∈ C(Y, W)

along with a collection

{Φa: a ∈ Σ} ⊆ C(X , Z)

of channels such that

ΞA←B=

X

a∈Σ

Φa⊗ Ψa, (52)

then ΞA←B is a one-way left LOCC channel.

3. The channel Ξ is an LOCC channel if it is equal to a finite composition of one-way left and one-way right LOCC channels.

We denote the set of all LOCC channels with input X ⊗ Y and output Z ⊗ W by LOCC(X : Y, Z : W). The colon here denotes the splitting between the parties. One side of the colon is Alice’s local system the other is Bob’s local system.

A good example of an LOCC channel is quantum teleportation in Example 1.1. In Figure 1 one can see that both parties apply only local operations and classical communication. In Example 2.3 we show how this protocol conforms with the Definition 2.2 of an LOCC channel.

Example 2.3. In quantum teleportation the first requirement is a specific input state: ρA0_AB = (|ψi

A0⊗

|Φ+_i

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|ψi_A0 by LOCC. As in the definition, both Alice and Bob have their own set of superoperators. Alice’s

superoperators are given by

Φi(ρ) = |φii hφi| ρ |φii hφi| (53)

with {|φii} the Bell basis as given in Example 1.1. Note that any Φi is a completely positive map and

thatP

i|φii hφi|φii hφi| = I hencePiΦi∈ C(A0⊗ A).

Bob’s set of channels applies a unitary rotation dependent on Alice’s measurement outcome:

Ψi(ρ) = UiρUi† (54)

with Ui ∈ UP auli from Example 1.1.

Putting both sets of superoperators together we get the LOCC channel

Ξ =X

i

Φi⊗ Ψi (55)

which perfectly implements teleportation as described in Example 1.1.

2.4.1 Properties of LOCC channels

The local channels that parties apply are generally not restricted in any way. This turns out to be quite an important feature in LOCC. It allows us to change any local behaviour of an LOCC channel.

Lemma 2.4. Let ξA→B ∈ LOCC(CdA,1 : CdB,1, CdA,2 : CdB,2) be a one-way LOCC channel from A to

B and let ζpost,A, ζpost,B and ζpre,A, ζpre,B be local pre- and post-processing channels for A and B,

respectively. Then the channel

ΞA→B = (ζpost,A⊗ ζpost,B) ◦ ξA→B◦ (ζpre,A⊗ ζpre,B) (56)

is a one-way LOCC channel.

Proof. By Definition 2.2, ξA→B is of the form

ξA→B =

X

a∈Σ

Φa⊗ Ψa (57)

with Σ some set and Φa ∈ CP(CdA,1, CdA,2) and Pa∈ΣΦa ∈ C(CdA,1, CdA,2), and Ψa ∈ C(CdB,1, CdB,2).

Note that ΞA→B can be written as

ΞA→B =

X

a∈Σ

(ζpost,A◦ Φa◦ ζpre,A) ⊗ (ζpost,B◦ Ψa◦ ζpre,B). (58)

We are left to prove that {(ζpost,A◦ Φa ◦ ζpre,A) : a ∈ Σ} forms an instrument and that for any a,

(ζpost,B◦ Ψa◦ ζpre,B) is a channel. The latter follows immediately from the fact that any composition of

channels is again a channel.

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we have:

X

a∈Σ

ζpost,A◦ Φa◦ ζpre,A= ζpost,A◦

X

a∈Σ

Φa

!

◦ ζpre,A (59)

which by composition of channels is again a channel. From this it follows that ΞA→Bis a one-way LOCC

channel.

The intuition behind LOCC, as said before, is that first one party applies a generalized measurement and shares its outcome classically with the other party. Then the other party can react by applying a local channel based on the measurement outcome. This intuition can be made more explicit.

Lemma 2.5. Any one-way LOCC channel can be written as a combination of local channels and a classical messaging channel as defined in equation (42).

