Dipartimento di Fisica “E. Fermi”
Corso di Laurea in Fisica Teorica
Tesi di Laurea Magistrale
Degradation of Entanglement
in Markovian Noise
Canditato:
Dario Gatto
Relatore:
Prof. Vittorio Giovannetti
Correlatore:
Dr.ssa Antonella De Pasquale
First and foremost I would like to thank my thesis advisor, Prof. Vit-torio Giovannetti. I was brought to him merely by our common interest in quantum information, but I found a mentor with whom I share so much of my view of physics. His extensive expertise is surpassed only by his physical intuition, and thanks to his constant and careful guidance I have grown so much as a scientist.
The next person in the list has to be my co-advisor, Dr. Antonella De Pasquale. She was the first to notice my inclinations and recognize my efforts. Not only she took the time to go through all my drafts, always re-sponded promptly to emails, and was always kind to me: most importantly, she never stopped encouraging me.
As this adventure in a new city comes to a closing, I can not help looking back. I want to thank my friends from home for not having forgotten me: thank you Aldo, Ferdinando, Francesca, and Maria for your visits through-out these years. The strength of the bonds we formed as children fills me with joy, and comfort.
Writing a thesis was a stressful task, the intensity of which I could not anticipate. Those who were close to me know it better than anyone. Among them, special thanks go to Giorgio and Nicola. In these years they have taken both the role of the friend, and of the colleague. Too many times they had to come to my crib and take me out for a beer. On the other hand, it was the inspiration I took from conversations with them (which could literally go on for hours) that made me break through many impasses in this career, including part of this thesis. Thank you for everything: I will never forget that afternoon in Marina di Pisa, and the taste of anchovies. Also, one special mention goes again to Maria, who was the only one to believe I could solve equation (4.26), and one goes to Manculo, for helping me out with numerics.
Last but not the least, I want to thank my family, and especially my parents, for having always been a pillar of support. It was only thanks to their efforts that I could make it to this point. You were my safety net every time I fell, so thank you, and I hope I have made you proud.
Acknowledgements ii
Introduction and Motivations v
1 Quantum Correlations: Bipartite Entanglement 1
1.1 The Space of States . . . 1
1.2 Measurements in Quantum Mechanics . . . 4
1.3 Composite Quantum Systems and Bipartite Entanglement . . 6
1.3.1 Composite Systems of Pure States . . . 6
1.3.2 Composite Mixed Systems and Purifications . . . 8
1.3.3 Separability Criteria and Entanglement Measures . . . 11
2 Quantum Evolution: Channels and Dynamical Semigroups 17 2.1 Quantum Channels . . . 17
2.1.1 Stinespring and Kraus Representations . . . 17
2.1.2 Axiomatic Representation and the Choi-Jamio lkowski Correspondence . . . 20
2.1.3 Stationary States . . . 23
2.1.4 Qubit Channels . . . 23
2.2 Quantum Dynamical Semigroups . . . 24
2.2.1 Microscopic Derivations . . . 26
2.2.2 Stationary States and Accumulation Points . . . 27
2.2.3 Resolvent Expansion . . . 27
3 Entanglement-Breaking Channels: Theory and Experiments 29 3.1 Theory of Entanglement-Breaking Channels . . . 29
3.2 Entanglement-Breaking Order and Amendable Channels . . . 32
3.2.1 Experimental Test of Entanglement Restoration . . . 34
3.3 Cut-and-Paste Entanglement Restoration Protocols . . . 35
3.3.1 Experimental Tests of Entanglement Restoration . . . 37
4 Entanglement Transmission Time 39 4.1 Entanglement-Breaking Conditions . . . 39
4.1.1 Invariance Under Unitary Conjugation . . . 41
4.2 The Bit-Flip Channel . . . 42
4.2.1 Introduction of the Driving . . . 44
4.2.2 Inequivalent Drivings . . . 49
4.3 The Amplitude Damping Channel . . . 53
4.3.1 Introduction of the Driving . . . 54
4.3.2 Inequivalent Drivings . . . 57
5 Conclusions 59
A Proof of the Structure Theorem for Quantum Dynamical
Semigroups 61
B Newton’s Tangents Method 65
Motivations
Quantum information theory describes how information can be encoded in quantum systems, and how that information can be processed by means of quantum operations. A theory of information based on quantum me-chanics naturally extends classical information theory [17]: the new theory features quantum generalizations of classical notions such as codes [46,41], sources, channels [34, 43], as well as two complementary kinds of informa-tion – classical informainforma-tion and quantum entanglement. The foundainforma-tions of the theory are now established [6], but the idea of quantum computation goes back at least to Feynman [22].
Since the introduction of the transistor, the computer industry has ex-perienced an incredible growth rate. Moore’s empiric law states that the number of transistors in a microprocessor will double every eighteen months: a trend which has been in fact been observed for over fifty years, but can not hold indefinitely. Today, a typical scale for a transistor is 14 nm. At some point, as we learn to manufacture smaller and smaller transistors, electrons will simply begin to tunnel through them. Overcoming this fundamental limitation is a challenge the physicists of our century will eventually have to face.
What is more interesting, a fully implemented quantum computer has the potential to outperform any classical computer [18, 28]. A quantum algorithm can efficiently simulate quantum many body-systems, whereas on a classical computer the same simulations are often intractable [22]. Fur-thermore, the problem of factorization of large integers in prime factors, for which no classical efficient algorithm is known, becomes feasible on a quantum computer [57], with serious implications for security protocols. Applications of quantum information, however, are not limited to quantum computation: today they include quantum communication [27], quantum metrology [24, 25], and quantum cryptography [5]. All these features de-pend on the possibility to cleverly exploit inherently quantum properties such as uncertainty, interference, and entanglement.
Entanglement, in particular, is one of the most striking features of quan-vii
tum mechanics. Indeed, entangled composite systems exhibit correlations with no classical counterpart. Entangled states were used by Einstein, Podolski and Rosen in an attempt to derive a paradox which could enlighten the inconsistency of quantum mechanics, in favor of a theory of hidden vari-ables [21]. Later, Bell and others worked out a set of inequalities which any such theory should respect [2,52]. However, by measuring the correlations of an entangled composite system, it was shown that the Bell inequalities are, in fact, violated [23]. After the first Bell test many other similar tests followed over the years, including the spectacular satellite distribution of entangled photon pairs over more than 1200 km, recently achieved by [64].
In quantum information, entanglement is a resource. Entanglement has been shown to be extremely successful in order to boost and overcome the limits of many classical protocols dealing with computational speed up, com-munication efficiency or metrological accuracy [50]. Furthermore, there exist some protocols which could not be even conceived without recurring to en-tangled states, an example is given by superdense coding [7,48], or by the outstanding procedure of quantum teleportation [3,9].
However, entanglement is an extremely fragile resource and needs to be actively protected [4]. Conventional data processing operations can destroy or at least corrupt entanglement, and even using quantum communication lines degrade it – typically exponentially fast. The ability to preserve and manipulate entangled states is crucial for quantum computers, being respon-sible both for their unconventional power and the difficulty of building them. In this thesis we study the degradation of entanglement in a composite two-qubit system subject to a Markovian, time homogeneous noise [47, 1, 42] which, for instance, can model the transmission of photons through an op-tical fiber. Specifically, we focus on the study of the maximum time t∗ for which, under the action of such noise models, the system of interest is capable to preserve the entanglement it has initially established with an ex-ternal ancilla. In particular we analyze how such quantity depends upon the relative strength of the dissipative and coherent (that is, Hamiltonian) components of the dynamical generator which describes the system evolu-tion. For the models we have considered, a non-trivial behavior is observed. In particular, in some cases it does happen that t∗ increases with the above mentioned relative strength, meaning that the higher is the dissipative con-tribution to the noise, the longer is the associated entanglement survival time.