Proof. Without loss of generality, we prove this for right one-way LOCC channels (a similar proof works also for left one-way LOCC channels). Let Ξ ∈ LOCC(A : B) be a right one-way LOCC channel. By definition Ξ is of the form

Ξ =X

a∈Σ

φa⊗ ψa (60)

with {φa | a ∈ Σ} an instrument and ψa a channel. We split the proof up into multiple steps. First, we

need to extend the local dimensions to allow for explicit messaging. We can create a fresh local classical register MA with underlying space MA = CΣ to hold the message by using Kd_∅→0 ∈ C(C, MA), where

the superscript indicates the size of the added space and the subscript ∅ → 0 means that the new register is initialized to |0i_M

A where 0 denotes some arbitrary element of Σ. We also apply a similar map on

Bob’s side to create another message register MB initialized to |0iMB. We can later use TrMA and TrMB

to remove these registers.

The second step is to transform Alice’s instrument into a local channel using the intuition of equa-tion (46). We can define a local channel Φ ∈ C(A ⊗ MA)

Φ = X

a,j∈Σ

φa⊗ Kj→a (61)

such that applying Φ to Alice’s local system leaves her with

Φ(ρA⊗ |0i h0|_M_A) =

X

a∈Σ

φa(ρA) ⊗ |ai ha|_M_A. (62)

After this local pre-processing on Alice’s side, a classical message channel Θ ∈ C(MA⊗ MB) is applied

on registers MA and MB, see equation (42). This models the one-way LOCC step in which Alice and

Bob communicate. The other steps are pre- and post-processing steps. After the message channel Bob applies the local channel Ψ ∈ C(B ⊗ MB) defined as

Ψ =X

a∈Σ

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This results in the final one-way LOCC channel Ξ0 ∈ LOCC(A : B) given by

Ξ0= (TrMA⊗ TrMB⊗IA,B) ◦ (IA,MA⊗ ΨB,MB) ◦ (IA,B⊗ ΘMA,MB) (64)

◦ (ΦA,MA⊗ IB,MB) ◦ (K

d

∅→0,MA⊗ K

d

∅→0,MB)

which perfectly implements the original channel Ξ.

Remark. This shows that one-way LOCC can be written as two consecutive qc-channels with a message channel in between. This classical message is transmitted by ΘMA,MB which also retains the message

with Alice. This is the same as if we would create a classical register and give both parties access to it, removing ΘMA,MB. This could be an alternative definition of LOCC.

Any LOCC channel can be written as a concatenation of one-way LOCC channels. However, this representation is not unique as the intermediate dimensions of the one-way LOCC channels can differ. One one-way LOCC channel can expand the dimensions massively and another can shrink them back down again. However, we can always find a representation in which all the intermediate dimensions are the same.

Lemma 2.6. Let Ξ ∈ LOCC(A : B), then there is a ˜Ξ ∈ LOCC(A : B) such that for some n

˜

Ξ = (FA⊗ FB) ◦ γA→B,n◦ · · · ◦ γB→A,1◦ (JA⊗ JB) (65)

where FA, FB, JA and JB are local channels and {γA→B,i, γB→A,i} ⊂ LOCC(CdA⊗ CdB, CdA⊗ CdB)

are one-way right and one-way left LOCC channels with the same in- and output dimensions, such that for any ρAB, Ξ(ρAB) = ˜Ξ(ρAB).

Proof. From Definition 2.2 it follows that for some n, Ξ can be written as a concatenation of one-way LOCC protocols:

Ξ = ξA→B,n◦ ξB→A,n−1◦ · · · ◦ ξA→B,2◦ ξB→A,1 (66)

where ξA→b,i∈ LOCC(CdA,i⊗ CdB,i, CdA,i+1⊗ CdB,i+1) with dA,iand dB,i the in- and output dimensions

of the intermediate linear spaces. Note also that the arrows only indicate if it is a right or left one-way LOCC channel.

Lemma 2.4 shows that we can add any local pre- and post-processing channels to a one-way LOCC channel. Note that dA,1, dA,n are the local in- and output dimensions at A respectively and dB,1, dB,n

the local in- and output dimensions at B respectively. Now let dA∈ N be the smallest integer such that

∀i, dA,i|dAand let dB ∈ N be the smallest integer such that ∀i, dB,i|dB. Then dAwill be the dimension of

the linear space at A and dB the dimension of the linear space at B. The idea is to explicitly construct

a new set of one-way LOCC channels ˜ξA→B

i ∈ LOCC(CdA⊗ CdB, CdA⊗ CdB) by adding extra steps of

local pre- and post-processing to the original set of one-way LOCC channels.

Local pre- and post-processing will consist of adding and removing registers. Adding a register is done by Kd

∅→0, where d denotes the dimension of the added register. Removing a register is done by

taking the partial trace. We will use Trd to denote the partial trace of the added d dimensions. If we

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of consecutive one-way LOCC maps, we can rewrite Ξ in the following form: Ξ = Tr dA dA,n ⊗ Tr dB dB,n ◦ K dA dA,n ∅→0⊗ K dB dB,n ∅→0 ! ◦ ξA→B,n◦ Tr dA dA,n−1 ⊗ Tr dB dB,n−1 ◦ · · · ◦ K dA dA,2 ∅→0⊗ K dB dB,2 ∅→0 ! ◦ ξB→A,1◦ Tr dA dA,1 ⊗ TrdB dB,1 ◦ K dA dA,1 ∅→0⊗ K dB dB,1 ∅→0 ! . (67)

Now we define ˜ξA→B,i to be

˜ ξA→B,i= K dA dA,i+1 ∅→0 ⊗ K dB dB,i+1 ∅→0 ! ◦ ξA→B,i◦ Tr dA dA,i ⊗ TrdB dB,i (68)

which by Lemma 2.4 are again one-way LOCC but now with the property of having the same in- and output dimensions. Now define ˜_{Ξ ∈ LOCC(C}dA,1_{⊗ C}dB,1_{, C}dA,n_{⊗ C}dB,n_{) to be}

˜ Ξ = Tr dA dA,n ⊗ Tr dB dB,n ◦ ˜ξA→B,n◦ · · · ◦ ˜ξB→A,1◦ K dA dA,1 ∅→0⊗ K dB dB,1 ∅→0 ! . (69)

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3 Timeless LOCC

Given the fundamental importance of LOCC, we want to investigate a particular subset of LOCC channels, the time-independent or timeless LOCC channels. In this chapter we define timeless LOCC channels and show that they are a subset of LOCC. At the end of the chapter we prove that any LOCC channel can be written as a timeless LOCC channel.

3.1 Definition of timeless LOCC

We want to first develop some intuition about what timeless LOCC is. A timeless LOCC channel is an LOCC channel with no dependence on time. It might not be directly clear what is meant by the time dependence of an LOCC channel, since the standard definition LOCC does not have an explicit parameter that indicates time. However, Definition 2.2 has some notion of progression through time. Any LOCC channel can be written as concatenation of one-way LOCC channels. This can be seen as a step-wise process in which two parties send messages, process information and react to one another. These steps can be seen as a progression through time. At any of these time steps a new and different one-way LOCC channel can be applied.

This degree of freedom is taken away in timeless LOCC by allowing only two one-way LOCC channels, one right and one left. The parties are allowed to repeat the channels as often as they want. To make this more general, we allow additional local pre- and post-processing.

Definition 3.1. A channel Ξ ∈ C(X ⊗ Y, Z ⊗ W) is in TLOCC(X : Y, Z : W) if there exist channels JX→A, JY →B, FA→Z, FB→W and Φ ∈ LOCC→(A : B), Ψ ∈ LOCC←(A : B) and T ∈ N such that

Ξ = (FA→Z⊗ FB→W) ◦ (Ψ ◦ Φ)◦T ◦ (JX→A⊗ JY →B).

Remark. Here we denote LOCC→(A : B) as right one-way LOCC and LOCC←(A : B) as left one-way

LOCC

In this definition pre-processing is done by JX→Aand JY →B and post-processing is done by FA→Z

and FB→W. Notice that Ψ and Φ are dimension retaining. This is necessary to be able to repeat them

many times. For constructing general TLOCC channels that do not retain dimensions local pre- and post-processing is necessary. Pre-processing also allows the two one-way LOCC channels to work on a much larger space than the input space. As it will turn out, this is a necessary ingredient to build general LOCC channels.