The structure of the thesis is as follows. The first chapter introduces the reader to the space of quantum states, with an empashis on how entangle-ment is defined and how it can be quantified and measured. In the second chapter the discussion continues with quantum channels, which provide the correct tool for describing noise in a quantum communication line. The notion of quantum dynamical semigroups, which give the description of a Markovian noise, is also introduced. The third chapter focuses on the class
entanglement-breaking channels, which from the point of view of quantum correlations represent the most undesirable form of noise possible, as the name suggests. A brief discussion on the related topics of amendable chan-nels and entanglement restoration protocols concludes the chapter. In the fourth chapter the original results are presented: we study the entanglement transmission time, defined as the time at which entanglement is completely broken for the bit-flip and and the amplitude damping channel. The the-sis finally ends with a conclusive section where we summarize the results obtained.
Quantum Correlations:
Bipartite Entanglement
The theory of density matrices is at the core of the field of quantum information [50, 44, 61]. The associated formalism is a natural choice for describing open quantum dynamics [8], and thus is particularly suitable in order to grasp the effects of external noise on a quantum system. In this chapter we will outline the basics of the theory, introducing the notion of bipartite entanglement, that is, quantum correlations.
We will then discuss how the entanglement content of a given quantum system can be quantified and measured [54]. While we are far from having a full-fleged theory of entanglement, a wide arrangement of separability cri-teria have been developed, allowing in some simple but physically relevant cases to conduct experimental tests with current technology [15].
1.1
The Space of States
According to the prescriptions of quantum mechanics, to every physical system is associated a Hilbert spaceH. If the system under consideration is isolated, the state of the system is completely described by a unit vector |ψi ∈ H. Unit vectors differing only by a complex phase actually represent the same state. They are known as pure states and represent a limited class of states corresponding to the physical situation in which our knowledge of the system is completely determined.
In general, physical systems may suffer the presence of external noise, and thus they will no longer be in a well-defined state, but rather in one of the pure states|ψ1i, ..., |ψni. If p1, ..., pn are the (classical) probabilities as-sociated to such states, the system is henceforth described by the ensemble given by the following collection of couples
E = {(p1,|ψ1i), ..., (pn,|ψni)}. (1.1) 1
In this case we say that the state of the system is represented by a statistical mixture of pure states, or equivalently the system is said to lay in a mixed state. For example, the state of the system may describe the statistics of a given experiment. So if one measures the observable Θ and the state of the system is described by the ensembleE, the expectation value for the measured observable Θ reads
hΘi = n X
i=1
pihψi|Θ|ψii. (1.2) Such slightly involved procedure can be simplified by describing the sys-tem with a linear operator ρ, the so-called density matrix. In particular, if the state is pure, the associated density matrix reads
ρ =|ψihψ|, (1.3)
which is no longer affected by the redundancy|ψi ∼ eiθ|ψi. On the contrary, if the system is given by the ensemble (1.1), we have
ρ = n X
i
piρi, with ρi =|ψiihψi|. (1.4) It can be shown [50,8] that a given density matrix ρ can be obtained from different ensembles of pure states. Namely, ifE0={(q
1,|ϕ1i), ..., (qn,|ϕni)} is another ensemble, E0 gives rise to the same density matrix associated to E if and only if √p i|ψii = n X j=1 uij√qj|ϕji, (1.5) where the uij’s are the entries of a unitary matrix [56,39]. Neverthless, in both cases the expectation values (1.2) are are equivalently computed as
hΘi = tr(Θ ρ). (1.6)
In this sense the description of physical systems in terms of density matrices is extremely convenient.
A generic density matrix ρ always verifies the conditions of 1. Semidefinite positivity (ρ≥ 0):
hx|ρ|xi ≥ 0 ∀ |xi ∈ H, (1.7) 2. Trace unitarity:
tr(ρ) = 1. (1.8)
Definition 1. A positive semidefinite linear operator of unitary trace is called a density matrix. The set of all density matrices
S(H) := {ρ : H −→ H linear | ρ ≥ 0, tr(ρ) = 1}, (1.9) is called the space of states1.
The density matrix description of quantum mechanics provides a simple way of dealing with mixed states. It is worth noticing that the space of states S(H) (unlike H itself) is not a vector space, however, it is compact and convex (see Chapter 2).
It would be desirable to have a simple way of deciding, given a density matrix, whether it describes a pure or a mixed state. An answer is provided by the following criterion:
Theorem 1. All ρ∈ S(H) verify the inequality
tr(ρ2)≤ tr(ρ) = 1, (1.10) and the inequality is saturated if and only if ρ is pure [50].
By introducing the Hilbert-Schmidt, or Frobenius inner product be-tween two operators Θ and Ξ on the Hilbert spaceH
(Θ|Ξ) := tr(Θ†Ξ), (1.11) relation (1.10) reads
||ρ||2 = (ρ|ρ) = tr(ρ2)≤ 1, (1.12) implying that pure states lie on the boundary of the space of states.
A qubit is a term referring to any 2-level quantum system with two com-pletely distinguishable states [11, 49]. The qubit is the quantum analog of the bit, and represents the smallest possbile chunk of information. However, unlike a classical bit, which can only exists in one of the two logical states 0, 1, a qubit can exist not only in the quantum states{|0i, |1i}, but in any superposition of them. For a survey of possible physical implementations, refer to [50]. The underlying Hilbert spaceH of a qubit is always isomorphic to C2. In particular, choosing as operator basis{I, X, Y, Z}, where I is the identity and where X, Y , Z are the Pauli matrices
X :=0 1 1 0 , Y :=0 −i i 0 , Z =1 0 0 −1 , (1.13)
we find that any quantum state can always be represented in the form ρ = I + ~n· ~σ
2 , (1.14)
1To avoid confusion we will refer to
H as “the underlying Hilbert space”, reserving the term “space of states” for S(H), the set just defined.
|0i
|1i
X
Y
Z
Figure 1.1: The Bloch sphere. The computational basis vectors are repre-sented: |0i and |1i. In the Bloch sphere, orthogonal pure states are antipo-dal boundary points, and convex combination of density matrices directly correspond to segments.
where ~σ = (X, Y, Z) and by Hermicity, ~n must be a vector of real numbers. The vector ~n is called the Bloch vector, and it characterizes a density matrix completely. By imposing (1.10) we find immediately that the Bloch vector satisfies
|~n|2 ≤ 1, (1.15)
with equality if and only if ρ is pure. In other words, in the case of a qubit the space of states is a unit 3-ball, usually called the Bloch sphere2 — see
figure (1.1).
The Bloch sphere provides a useful visual representation of the physics of a qubit. It can be used, for instance, to follow a quantum algorithm as a sequence of discrete rotations. Another application which we will see in Chapter2 is the visualization of an arbitrary noise model.
Futhermore, the components of the Bloch vector correspond to mean values of physically relevant operators
n1= tr(X ρ), n2 = tr(Y ρ), n3= tr(Z ρ), (1.16) and can hence be determined experimentally (the resulting process being known as quantum state tomography)[50,15].
1.2
Measurements in Quantum Mechanics
Quantum mechanics prescribes that to every observable should be as-signed a self-adjoint operator Θ. The possible outcomes of the measure-2When the physical implementation of the qubit is the polarization of a photon, the
ment described by Θ correspond to its (real) eigenvalues{λi}ri=1, so that its spectral decomposition reads
Θ = r X
i=1
λiΠi, (1.17)
where the Πi’s are the corresponding orthogonal projectors fulfilling the completeness relation
r X
i=1
Πi= I. (1.18)
Let us suppose our quantum system to be in the state|ψi. If the measure-ment gave the result λi, the state after the measurement, by prescription, is given by the projection on the i-th eigenspace
|ψ0i = Πi|ψi phψ|Πi|ψi
, (1.19)
while the probability of the event is given by
pi =hψ|Πi|ψi. (1.20) The measurement described above is called projective measurement. Projective measurements are sufficient to describe ideal measurements in closed quantum systems. For instance they exhibit the property of being repeatable. In fact, if a projective measurement is applied again an instant after it has taken place, the state will not change
|ψi −→ |ψ0i = Πi|ψi phψ|Πi|ψi −→ |ψ00i = Πi|ψ 0i phψ0|Π i|ψ0i =|ψ0i, (1.21) and the measurement will result in the same outcome with probability 1.
However in order to account for interactions with an external environ-ment, or for noisy fluctuations, the theory of generalized measurements is more adequate [50, 8]. According to this prescription, a measurement is described by a collection of operators {Ai}ri=1, labeled by an index i corre-sponding to the possible outcomes λi of the measurement, and subject to the completeness condition
r X
i=1
A†iAi = I. (1.22) Suppose the quantum system of interest to be in the state ρ. If the outcome corresponding to the index i occurred in the measurement, the state after the measurement is, by prescription,
ρ0= AiρA † i tr(A†iAiρ)
Furthermore, the probability for the outcome i to have occurred is
pi= tr(A†iAiρ), (1.24) so that the completeness condition ensures the normalization of probabilities
r X
i=1
pi= tr(ρ) = 1. (1.25)
The measurement described above is called a generalized measure-ment. This formalism is capable of describing several physical situations, such as approximate, indirect [8], and of course projective measurements as a special case by simply identifying the operators Ai with the projectors Πi. In many cases of physical interest, one is only interested into the statistics of the measurement outcomes, but not into what happens to the state of the system after the measurement. In this situation the formalism of generalized measurements can be simplified. Indeed by close inspection of (1.24) we observe that it can written as
pi= tr(Piρ), (1.26) with the Pi’s being the positive operators
Pi = A†iAi, (1.27) which by construction fulfill the normalization condition
r X
i=1
Pi = I. (1.28)
Motivated by this observation, let us give the following definition. Definition 2. A collection of positive operators {Pi}ri=1 satisfying the nor-malization condition (1.28) is called a positive-operator-valued mea-sure, or POVM.
The formalism of POVMs is a simple mean for describing a general-ized quantum measurement, when the knowledge the state after the mea-surement has taken place is not important [50, 8]. POVMs fall naturally under the category of generalized measurements. One important exam-ple of POVMs in quantum mechanics is the distinction between two non-orthogonal states [51].
1.3
Composite Quantum Systems and Bipartite
Entanglement
1.3.1 Composite Systems of Pure States
A composite quantum system is a system composed by more than one subsystem. The simplest scenario corresponds to a bipartite system AB, in which the two subsystems can be even hold by distant parties, typically dubbed Alice and Bob in the computer science tradition.
Suppose A and B are uncorrelated, and let A be in the state |ϕiA and B be in the state|ϕ0i
B. Then by prescription, the state of the total system is
|ψi = |ϕiA|ϕ0iB. (1.29) The state |ψi belongs to the total Hilbert space HA⊗ HB. However states of the form (1.29) do not exhaust all possible states. Such states are called factorized. The principle of superposition demands that also states of the form |ψi = n X i,j=1 wij|ϕiiA|ϕ0jiB, (1.30) to be physical states. A pure state which cannot be written as a factorized state is called entangled.
The constituents of an entangled composite state are correlated, in that a measurement on a part of the composite system, say, performed by Alice’s party, instantaneously affects the state of the whole system, and therefore the outcome of a measurement performed by Bob’s party will somehow depend on the outcome of the measuremement previously performed by Alice, and vice versa. This is the famous “spooky action at a distance” described by Einstein [21]. As mentioned in the introduction, this is a purely quantum mechanical form of correlation, much stronger than what would be classically possible. For instance, it is possible to violate the Bell inequalities using such correlations [2,12,23], whereas by Einstein’s principle of local realism, no such violation should be possible.
A natural question to ask is: given a bipartite quantum state, is there any criterion for deciding whether it is entangled or not? In the case of pure states, the Schmidt decomposition provides a complete answer [50].
Theorem 2 (Schmidt decomposition). Let |ψi ∈ HA⊗ HB be a bipartite pure state, let nA = dimHA, nB = dimHB, n = min{nA, nB}. There exists orthonormal bases{|ϕiiA}nAi=1 of HA and{|ϕ0iiB}nBi=1 of HB, and non-negative numbers {si}ni=1 such that
|ψi = n X i=1 si|ϕiiA|ϕ0iiB, n X i=1 s2 i =||ψ||2 = 1. (1.31)
Proof. Let {|eiiA}ni=1A be an orthonormal basis of HA, let {|e0iiB}ni=1B be an orthonormal basis ofHB. For suitable complex coefficients wij, we have
|ψi = nA X i=1 nB X j=1 wij|eiiA|e0jiB. (1.32)
Let W denote the matrix which entries are the wij’s. By the singular value decomposition [35,50], it holds
W = U†ΣV, (1.33)
where U, V are unitary matrices, and Σ is a diagonal matrix with non-negative entries. Its diagonal entries si are called the singular values of A. Applying the singular value decomposition, we can write
|ψi = nA X i=1 nB X j=1 n X k=1 u∗kiskvkj|eiiA|e0jiB, (1.34)
and by virtue of the unitarity of U, V , the vectors |ϕiiA= nA X j=1 uij|ejiA, |ϕ0iiB = nB X j=1 vij|e0jiB, (1.35)
form orthonormal bases ofHA,HB respectively. This completes the proof.
Please notice how the orthogonal bases mentioned in the statement of the theorem depend on the state, in general.
Factorizable states are exactly those which have only one non-zero sin-gular value. The number of non-zero sinsin-gular values is called Schmidt number, and it is simple to see that it does not depend on the bases chosen in the proof. Entangled pure states are exactly those with Schmidt number greater than 1.
For instance by making use of the computational basis we can build the maximally entangled state3
|Ωi = √1 n n X i=1 |eii|eii, (1.36) which attains the maximal Schimdt number, n [62].
3It is actually possible to build other maximally entangled states, but for the sake of
1.3.2 Composite Mixed Systems and Purifications
Consider again a bipartite quantum system AB. If A is in the state ρA (which in general might be mixed) and B is in the state ρB, and if A and B are uncorrelated, then the total system is described by the density matrix
ρ = ρA⊗ ρB. (1.37)
Once more, states the form (1.37) do not exhaust all possible states. Such states are called factorized. Indeed, the systems A and B might be correlated. Let us give the following definition.
Definition 3. Letρ ∈ S(HA⊗ HB) be a bipartite quantum state. If ρ can be written as a convex combination of factorizable states
ρ =X i
piρA,i⊗ ρB,i (1.38) it is called a separable state. If a state is not separable, it is called en-tangled. Here thepi’s are positive numbers which sum to one.
One of the advantages of the description of quantum states with density matrices is that it allows to discard very simply the degrees of freedom of one of the parties (which might represent the environment).
Definition 4. LetΘ be a linear operator acting onHA⊗HB. Let{|eiiA}nAi=1 be an orthonormal basis ofHA, let{|e0iiB}ni=1B be an orthonormal basis ofHB. The partial traces of Θ with respect to HA and HB are the operators
trA(Θ) := nA X i=1 hei|Θ|eiiA, trB(Θ) := nB X i=1 he0i|Θ|e 0 iiB. (1.39) Notice that the partial traces do not depend on the bases chosen. If Alice is only interested in her local observables, that is, observables of the form
ΘA= Θ⊗ I, (1.40)
then all expectation values can be computed as
hΘAi = tr((Θ ⊗ I) ρ) = trA(trB(Θ⊗ I) ρ) = trA(Θ trB(ρ))
= trA(Θ ρA), (1.41)
where ρA= trB(ρ) is called reduced density matrix. In fact, the partial trace of a density matrix is still a density matrix [50], as it describes all the statistics of local experiments. Bob can define a reduced density matrix ρB in the same way.
Consider now a quantum system A, and suppose it is in a mixed state ρA. Let us imagine to enlarge the Hilbert spaceHAto a composite Hilbert space HA⊗ HB. It turns out that in this new space it is always possible to represent ρA as a subsystem of a composite pure state|ψiAB.