What is lost by this definition is the versatility of the channel – it cannot change over time, only run for a longer time. It is also not directly clear if combining two TLOCC protocols produces another TLOCC protocol. At first sight this seems to break the time symmetry that TLOCC enforces, as after some time the channel changes from one to the other. As it turns out, the freedom gained with pre-and post-processing is enough to generate any finite-round LOCC channel pre-and hence TLOCC is closed under composition, albeit without a clear way to express the composed channel in the desired form. Pre-processing allows both parties to enlarge their local dimensions. This extra space can be used for constructing a shared clock, which can be used to keep track of how long the protocol has run for. This is illustrated in Example 3.2. Once built, this clock can be used to keep track of how far the channel has progressed and can be used to build more general channels.

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Example 3.2. A clock is built in two steps. First, we build the two local dimensions using the extension channel K_∅→0∈ C(C, L(C)), which creates a memory register for both parties. We denote these registers CA and CB for the clock register of Alice and the clock register of Bob. Secondly, let Ki→i⊕1∈ L(L(C))

as in equation (106), with i = i mod n + 1. K_i→i+1 act as a number increasing operator which makes the clock tick. With this define two one-way LOCC channels:

Φ→= n X i=0 K_i→i+1,C A⊗ ICB, (70) Φ←= n X i=0 ICA⊗ Ki→i+1,CB. (71)

Now let Ξ be an TLOCC protocol such that:

Ξ = (TrCA⊗ TrCB) ◦ (Φ←◦ Φ→)

n−1

◦ K_∅→0⊗ K_∅→0 (72)

Note that we indicate the clock registers as CA and CB. Also note that TrCAand TrCB are used to remove

the clock registers in the end. This channel can be considered the clock channel as it implements a shared clock between Alice and Bob. The added memory CA, CB keeps track of how many times the channel

Φ←◦ Φ→ has been applied, giving both parties a handle on the progression of time. Notice that time is

now explicitly encoded into a state. After i repetitions we have ρCA= |ii hi| = ρCB.

3.2 From LOCC to timeless LOCC

This clock structure shows how a time-invariant channel can keep track of time. We can use this to build any LOCC channel out of an TLOCC channel.

Theorem 3.3. Let Γ ∈ LOCC(A : B) be a finite round LOCC channel between two parties, Alice and Bob. There is an TLOCC channel Ξ ∈ TLOCC(A : B) such that Γ(ρAB) = Ξ(ρAB) for any state ρAB.

Proof. By Lemma 2.6 Γ has a decomposition in one-way LOCC channels with the same input- and output-dimensions: Γ = Tr dA dA,n ⊗ Tr dB dB,n ◦ γA→B,n◦ · · · ◦ γB→A,1◦ K dA dA,1 ∅→0⊗ K dB dB,1 ∅→0 ! , (73)

with γA→B,i∈ LOCC(CdA : CdB). Note that dA,1and dB,1, and dA,nand dB,nare local input and output

dimensions of Γ respectively.

The idea of the proof is to give both parties a local clock in which they keep track of time. To do this, we first need to change the channel to extend the local dimensions:

Γ = Tr dA dA,2n ⊗ Tr dB dB,2n ⊗ (TrCA⊗ TrCB) ◦ [γA→B,n⊗ ICA,CB] ◦ · · · ◦ [γB→A,1⊗ ICA,CB] (74) ◦ " K dA dA,1 ∅→0⊗ K dB dB,1 ∅→0 ! ⊗ Kn ∅→0,CA⊗ K n ∅→0,CB # .

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To simplify notation, we now define the pre- and post-processing channels for this TCLOCC channel: JX→A,CA⊗ JY →B,CB = K dA dA,1 ∅→0⊗ K n ∅→0,CA ! ⊗ K dB dB,1 ∅→0⊗ K n ∅→0,CB ! , (75) FA,CA→W ⊗ FB,CA→Z = Tr dA dA,2n ⊗ TrCA ⊗ Tr dB dB,2n ⊗ TrCB . (76)

Note that any LOCC channel can be decomposed into an even number of one-way LOCC channels. We could always add an extra identity channel to make it an even decomposition. Also note that [I ⊗ (TrCA⊗ TrCB)] ◦I ⊗ K

n

∅→0,CA⊗ K

n

∅→0,CA = I.