Theorem 3 (Purifications). Let ρA∈ S(HA) be a mixed state. Then there exists a Hilbert spaceHB and a pure state |ψiAB ∈ HA⊗ HB such that
trB(|ψiABhψ|) = ρA. (1.42) Proof. Let ρA= r X i=1 pi|eiiAhei|, (1.43) be the spectral decomposition of ρA. Using the orthonormal basis vectors {|eii}ri=1, let us define a Hilbert space HB as a copy of the vector space generated by them
HB = Span(|e1i, ..., |eri ). (1.44) Then, if we define the pure state
|ψiAB = r X i=1 √ pi|eiiA|eiiB, (1.45) we have trB(|ψiABhψ|) = r X i,j=1 √p ipj|eiiAhej|δij = ρA. (1.46)
The pure state|ψiAB is called a purification of ρA[50]. The operation of purification can be thought of as a sort of inverse operation of the partial trace. For instance if we trace the pure state
|ψiAB = |0iA|0iB
+|1iA|1iB √
2 , (1.47)
with respect to B, we obtain the maximally mixed state 1
2(|0iAh0| + |1iAh1|) = 1
2I, (1.48)
and if we purify this mixed state, we obtain back the pure state|ψiAB. It is implicit in the proof that r, the dimension of HB, is equal to the rank of ρA, so that the purification of a pure state (a rank-1 projector) is trivial. Of courseHB can be chosen so that its dimension is greater than r, but when it is chosen so that it is exactly equal to r,HB is called minimal dilation space.
One useful application of purifications is the following result [50], of which we will make use in the next chapters.
Proposition 1. Let|ψiAB ∈ HA⊗ HB be a pure state with reduced density matrixρB of rank lower than n, where n = dimHA= dimHB. Then,
|ψiAB = (I⊗√n ρB)|Ωi, (1.49) where |Ωi is the maximally entangled state.
Proof. By taking explicitly a partial trace, we find trB(|ψiABhψ|) =
n X
i,j=1
δij√ρBEij√ρB = ρB. (1.50) Where Eij = |iihj|. Vice versa, consider the spectral decomposition of ρB ρB = n X i=1 pi|eiiBhei|, (1.51) we have that its square roots reads
√ ρB = n X i=1 √ pi|eiiBhei|. (1.52) We can retrieve|ψiAB via purification
|ψi = n X i=1 √p i|eiiA|eiiB, (1.53) and again, by direct computation we find
(I⊗√n ρB)|Ωi = n X
i,j=1
δikδjk√pk|eiiA|ejiB=|ψiAB. (1.54)
1.3.3 Separability Criteria and Entanglement Measures Unfortunately, there is no known analog of the Schmidt decomposition for mixed states. In order to classify entangled mixed state, let us give a definition.
Definition 5. LetΛ be a linear map from the vector space of linear operators onH into itself. Λ is called positivity-preserving, or positive, if it sends positive semidefinite operators into positive semidefinite operators, that is
Λ : S(H) −→ S(H) . (1.55) Furthermore, Λ is said to be completely positive if any extension to a larger Hilbert space, that is id⊗ Λ, is positivity-preserving.
In the literature, linear operators acting on the space of linear operators are referred to as super-operators, to avoid confusion. By construction a completely positive super-operator is also positive. On the contrary there exist positive super-operators which are not necessarily completely positive. An example is given by the transposition [62]
T : ρ−→ ρT. (1.56)
With this notion in mind, we can formulate the following.
Theorem 4 (Horodecki criterion). Let ρ ∈ S(HA⊗ HB) be a bipartite quantum state. The state ρ is separable if and only if for all positivity-preserving super-operators
Λ : S(HB)−→ S(HA), (1.57) we have
(id⊗ Λ)(ρ) ≥ 0. (1.58) This is a fundamental result, which was proved in [36]. The theorem gives a theoretical answer to the bipartite separability problem, but it does not provide a computational criterion for checking whether a state is entan-gled or not, as the number of super-operators one needs to consider grows extremely fast with n. Consider, however, the following result.
Proposition 2. Let ρ∈ S(HA⊗ HB) be a bipartite quantum state. Let Λ : S(HB)−→ S(HA) (1.59) be a positivity-preserving super-operator. Ifρ is separable, then
(id⊗ Λ)(ρ) ≥ 0. (1.60) Proof. By hypothesis, ρ can be written as a convex combination of product states ρ =X i piρi,A⊗ ρi,B, (1.61) then, we have (id⊗ Λ)(ρ) =X i piρi,A⊗ Λ(ρi,B), (1.62) and since Λ is positivity-preserving, this is a positive semidefinite opera-tor [36,35].
This proposition can be used as a necessary criterion for entanglement, for instance by choosing Λ = T , the transposition. The criterion obtained in this way is called the Peres-Horodecki criterion, or also positive partial transpose (PPT) test[36,53].
In what follows, an operator Θ partially transposed with respect to Al-ice’s or Bob’s party will be denoted ΘTA or ΘTB, respectively. Of course it holds (ΘTA)TB = (ΘTB)TA = ΘT.
Choosing different Λ’s gives different entanglement criteria, but all of them will be, in general, only necessary. At the present day we have no complete theory of entanglement, and the determination of the separability of a given state is no easy task, both theoretically and experimentally [38]. However, in the qubit case the situation is more favorable.
Theorem 5. Letρ∈ S(HA⊗HB) be a bipartite quantum state. If dimHA· dimHB≤ 6, the PPT test is also sufficient.
Proof. By the Størmer-Woronowicz theorem [59,63], any positivity-preserving super-operator Λ has the structure
Λ = Φ1+ Φ2◦ T, (1.63) where T is the transposition, and Φ1, Φ2 are completely positive super-operators. If ρTB is positive semidefinite, then
(id⊗ Λ)(ρ) = (id ⊗ Φ1)(ρ) + (id⊗ Φ2)(ρTB)≥ 0, (1.64) for all Λ positivity-preserving. The thesis follows by the Horodecki criterion.
The PPT test thus provides a complete answer to the separability prob-lem for a 2-qubit system [36,53]. The test has a clear computational meaning and, at least for our purposes, it has enough power to experimentally certify the presence of entanglement, provided quantum state tomography has been carried out [50,15].
We have seen that there are states which are “more entangled” than others, like the the maximally entangled state |Ωi introduced earlier. It is useful to introduce some functionals which make this idea precise, and possibly, which have a well-defined operational meaning.
Entanglement measures are important quantities which address this is-sue [54].
Definition 6. An entanglement measure E is a functional
E : S(H) −→ R, (1.65)
satisfying the requirements
1. E(ρ)≥ 0 ∀ρ ∈ S(H), and equality if and only if ρ is separable. 2. E(ρ) = E(U⊗ V ρ U†⊗ V†) ∀ U, V unitary.
These requirements are very reasonable from a physical point of view. The first requirement asks E to be positive on the set of operators which actually represent physical states. The second requirement asks E to be con-stant under unitary local operations, which can never break entanglement. The third requirement asks E to be non-increasing under general local op-erations, or when there is a classical communication line between Alice and Bob. Here by “operation” we mean a generalized quantum measurement, or a physical quantum evolution (see Chapter2).
For the case of a pair of qubits, the concurrence is one of the best known examples of an entanglement measure.
Definition 7. Letρ∈ S(HA⊗ HB) be a bipartite quantum state. Consider the operator
C =q√ρ ˜ρ√ρ, (1.66) where ρ = Y˜ ⊗ Y ρT Y ⊗ Y . Let λ
1, λ2, λ3, λ4 be the eigenalues of C put in decreasing order. Then, the concurrence is the quantity
C(ρ) := max{ λ1− λ2− λ3− λ4, 0}. (1.67) It can be shown that this is a good entanglement measure [38].
Another entanglement measure, often simpler to compute, can be based upon the partial transpose test.
Definition 8. Let ρ ∈ S(HA ⊗ HB) be a bipartite quantum state. The negativity of entanglement is the quantity
N (ρ) = ||ρ TA||
1− 1
2 , (1.68)
where ||Θ||1 is the trace norm of the operator Θ ||Θ||1:= tr(
√
Θ†Θ). (1.69)
It is simple to check that this functional satisfies the first two require-ments of a good entanglement measure. The only non-trivial task is checking that the third requirement is satisfied: details can be found in [60]. The com-putational simplification lies in the fact that the negativity can be expressed in terms of the eigenvalues{λi}ri=1of the state ρTA as
N (ρ) = r X i=1 |λi| − λi 2 , (1.70)
that is, it is the sum of the negative parts of the eigenvalues [60].
Finally, it is worth mentioning the entanglement of formation for its great theoretical interest.
Definition 9. Let ρ ∈ S(HA⊗ HB) be a bipartite quantum state, and consider an ensemble forρ
ρ = n X
i=1
pi|ψiihψi|, (1.71) the entanglement of formation is the quantity
EF(ρ) := min (pi,|ψii) n X i=1 piS(ρB), (1.72) where the minimum is taken over all possible ensembles forρ, S is the von Neumann entropy
S(ρB) =− X
j
qjlog qj, (1.73) and the qj’s are the eigenvalues ofρB.
The entanglement of formation is often difficult to compute due to the complicated minimization it involves. Nevertheless, it is an important func-tional with a physical meaning: it quantifies the “cost” needed to prepare many copies a quantum state [4].
Quantum Evolution:
Channels and Dynamical
Semigroups
After having introduced some fundamental notions about quantum states and the structure of state space, in this chapter we will focus on how quan-tum states change over time. It is well known that the Scr¨odinger equation describes such evolution, but this is true only in the case of a perfectly iso-lated system. Since we are interested in studying the effects of noise on a qubit, this hypothesis needs to be removed [50,8]. This leads to the notion of quantum channel [43, 34], a mathematical object introduced in order to describe any noise model tampering the system.
Quantum channels in general can be very complicated objects. Here-with we will focus on a particular class of quantum channels, associated with the so-called quantum dynamical semigroups [1]. Physically, they rep-resent a memory-less noise, and provide a quite faithful description for the system dynamics whenever the system is weakly coupled to a very large environment. As we will see, in these evolutions both a unitary (that is, Schr¨odinger-like) and a dissipative contributions are present at the same time.
2.1
Quantum Channels
2.1.1 Stinespring and Kraus Representations
By the prescriptions of quantum mechanics, the evolution of a closed quantum system is given by the solution to the Schr¨odinger equation, which reads
i d
dt|ψi = H|ψi, (2.1)
where H is an observable called the Hamiltonian of the system, and we have set ~ = 1. Thus the evolution is given by the action of a unitary operator on the initial state
|ψ(t)i = U(t, t0)|ψ0i, (2.2) where|ψ0i is the state at the initial time t0. If H is time-independent, the unitary operator U (t, t0) is simply given by
U (t, t0) = e−iH(t−t0), (2.3) otherwise it is has a more involved expression in terms of the so-called time-ordered productT U (t, t0) =T exp −i Z t t0 H(s)ds . (2.4)
If more generally we describe the system with a density matrix ρ, its evolution is retrieved by solving the von Neumann equation
˙ρ =−i[H, ρ] , (2.5)
and thus
ρ(t) = U (t, t0) ρ0U†(t, t0) =:U(ρ0), (2.6) being ρ0 the state of the system at t = t0. The mapping defined by equa-tion (2.6) is called a unitary evolution.
How can we describe the temporal evolution of an open quantum sys-tem? Let us allow the system of interest, A, to interact with an external environment, E. Suppose then that at the beginning of the evolution the global state of A and E is described by the joint density matrix
ρA⊗ |ψiEhψ|, (2.7)
where ρA and |ψiE represent the initial configurations of A and E, respec-tively (the latter assumed to be pure without loss of generality via a dilata-tion of the environment Hilbert spaceHE). This extended system can then be considered closed, and will evolve according to equation (2.6), for an ap-propriate unitary matrix U acting on bothHAandHE. In order to retrieve the state of the system of interest, we must trace away the environment degrees of freedom, thus yielding the evolved density matrix
ρ0A= trE(U ρA⊗ |ψiEhψ| U†) =: Φ(ρA). (2.8) For a fixed choice of the initial state of the environment, equation (2.8) defines a mapping Φ on the set of density matrices S(HA) which generalizes equation (2.5) to the case of open dynamics. More precisely, equation (2.8) is called Stinespring representation of the super-operator Φ, which in turn is called a quantum channel. Actually, the Stinespring representation is not the only possible way to introduce Φ. An alternative representation is provided by the so-called Kraus form, or operator-sum form.
Theorem 6. Let Φ be a super-operator acting on S(HA). The super-operatorΦ can be cast in Stinespring form
Φ(ρA) = trE(U ρA⊗ |ψiEhψ| U†), (2.9) if and only there exists a set of operators{Ki}si=1such thatΦ can be written in Kraus form, Φ(ρA) = s X i=1 KiρAKi†. (2.10) Furthermore, the Ki’s satisfy the normalization condition
s X
i=1
Ki†Ki = I. (2.11) Proof. Let Φ be in Stinespring form
Φ(ρA) = trE(U ρA⊗ |ψiEhψ| U†). (2.12) Define an orthonormal basis{|eii}ni=1E of HE, and the linear operators
Ki =Ehei|U|ψiE, (2.13) which act on HAonly. Then, the action of Φ on ρAis given by
Φ(ρA) = nE X i=1 KiρAK † i, (2.14)
and the Ki operators satisfy the constraint nE
X
i=1
Ki†Ki =Ehψ|U†U|ψiE = I. (2.15) Vice versa, let Φ be in Kraus form (2.10). For each i = 1, .., s let us choose an orthonormal vector |eii, for instance using the computational basis. Consider the Hilbert space generated by these vectors
HE = Span(|e1i, ..., |esi), (2.16) and let us define the linear map
V = s X
i=1
Ki⊗ |eii. (2.17) Since Φ verifies the normalization condition, V is an isometry, that is
and the expression of the super-operator Φ can be recast into
Φ(ρA) = trE(V ρAV†). (2.19) If we further dilatateHE so that is has dimension n2A, where nA= dimHA, we can always embed V so that it acts on a tensor product [62]
V = U (I⊗ |ψiE), (2.20) for an appropriate n3
A× n3A unitary matrix U and a suitable unit vector |ψiE ∈ HE. Thus, Φ is in the form (2.8), as claimed.
As already mentioned, equation (2.14) is called Kraus representation of a quantum channel. The Ki’s are called Kraus operators [58,30,31], and they represent transition “amplitudes” towards pure states of the envi-ronment. Implicit in the proof of the theorem is the fact that the number of Kraus operators, s, can always be chosen to be lower than n2
A, where nA= dimHA.
The set of Kraus operators{Ki}si=1is not unique. Indeed, any other set of nonnull Kraus operators representing the same evolution is given by [62]
Wi = nE X
j=1
uijKj, (2.21)
where the uij’s are the elements of a unitary matrix. Such property is known as unitary freedom in operator-sum representation.
2.1.2 Axiomatic Representation and the Choi-Jamio lkowski Correspondence
Let us explore the properties of quantum channels. From both repre-sentations we can see that a quantum channel is always trace-preserving, that is
tr(Φ(Θ)) = tr(Θ) = 1 ∀ Θ acting on HA. (2.22) From the Kraus representation it is immediately evident that quantum channels are positivity-preserving super-operators, that is, they send positive semidefinite operators into themselves (see Definition 5). What is more, a quantum channel remains positivity-preserving even if it acts only on a part of a composite quantum system, as a quantum channel in Kraus form verifies the much stronger requirement of complete positivity.
Proposition 3. LetΦ be a super-operator acting on S(H). If Φ can be cast in Kraus from (2.10), then it is completely positive.
Proof. By invoking the spectral decomposition of ρ ρ = n X i=1 pi|eiihei|, (2.23)
we can reduce ourselves to proving the statement on pure states only. If dimH = n, by Proposition1 any such state can be written in the form
|ψi = (I ⊗√n ρB)|Ωi, (2.24) so we have that
(id⊗ Φ)(|ψiABhψ|) = I ⊗ Φ(√n ρB|ΩihΩ|√n ρB), (2.25) which is positive semidefinite, by the positivity preservation of Φ. This completes the proof.
Given a super-operator Φ, we can associate to it an operator acting on H ⊗ H (the so-called Liouville space)
ρΦ:= (id⊗ Φ)(|ΩihΩ|) = 1 n n X i,j=1 Eij ⊗ Φ(Eij), (2.26)
where Eij := |iihj| is called an elementary matrix. A matrix repre-sentation of ρΦ is simply given by arranging the matrices Φ(Eij) block-wise in an n2 × n2 matrix. If Φ is represented in the basis of the elementary matrices, this only accounts to a permutation of matrix entries.