As γA→B,i⊗ I is a one-way LOCC channel, it can be decomposed into local operations and a classical

message from some alphabet Σi:

γA→B,i⊗ ICA,CB = X a∈Σi Φi_a⊗ ICA⊗ Ψ i a⊗ ICB. (77)

Note that this is very similar in the case B → A but with the roles of Φ and Ψ reversed.

The trick is now to combine the intuition of Example 3.2 and the decomposition in one-way LOCC channels to create two one-way LOCC channels that implement Γ if repeated many times. The two one-ways LOCC’s are:

ξA→B= i=n+1

X

i=1 a∈Σi

ΦA→B,i_a ⊗ K_i−1→i⊗ ΨA→B,i

a ⊗ ICB, (78) ξB→A= i=n+1 X i=1 a∈Σi ΨB→A,ia ⊗ ICA⊗ Φ B→A,i a ⊗ Ki−1→i. (79)

Note that i runs from 1 to n + 1 and ΨA→B,n+1a = ΦA→B,n+1a = ΨB→A,n+1a = ΦB→A,n+1a = I. This is

necessary to build a valid channel.

It is clear that ΨA→B,ia ⊗ ICB is a channel, hence all that is left to show is that {Φ

A→B,i

a ⊗ Ki−1→i |

i ∈ {1, . . . , n + 1} and a ∈ Σi} is an instrument. By definition we know thatP_a∈Σ_iΦA→B,ia is a channel

and that K_i−1→i is completely positive. If Pn+1

i=1 Ki−1→i is a channel

Pn+1 i=1 P a∈ΣiΦ A→B,i a ⊗ Ki−1→i

Hence {ΦA→B,ia ⊗ Ki−1→i | i ∈ {1, . . . , n + 1} and a ∈ Σi} is an instrument and therefor ξA→Band ξB→A

are both one-way LOCC channels. Now note what happens if we apply ξB→A once to the state:

ξB→A(ρAB⊗ |0i h0|CA⊗ |0i h0|CA) =

X

i

γB→A,i(ρAB) ⊗ |ii hi − 1| |0i h0|CA|i − 1i hi| ⊗ |0i h0|CB

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As we repeat this, the clock keeps increasing and the one-way LOCC channels γB→A,i are applied in

correct order.

To conclude the proof, define Ξ ∈ TLOCC(A : B) as

Ξ = FA,CA→W ⊗ FB,CB→Z◦ (ξA→B◦ ξB→A)

n

◦ JX→A,CA⊗ JY →B,CB. (82)

The final output of Ξ is

Ξ(ρAB) = Tr dA dA,2n ⊗ Tr dB dB,2n ⊗ (TrCA⊗ TrCB)

Γ(ρAB) ⊗ |0i h0| ⊗ |0i h0| ⊗ |ni hn|_C_A⊗ |ni hn|_C_B ,

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which coincides with the original output Γ(ρAB).

As any TLOCC channel is clearly an LOCC channel this proves that TLOCC = LOCC.There are two interesting effects happening: the time dependence of an LOCC channel is encoded into physical dimensions, and these dimensions are classical and remain such throughout the protocol. This suggests that there might be a simpler definition for TLOCC. In effect both ξA→B and ξB→A could be considered

qc-channels. We have seen a similar construction for one-way LOCC channels. In Lemma 2.5 we showed how we can use two local qc-channels and a classical message channel to perform a one-way LOCC channel. TLOCC could be constructed in a similar fashion.