The Choi-Jamiolkowski operator ρΦ encodes all information on the super-operator Φ, in fact if we let
Θ = n X i,j=1 ΘijEij, Ξ = n X i,j=1 ΞijEij, (2.27) we have n tr((ΞT ⊗ Θ) ρΦ) = n X i,j,k,h=1 ΘijΞkhhj|Φ(Ekh)|ii = n X i,j,k,h=1 ΘijΞkhhk|hj|ρΦ|hi|ii, (2.28)
from which we conclude that
Therefore, we can represent the super-operator Φ in terms of its associ-ated Choi-Jamio lkowski operator ρΦ in the following way
Φ(Θ) = n X
i,j,k,h=1
nhk|hi|ρΦ|hi|ji EikΘEhj. (2.30)
The importance of the Choi-Jamiolkowski correspondence stems from the following result.
Theorem 7(Choi-Jamiolkowski). Φ is a completely positive super-operator if and only if ρΦ is a positive semidefinite operator.
Proof. Let Φ be completely positive. Then the operator
ρΦ= (id⊗ Φ)(|ΩihΩ|), (2.31) is trivially positive semidefinite, as it is given by the action of id⊗ Φ on the positive semidefinite operator|ΩihΩ|.
Vice versa, let ρΦ be positive semidefinite. To avoid confusion, we will denote vectors in the Liouville space with the notation |Ψii ∈ H ⊗ H. Let the spectral decomposition of ρΦ be
ρΦ = s X
i=1
pi|ΞiiihhΞi|, (2.32)
then upon defining the operators Ki =√npi
n X
j,k=1
hk|hj| Ξiii Ejk, (2.33) and making use of equation (2.30), the super-operator Φ reads
Φ(Θ) = s X
i=1
KiΘ Ki†, (2.34)
in other words, Φ is explicitly written in Kraus form, and therefore it is completely positive.
Motivated by the last characterization theorem, we introduce the follow-ing definition:
Definition 10. A quantum channel is a completely positive, trace-preserving, linear super-operator.
The last definition is the so-called axiomatic representation of a quan-tum channel. Indeed, we could have defined a quanquan-tum evolution in such a way to begin with, motivating the requirements on the super-operator Φ on physical grounds [50,8].
To sum up, we have provided three equivalent representations of a quan-tum evolution: the Strinespring representation, which makes it clear that we are actually representing on open quantum system, the Kraus represen-tation, which makes no explicit use of external degrees of freedom, and the axiomatic representation, which allows us to explicitly test a super-operator for physicality, by simply testing the Choi-Jamio lkowski state for positivity. 2.1.3 Stationary States
We noted in Chapter 1that the space of states is a compact convex set. An important physical fact stems from the following theorem, of which we give a simplified version.
Theorem 8(Brouwer’s fixed point). Any continuous function from a com-pact convex set into itself admits a fixed point.
A proof of the theorem (which is a fundamental result of great generality) can be found in [29]. A fixed point for the channel Φ is a state ρ such that Φ(ρ) = ρ, therefore a fixed point represents a stationary state. A super-operator Φ is called unital if it fixes the identity, that is, if Φ(I) = I.
A stationary state ρ∞ is called a relaxation state if it satisfies the additional property
lim n→∞Φ
n(ρ) = ρ
∞, ∀ρ ∈ S(H), (2.35) where Φn represents the n-fold application of the map Φ. As we will see in Chapter 4, the stationary states have encoded some information of the asymptotic entanglement-breaking properties of a channel.
2.1.4 Qubit Channels
In the case of one-qubit systems it is quite useful to introduce a fur-ther representation of quantum channels. To this end, let us fix the basis {I, X, Y, Z} to represent all the operators involved. This means that we choose to represent a qubit with a Bloch vector (see Chapter1)
ρ = I + ~n· ~σ
2 , (2.36)
and the Kraus operators as
Ki = aiI + 3 X
i=1
where ai and bij are complex coefficients. A straightforward calculation shows that the Bloch vector is transformed by a quantum channel as follows
ni 7−→ n0i = 3 X
i=1
Tijnj+ ci, (2.38)
where the matrix T and the vector c are given by
Tij = 4 X k=1 " bkib∗kj+ b∗kibkj+ |ak|2− 3 X h=1 |bkh|2 ! δij + i 3 X h=1 ijh(akb∗kh− a ∗ kbkh) # , (2.39) ci = 2i 4 X l=1 3 X j,k=1 ijkalja∗lk. (2.40)
The relation (2.38) provides a geometric representation of a qubit channel [50]. Namely, a qubit channel is a linear transformation of the Bloch vector, followed by a translation: a so-called affine transformation. Such a transformation deforms the unit sphere into an ellipsoid which is contained in the initial Bloch sphere [55]. Furthermore, unitary evolutions correspond to rotations of the Bloch sphere [50].
Through the Bloch sphere representation, all the elements developed so far come together in a very intuitive picture. Clearly, a quantum channel is unital if and only if it does not translate the Bloch sphere, in which case the completely mixed state, which corresponds to the center of the Bloch sphere, is a fixed point, or a stationary state. Any evolution which actually involves some form of noise – that is, any quantum channel which is not a unitary evolution – can be visualized shrinking the Bloch sphere into the stationary states (by Brouwer’s theorem there will always be at least one such state). Otherwise, if the evolution pertains to a closed, isolated system, the Bloch sphere will simply rotate around an axis of stationary states passing through the origin.
2.2
Quantum Dynamical Semigroups
From the previous section it follows that the open system dynamics of a quantum system A can be characterized by assigning a one-parameter family of quantum channels {Φt}t≥0 such that, given the state ρ0 of the system A at time t = 0, its evolved counterpart at time t≥ 0 is obtained as
Apart from the trivial requirement that at t = t0the element Φt0,t0 corre-sponds to the identity map (no evolution occurring), and possibly continuity of its elements, the properties of the family {Φt}t≥0 strongly depends upon the kind of coupling the system experiences with its environment. Yet an interesting subclass of open quantum processes are those which fulfill the following constraint
Φt2,t1◦ Φt1,t0 = Φt2,t0, (2.42) where the symbol “◦” denotes the composition of super-operators. Equa-tion (2.42) is known as inhomogeneous semigroup property and it ap-plies to those systems which, roughly speaking, interact with an environment characterized by a fast intrinsic dynamics and exhibiting a Markovian char-acter (see more on this later). In what follows we shall focus on a special subset of these processes which also exhibits invariance under time transla-tion, that is,
Φt2,t1 = Φt2−t1,0 =: Φt2−t1, (2.43) for all t2 ≥ t1. For these families equation (2.42) reduces to the homoge-neous semigroup property [42].
Φs◦ Φt= Φt+s. (2.44) Definition 11. A quantum dynamical semigroup is a continuous family of completely positive, trace-preserving super-operators such that the initial-time super-operator is the identity, and that satisfies the semigroup property. It turns out that a quantum dynamical semigroup is not only continuous, but also differentiable [62]. Therefore, upon defining the super-operator
L := dΦdtt
t=0, (2.45)
we can write
Φt= eLt. (2.46)
The super-operatorL is called the generator of the dynamics, or Lind-bladian operator. In the case of inhomogeneous quantum dynamical semi-groups the Lindbladian becomes time-dependent, and equation (2.46) is re-placed by the expression
Φt0,t=T exp Z t
t0L(s) ds
. (2.47)
The evolution of a quantum system described by a quantum dynamical semigroup can thus be obtained solving the Markovian quantum master equation
˙ρ =L(ρ), (2.48)
which generalizes equation (2.5), pertaining to a closed quantum system. It has been shown that the generator of a quantum dynamical semigroup always presents a well-defined structure, called Lindblad form [26,47].
Theorem 9. Let H be a Hilbert space of finite dimension n. Let L be the generator of a quantum dynamical semigroup. L can always be cast in Lindblad form
L(ρ) = −i[H, ρ] + D(ρ), (2.49) whereH is a Hermitian operator, representing the Hamiltonian contribution to the dynamics. D is called the dissipator super-operator
D(ρ) = n2−1 X i=1 γi Liρ L†i − 1 2L † iLiρ− 1 2ρ L † iLi , (2.50) the Li’s are called Lindblad operators, and theγi’s are non-negative param-eters.