3.3 Alterative definition of TLOCC

We could define TLOCC as the repetition of two qc-channels. We do this in a similar fashion as Lemma 2.5. The only difference is that we allow both qc-channels to interact with the same classi-cal register. This leaves us with the following definition:

Definition 3.4. A channel Ξ ∈ C(A ⊗ B) is in TLOCC(A : B) if there exists an extending channel^ K_∅→0∈ C(C, M) and two qc-channels Φ ∈ C(A ⊗ M) and Ψ ∈ C(B ⊗ M) and a T ∈ N such that

Ξ = TrM◦(ΨBM◦ ΦAM)◦TK∅→0,M. (84)

One can see that TLOCC is the same as TLOCC by studying ξ^ A→B and ξB→A from the proof of

TLOCC = LOCC. Applying ξA→B once consists of performing a channel γA→B,i, conditioned on the

local clock being in state i, and then increasing the local clock:

ξA→B= n+1

X

i=1

γA→B,i⊗ Ki−1→i,M. (85)

By using Lemma 2.5, we can rewrite this in the following form:

ξA→B= n+1 X i=1 TrMM◦(Ψ i BMM◦ Φ i AMM) ◦ K∅→0,MM ⊗ Ki−1→i,MC. (86)

Here we split the classical message space into two parts, MM and MC, to accommodate the message

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classical register is created at the start of the channel. Then we can rewrite ξA→B as TrMM◦ ξA→B◦ K∅→0,MM = TrMM◦ n+1 X i=1 Ψi_BM M⊗ Ki−1→i,MC ◦Φi_AM M⊗ Ki−1,MC ◦ K_∅→0,MM. (87)

If we are repeating this channel, we do not need to trace away and rebuild the message register MM

because ΦAMM overwrites whatever message is in MM. A similar decomposition exists also for ξB→A:

TrMM ◦ ξB→A◦ K∅→0,MM = TrMM◦ n+1 X i=1 Ψi_AM M⊗ Ki,MC ◦Φi_BM M⊗ Ki,MC ◦ K_∅→0,M_M, (88) where we used Φi

BMM to denote Bob’s local instruments and Ψ

i

AMM to denote Alice’s local channels.

Since we are repeating ξB→A◦ ξA→B, we only need to increment i once during such a repetition. By

putting these two together we get

TrMM◦ ξB→A◦ ξA→B◦ K∅→0,MM (89) = TrMM◦ n+1 X i=1 Ψi_AM M ⊗ Ki,MC ◦Φi_BM MKi,MC ◦Ψi_BM M⊗ Ki−1→i,MC ◦Φi_AM MKi−1,MC ◦ K_∅→0,MM.

By grouping the correct terms together we find the two qc-channels

ΦAMMMC = n+1 X i=1 (Φi_AM_M◦ Ψi AMM) ⊗ Ki−1, (90) ΨAMMMC = n X i=1 Φi_BM_M◦ Ψi BMM ⊗ Ki−1→i,MC, (91) where Ψ0 AMM = I = Φ n

AMM are the identity channels. For every repetition of the qc-channel the parties

first react to the received message. After that they apply an instrument and send and change the classical message register to the measurement outcome. With these two operators we get

(FA,CA→W ⊗ FB,CB→Z) ◦ (ξA→B◦ ξB→A)

n

◦ (JX→A,CA⊗ JY →B,CB) (92)

= TrMM,MC◦ (ΨBMMMCΦAMMMC)

◦T_{◦ K}

∅→0,MMMC.

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4 The Lindblad picture

In quantum mechanics one of the most important and well-known equations is the Schr¨odinger (1926) equation. It describes the time evolution of states as governed by the energy of a system. In a closed system this is done by a Hamiltonian H ∈ L(H), a Hermitian operator, H† _{= H, that measures the}

energy of a system. The most general form for pure states is

i~d

dt|ψi = H(t) |ψi . (93)

If assumed that the Hamiltonian is time independent, H(t) = H for ∀t, and set ~ to 1 we can solve and get the general solution

|ψ(t)i = e−iHt|ψ(0)i . (94)

which describes unitary evolution. This can be extended to unitary evolution of density matrices

ρ(t) = e−iHtρ(0)eiHt. (95)

which is known as the von Neuman equation.

The Schr¨odinger equation describes the evolution of a closed quantum system, in which information can never be destroyed or created. There are however plenty of examples of channels that destroy information. Take for instance the dephasing channel:

D(ρ) =

n

X

i=0

|ii hi| ρ |ii hi| (96)

This channel destroys all quantum information in the system. In general it is impossible to do any kind of measurement using Hamiltonian evolution. A continuous quantum channel should be able to lose information and do measurements.