We provide a proof of this result in the Appendix. It will be useful for the following to observe that the unitary part of the dynamics can always be eliminated by passing to the interaction picture. Indeed, by considering the evolution of the interaction picture density matrix
ˆ
ρ(t) = eiHtρ(t)e−iHt, (2.51) we find the Markovian quantum master equation
d dtρ(t) =ˆ n2−1 X i=1 γi ˆ Li(t) ˆρ(t) ˆLi † (t)−1 2Lˆi(t) ˆLi † (t) ˆρ(t)−1 2ρ(t) ˆˆ Li(t) ˆLi † (t) , (2.52) where also the Lindblad operators have been transformed, ˆLi(t) = eiHtLie−iHt, that is, we have eliminated the unitary part of the dynamics, at the cost of making the Lindbladian time-dependent.
2.2.1 Microscopic Derivations
Let us briefly discuss some physical conditions under which the evolu-tion of an open quantum system can be described by a quantum dynamical semigroup [8,16].
A possible way to derive a Markovian quantum master starting from a purely Hamiltonian dynamics of the system under investigation and the environment is by means of the weak-coupling limit and the Born-Markov approximation [20]. Ultimately this consists in assuming the system of inter-est to be coupled weakly to a very large environment, in such a way that the state of the environment is negligibly affected by the interaction1.
Further-more, the evolution of the system of interest is supposed to be memory-less, 1The fact that the evolution of the environment is neglected does not mean that there
are no excitations in the environment caused by the system. Rather, the Markovian approximation provides a description on a coarse-grained time scale, when environmental excitation decays are still not resolved.
which is justified if the time scale over which the state of the system varies is large compared to the time scale over which the environment correlation functions decay. Therefore, the relevant physical condition is that the en-vironment correlation time is small compared to the relaxation time of the system.
Following this analysis, the damping parameters γi can be seen to be related to the environment correlation functions: indeed, the damping pa-rameters play the role of relaxation rates towards different decay modes. The Hermitian operator H can not, in general, be identified with the local Hamiltonian of the free, unperturbed system, instead, under appropriate conditions it controls the phenomenon of the Lamb shift of the unperturbed energy levels of the free system.
2.2.2 Stationary States and Accumulation Points
The stationary states of a quantum dynamical semigroup can easily be found by solving the linear equation
L(ρ) = 0. (2.53)
It is worth stressing that a state ρ fulfilling equation (2.53), has the property of begin invariant under all the elements of the family, that is
Φt(ρ) = ρ ∀ t ≥ 0. (2.54) In many cases of physical interest, a semigroup is characterized by a unique stationary state ρ∞: when this happens, it turns out that such a special state is typically also an accumulation point of the corresponding dynamical evolution, that is
lim
t→∞Φt(ρ) = ρ∞ ∀ρ ∈ S(H). (2.55) In other words, ρ∞ becomes a relaxation state of the dynamical semigroup. If the environment is a heat bath at temperature T to which the system thermalizes, the relaxation states of a semigroup takes the form of a Gibbs state
ρ∞= 1 Z e
−βHA, (2.56)
where Z = tr(e−βHA) is the partition function, β = 1/kT is the inverse temperature, and k is the Boltzmann constant.
2.2.3 Resolvent Expansion
While the formal integration of a Markovian master equation is triv-ial, the explicit evaluation of the exponential (2.46) might be technically cumbersome. For this reason it is desirable to have at our disposal approx-imate solutions. A possibility on this direction is provided by the resolvent technique [65,62]:
Definition 12. Let Θ be a linear operator acting on H. The resolvent of Θ is the operator
RΘ(λ) = (I λ− Θ)−1, (2.57) for allλ∈ C such that I λ − Θ is invertible.
Notice that if λ is an eigenvalue ofL, the resolvent is not defined. Indeed, the resolvent operator as a function of λ has poles in the spectrum ofL [65]. The importance of the resolvent lies in the fact that it enables to use operator calculus methods to represent a quantum dynamical semigroup.
Theorem 10. Let L be the generator of a quantum dynamical semigroup {Φt}t≥0. Then it holds Φt= 1 2πi I eλtR L(λ) dλ, (2.58)
where the integration is taken on a closed path in the complex plane contain-ing the spectrum ofL, and furthermore it holds
RL(λ) = Z ∞
0
e−λtΦtdt, (2.59) whenever the real part of λ is positive.
The integral representation (2.58) is known as Dunford integral, while the inverse formula (2.59) is kwown as Hille-Yosida theorem [65,62]. The result involving the resolvent which makes it useful for approximations is the following.
Theorem 11. Let L be the generator of a quantum dynamical semigroup {Φt}t≥0. Then it holds [62]
Φt= lim n→∞(n/t)
nRn(n/t). (2.60) In fact, under a closer look, the content of the last theorem is nothing but the operator-theoretical version of Euler’s approximation
eLt= lim n→∞ id−Lt n −n . (2.61)
Another remarkable property of this approximation is that it preserves the fixed points order by order, in fact if ρ is a stationary state
(λ id− L) ρ = λ ρ, (2.62) from which it follows thatRL(λ)ρ = (1/λ)ρ, so that
Entanglement-Breaking
Channels: Theory and
Experiments
Entanglement-breaking channels are a particular class of quantum chan-nels [32]. They represent the most detrimental form of noise possible from the point of view of quantum correlations: as the name suggests, once such an evolution has taken place the entanglement shared between the system under investigation and any other (ancillary) system is completely lost. Once destroyed, quantum correlations cannot be generated by applying any (lo-cal) transformation [50]. Therefore, any error-correction protocol based on pre- and post-processing operations on the system, or on entanglement dis-tillation techniques is completely ineffective [50,62]. These channels are not at all rare: indeed, as we will see, after a sufficiently long amount of time even Markovian noise can induce an entanglement-breaking process.
However, not all hope is lost. There exists some entanglement-breaking channels, called amendable channels, whose detrimental effect can be greatly mitigated by introducing suitable “filters” [13]. In some cases, it results that by suitably manipulating two entanglement-breaking maps, the latter man-age to “correct” each other [14]. Such techniques, which will be extensively explained in this chapter, simply exploit the non-commutative character of operations in quantum mechanics.
3.1
Theory of Entanglement-Breaking Channels
Entanglement-breaking channels are defined by the fact that when acting on half of a bipartite system (such as one of the two photons of a polarization entangled state) the output state is separable, regardless of the input one. Definition 13. Let ρ ∈ S(HA⊗ HB) be a bipartite quantum state. A
quantum channelΦ is said to be entanglement-breaking if
(id⊗ Φ)(ρ), (3.1)
is separable ∀ρ ∈ S(HA⊗ HB).
Entanglement-breaking channels have been fully characterized: they can always be cast in a certain form, known as Holevo form. As will be clarified in the proof of the characterization theorem [55], the Holevo form leads to the interpretation of entanglement-breaking maps as measure-and-reprepear processes.
Theorem 12. Let ρ ∈ S(HA ⊗ HB) be a bipartite quantum state. The following statements are all equivalent
1. (id⊗ Φ)(ρ) is separable ∀ρ ∈ S(HA⊗ HB). 2. The Choi-Jamio lkowski stateρΦ is separable.
3. Φ can be cast in Holevo form, that is, there exists density matrices ρi and positive semidefinite operatorsPi such that
Φ(ρ) =X i
ρitr(Piρ), (3.2)
and satisfying the constraint X
i
Pi = I. (3.3)
Proof. 1 =⇒ 2). Obvious.