In a closed system this is impossible. Information cannot be destroyed and cannot flow out of the system. For a system to lose information it needs to be coupled to an environment. The information can than flow from the system to the environment resulting in the loss of information. The time evolution equation that arises from coupling a system to an environment is called the Lindblad (1976) equation.

In this chapter we will first derive the Lindblad equation. We will study how we can use the Lindbald equation to evolve qc-states. This involves adding extra constraints to the Lindblad equations which will derived. This is followed up by a discussion about Markovian channels, channels driven by a Lindbladian.

4.1 The Lindblad equation

Time evolution for open systems involves two main ingredients: a system to evolve and an environ-ment. Together the system and the environment form a closed system. Their time evolution is therefore described by the Schr¨odinger equation given as

d

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With S we denote the system to be evolved and with E we denote the environment. By tracing out the environment we end up with the evolution of only system S. It is not very clear what kind of differential equation we end up with after tracing out E.

To derive the leftover behaviour, we need to make the assumption that the evolution is Markovian. For an evolution to be Markovian we assume that the environment is memoryless. Information flowing into the environment will not flow back at a later point in time or even stronger, that there is no information flowing from the environment to the system at all. In general this is quite a reasonable assumption. Environments tend to be much larger than the system we are studying, hence any information flowing in it is quickly lost. The Markovian assumption greatly simplifies the time evolution. It ensures that the time evolution is homogeneous. This ensures time evolution of the form

ρ(t + s) = Ts(ρ(t)), (98)

with Tsthe quantum channel that propagates ρ(t) forward in time for a time s.

In a physical system time evolution is completely determined by a systems energy. As this should hold for any amount of time s, we should not just have one channel Ts but a family of channels {Ts}s,

one for every length of time. These channels should obey a simple composition rule:

Ts1◦ Ts2 = Ts1+s2. (99)

First evolving for a time s1 and then for a time s2should be the same as evolving for total time s1+ s2.

Besides that we assume that we are propagating forward in time. Effectively this means that s ≥ 0, with T0= I. Because of these two properties the family Ts forms a semi-group.

A physical time evolution is a continuous process, hence it is very natural to impose this continuity on the family {Ts}. Using the trace norm on the set of states D(H) we impose that Ts→ Ts0 converges

strongly: when s → s0then |Ts(ρ) − Ts0(ρ)| → 0 for all ρ ∈ D(H).

Michael Wolf (2012) proved in proposition 7.1 of his lecture notes, that a dynamical semi-group equipped with a notion of continuity, as stated above, is differentiable. Further more he showed that for finite H Ts is of the form Ts = esL, with generator L ∈ L(L(H)). L has to be traceless and time

independent to keep the evolution Markovian. Using this we can study infinitesimal time evolution of a state:

ρ(t + δt) = Tδt(ρ(t)). (100)

This equation can be expanded to

ρ(t) + δtd

dtρ(t) + O(δt

2_{) = I(ρ(t)) + δtL(ρ(t)) + O(δt}2_). ₍₁₀₁₎

From which we can derive

d

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If L is time-independent this gives the general solution

ρ(t) = eLtρ(0). (103)

From this point on we will be using proofs given in the lecture notes of Michael Wolf. Ts is a channel

and hence completely positive, from this it follows that

0 ≤ (etL⊗ I) |Ωi hΩ| = |Ωi hΩ| + (L ⊗ I) |Ωi hΩ| + O(t2),

where |Ωi is the maximally entangled state. Sending t → 0 and projecting on the orthogonal to the maximally entangled state P = I − |Ωi hΩ| gives:

0 ≤ P (L ⊗ I) [|Ωi hΩ|] P

By proposition 7.2, found in Michael Wolf’s notes, we can rewrite L as follows

L(ρ) = φ(ρ) − κρ − ρκ†_, ₍₁₀₄₎

φ∗(I) = κ + κ†, (105)

for some completely positive map φ ∈ CP(H) and some linear operator κ ∈ L(H). The second identity is found by using that L has trace 0 and that φ is completely positive and Hermitian-perserving. We can use this to find a definition for κ:

κ = 1 2φ

∗_{(I) + iH,} _{with H a Hermitian operator.} ₍₁₀₆₎

Using a Kraus representation of φ, we find the Lindblad equations:

φ(ρ(t)) =X a∈Σ LaρL†a, κ = 1 2 X a∈Σ L†_aLa+ iH, L(ρ(t)) = −i[H, ρ(t)] +X a∈Σ −1 2{L † aLa, ρ(t)} + Laρ(t)L†a. (107)

We call this operator L the Lindbladian. Notice that by construction the Lindbladian is indeed traceless and completely positive, as is required. In the rest of the thesis we will refer to the Hermitian matrix H as the Hamiltonian or Hermitian part of the Lindbladian. The linear operators La will either be referred

to as jump operators or the dissipative part of the Lindbladian.

4.2 Time evolution of quantum-classical states

As seen in the case of TLOCC, it will turn out that quantum-classical states, or qc-states, are a crucial ingredient of defining CLOCC. This chapter will focus on developing a more solid understanding of qc-states and the behaviour of Lindbladians evolving qc-states through time.

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A thorough derivation for classical states can be found in appendix A.6. We can use the intuition of that derivation to derive restrictions on the Lindbladian for time evolving qc-states.

We will do this in two steps. First, we find the structure for Lindbladians that retain the qc-structure of states. Secondly, we show that all Lindbladians restricted to qc-states can be written in that form.

Lemma 4.1. Let ρqc=P_ipiρi⊗ |ii hi| ∈ D(X , M) be a quantum-classical state. Here X is the quantum

system and M is the classical one. Let L ∈ L(L(X ⊗ M)) be a qc-Lindbladian with the following jump operators and Hamiltonian:

Lqc_ij = Mij⊗ |ii hj| (108)

Hqc=X

k

Hk⊗ |ki hk| . (109)

Any Lindbladian of this form retains the qc-structure of a qc-state for any t.

Proof. By equation(103) we have that at any t, ρqc(t) is given by:

ρqc(t) = eLt[ρqc(0)] = X n 1 n!(Lt) n [ρqc(0)] (110)

Hence if L(ρqc(0)) is a qc-state, so is ρqc(t) for any t. Since

L(ρqc) =

X

kl

+X

ijl

(Mij⊗ |ii hj|)(plρl⊗ |li hl|)(Mij† ⊗ |ji hi|)

−1 2[(M

†

=X

ij

ipi[Hi, ρi] ⊗ |ii hi| + pjMijρjMij† ⊗ |ii hi| −

1 2(pj{M

†

ijMij, ρj} ⊗ |ji hj|)

is a qc-state, ρqc(t) retains it’s qc-structure for any t.

Remark. Notice that if we take the trace over the quantum system we are exactly left with classical evolution as in appendix A.6.

hl| d dtρc|li = X j TrhMljρjM_lj† i pj− Tr h MjlρlM_jl† i pl

We can now show that when restricting to qc-states any Lindbaldian can be written in the form of equations (108) and (109).

Lemma 4.2. Let L ∈ L(L(X ⊗ M)) be a Lindbladian that for any t retains the qc-structure of a state. Then there is another Lindbladian ˆL ∈ L(L(X ⊗M)) with operators as given by Lemma 4.1 that restricted on qc-states acts the same as L.

From LOCC to continuous LOCC

Master Physics & Astronomy

Theoretical physics

Master’s thesis

From LOCC to continuous LOCC

A study of continuous classical-quantum interactions

by

Marten Folkertsma

10428666

December 2020,

60 EC,

Acknowledgements

Contents

List of Symbols

1

Introduction

2

Preliminaries

2.1

Quantum states

2.2

A notion of entanglement

2.3

Quantum channels

2.4

LOCC channel

3

Timeless LOCC

3.1

Definition of timeless LOCC

3.2

From LOCC to timeless LOCC

3.3

Alterative definition of TLOCC

4

The Lindblad picture

4.1

The Lindblad equation

4.2

Time evolution of quantum-classical states