2 =⇒ 3). By assumption ρΦ is separable, so there must exist unit vectors|vii, |wii and positive numbers pi such that
ρΦ = X
i
pi|viihvi| ⊗ |wiihwi|. (3.4)
Given the following linear map (applied on ρ) ˜
Φ(ρ) = nX i
the associated Choi-Jamio lkowski state coincides with ρΦ, (id⊗ ˜Φ)(|ΩihΩ|) = 1 n n X j,k=1 |ejihek| ⊗ ˜Φ(|ejihek|) =X i n X j,k=1
|ejihek| ⊗ pihek|viihvi|eji|wiihwi| =X i pi n X j,k=1
|ejihej|viihvi|ekihek| ⊗ |wiihwi| =X
i
pi|viihvi| ⊗ |wiihwi|. (3.6) Since the Choi-Jamio lkowski state completely characterizes a quan-tum channel [62], we have ˜Φ = Φ. Then, by consistency, it must hold
nX i
pi|viihvi| = I, (3.7) and by taking the partial trace of the Choi-Jamio lkowski state and imposing trace-preservation, it results
trB(ρΦ) = X i pi|viihvi| (3.8) = 1 n n X j,k=1
|ejihek| tr(|ejihek|) = 1
nI. (3.9) 3 =⇒ 1). If Φ is written in Holevo form, then for some complex coefficients aij, and for some operators ρi,A, ρi,B, we have that the state (id⊗ Φ)(ρ) = n X j,k=1 ajkρj,A⊗ Φ(ρk,B) =X i n X j,k=1 ajkρj,Atr(Piρk,B)⊗ ρi =X i trB(Piρ)⊗ ρi = X i piτi⊗ ρi, (3.10) is explicitly separable, where we have defined
pi:= tr(Piρ), τi :=
trB(Piρ) tr(Piρ)
The Holevo form gives an insightful description of entanglement-breaking channels, suggesting how such channels could be physically implemented. Holevo first introduced such channels in [32]. If two distant parties Alice and Bob were interested in implementing such a channel, they could proceed as follows:
1. Alice and Bob agree on a generalized measurement, described by the operators POVM operators Pi [50]. To the possible outcomes, labeled by the index i, they decide to associate the states ρi.
2. Alice measures the quantum state ρ, and sends the result i to Bob over a classical communication line.
3. Bob prepares the state ρi.
In this way, as the procedure is repeated, Bob will end up with a mixture of the states ρi with weights given by the probabilities of the outcomes labeled by i.
3.2
Entanglement-Breaking Order and Amendable
Channels
While establishing whether a map is entanglement-breaking or not is rel-atively simple, as it is enough to test the separability of the associated Chio-Jamio lkowski state, in general quantifying the level of noise associated to an arbitrary map is a quite non-trivial task typically attacked by computing some quite involved entropic functionals known as quantum capacities [33]. An alternative way to solve this problem is offered by a classification cri-terion of quantum maps [19]. Before introducing it let us observe that the set of entanglement-breaking channels form an ideal under composition of quantum channels. More precisely, given Φ entanglement-breaking and Φ0 another (non necessarily entanglement-breaking) quantum channel we have that both Φ◦ Φ0 and Φ0 ◦ Φ are also entanglement-breaking channels. The first of these properties trivially follows from the definition of entanglement-breaking maps. The second, instead has to do with the fact that no entan-glement can be created by local operations [50]. On the other hand, it is also possible that the composition of two non-entanglement-breaking maps is entanglement-breaking. In other words, the the combination of not-so-disturbing quantum channels can produce a rather detrimental evolution. This observation motivates the following:
Definition 14. LetΦ be a quantum channel. Φ is said to be entanglement-breaking of orderp if it is not entanglement-breaking, but becomes so after p channel uses:
Regular breaking channels correspond to order-1 entanglement-breaking channels. Clearly a channel of order p + 1 can be seen as “more noisy” than a channel of order p, as it need a higher amount of iteration to before it completely destroys the entanglement between the system and any other ancilla. Some channels might have an infinite entanglement-breaking order, that is, they may become entanglement-entanglement-breaking only af-ter the channel has been used infinitely many times. Such channels will be called asymptotically entanglement-breaking channels [43]. Trivial examples are the unitary evolutions, but we will see that they are not the only ones.
The notion of entanglement-breaking order suggests the possibility of mitigating the action of a noisy channel by interposing suitable intermediate gates in between subsequent channel uses.
Definition 15. Let Φ be an order-2 entanglement-breaking channel. Φ is said to be amendable if there exists a quantum channelF, such that Φ◦F◦Φ is not entanglement-breaking.
The mapF is called a filtering operation, since it is used between two channel uses of Φ so that finally entanglement will not be broken completely destroyed.
As an example of amendable maps, consider the following quantum chan-nels, Ση(ρ) = 2 X i=1 Ki(η) ρ Ki†(η), (3.12) Γp(ρ) = 1−p 2 ρ + p 2ZρZ, (3.13)
where p and η are positive parameters, and the Kraus operators Ki are given by K1(η) =1 0 0 √η , K2(η) =0 √ 1− η 0 0 . (3.14) The channels Γpand Ση are important noise models which will be extensively studied in Chapter 4. For the moment, the important fact is that neither channel is entanglement-breaking [13]. Furthermore, consider the unitary channel
Uθ = Rθρ Rθ, Rθ= cos(2θ) − sin(2θ) − sin(2θ) − cos(2θ)
. (3.15) The channels Φ =Uθ◦Γp and Ψ =Uθ◦Ση, formed by the subsequent ap-plication of a non-unitary and a unitary channel, are called rotated phase damping channel and rotated amplitude damping channel, respec-tively.
They are both examples of amendable maps: for appropriate values of the parameters, Φ and Ψ are known to be entanglement-breaking of order
Figure 3.1: Quantum optics experimental setup for the test of entanglement restoration in amendable channels. Courtesy of [13].
2 [13], but they can clearly be corrected simply my making use of an ap-propriate unitary filterF = Uϕ to cancel the rotation (3.15). A trivial case corresponds to ϕ = θ.
3.2.1 Experimental Test of Entanglement Restoration
The effectiveness of these filtering operations on amendable channels can be tested with a quantum optics experiment. The goal is to demonstrate that the entanglement transmission length associated to a channel Φ can be increased by applying the above mentioned techniques [13].
In such tests, photon polarization states are used as qubits. Entan-gled photon pairs are generated in two indistinguishable Type-II parametric down-conversion processes inside a PPKTP nonlinear crystal.
One of the two photons travels in free space directly into a quantum state tomography stage, implemented using half- and quarter-wave plates, and a polarizing beam splitter. The other one is sent through a single-mode fiber into a bulk optics setup that implements various quantum channels. In between the bulks a unitary filtering gateF can be interposed, which is implemented using a half-wave plate; eventually the photon enters the to-mography stage. Finally, the photons are revealed through two synchronized avalanche photo-detectors (see figure (3.1)).
In [14] two amendable channels have been implemented, namely a rotated phase damping channel and a rotated amplitude damping channel described in the previous section. The phase damping channel can be implemented us-ing only two suitably rotated half-wave plates and a randomized switch. The implementation of the amplitude damping channel instead requires a more sophisticated Sagnac interferometric setup. The interferometer is opened and closed by a single polarizing beam splitter: each trajectory is inter-cepted by an independent half-wave plate, and finally an unbalanced Mach-Zehnder interferometer allows to couple in the same trajectory the damped
(a) Schematics of the phase damping setup
(b) Schematics of the amplitude damping setup.
Figure 3.2: Single channel modules used in the experiment. Here the unitary channelU is denoted Λ. Courtesy of [13].
and undamped polarizations as they pass through a beam splitter (see figure (3.2)).
With this setup, the presence of entanglement can be detected by mea-suring the concurrence of the output photons via quantum state tomography. The results of the experiment performed in [13] are summarized in figure (3.3). In the experiment it was set: p = 0.65 and θ = π/8 for the rotated phase damping, and η = 0.66 and θ = π/4 for the rotated amplitude damp-ing. The concurrence was measured with and without the unitary filter Uϕ. A non-zero concurrence was expected near the values ϕ = ±π/8 and ϕ =±π/4, respectively. In both cases there was a good agreement between theory and experiment, showing the loss of entanglement in the expected regions of parameters, and subsequent revival thanks to unitary filtering operations. The discrepancies between the theoretical curve and the exper-imental points are due to the post-processing generation of the channel in the phase damping case, and to difficulties in coupling several spatial modes within a unique single-mode fiber at the end of the channel in the amplitude damping case.
3.3
Cut-and-Paste Entanglement Restoration
Pro-tocols
The entanglement-recovery technique discussed above is based on the intuition of an external filter. In what follows we will discuss a more coun-terintuitive scheme. Imagine that we have at our disposal two imperfect transmission lines, modeled by the channels Φ and Ψ. Suppose Φ and Ψ are
