DIPARTIMENTO DI FISICA “E. FERMI”
CORSO DI LAUREA MAGISTRALE IN FISICA
Tesi Di Laurea Magistrale
NON-CLASSICAL CORRELATIONS
IN QUANTUM STATES
Candidato:
Luca Rigovacca
Relatori: Prof. Vittorio Giovannetti Dr. Antonella De Pasquale
Anno Accademico 2013 - 2014 Luglio 2014
Abstract
Laurea Magistrale
Non-Classical Correlations in Quantum States Candidato: Luca Rigovacca
Relatori: Prof. Vittorio Giovannetti, Dr. Antonella De Pasquale
The presence of correlations which do not have a classical counterpart, shared among the subparts of a given system, is one of the best signatures of non-classicality for a quantum state. Although entanglement is the most remarkable among these correla-tions, even some separable (i.e. not entangled) mixed states can exhibit non-classical behaviours. After reviewing Quantum Discord, that has been the first correlation measure historically introduced, several other known quantifiers are presented. Particular attention is dedicated to the geometric measure known as Trace Distance Discord, that in this thesis has been evaluated on a new class of two-qubit states and whose maximum value on the set of separable states has been found when the tested subsystem is a qubit and the other one is infinite dimensional.
Starting from this scenario, a new measure of correlations is then introduced and characterized. It has a clear operational interpretation in the context of state dis-crimination: it quantifies the ability of a bipartite state to distinguish between the application or not of an unknown local unitary map acting on one subsystem, la-belled withA. The dependence upon the spectrum of such unitary map is discussed, showing in particular that the harmonic choice is optimal at least for pure states with dA = 3 and whichever dB (here dA and dB being the dimensions of the two
subsystems). It is proven that for a two dimensional subsystem A the maximum over the set of separable states is reached by classical-quantum ones: its value has been computed analytically when dB ≥ 3 and numerically when dB= 2.
A comparison among all the discussed measures is performed, plotting their value as a function of the parameter of the amplitude damping channel applied to the first subsystem. The result confirms how an appropriate local operation on the tested subsystem can enhance the amount of non-classical correlations.
Eventually, a possible generalization of the new measure to the set of bosonic Gaus-sian states for continuous variable systems is discussed, evaluating at last the pro-posed expression on two-mode squeezed thermal states.
First and foremost, I want to thank my thesis advisor Prof. Vittorio Giovannetti for his constant support. Not only he was always there to advice and guide me with his deep insight, but he also never stopped encouraging and believing in me. Thank you very much for your support and understanding throughout this past year.
I would also like to show my gratitude to Dr. Antonella De Pasquale for her sugges-tions and for having let me join the work during the paper preparation, reason for which I sincerely thanks Alessandro Farace, too. Interacting and discussing with the two of you made me really motivated and surely a good part of this thesis is also due to you. Many thanks also to Gerardo Adesso and Marco Piani for answering some of my questions.
Completing this thesis required more than purely academic support, and I cannot express all my thanks for all those people that keep being near to me, even after several years. Let me begin with my flatmates Matteo, Giulio and Giacomo that tol-erated me throughout this past year. A particular thanks to Giacomo, for suggesting the use of rearrangement inequality. I cannot forget all other friends that I met dur-ing these university years, particularly Federico, Marco, Leonardo, Luca, Federico, Lorenzo, Ludovico, Alvise, Paolo, Jinglei along with Pietro, Davide and Fabio that had already left Pisa for adventures abroad. After these compulsory names, there would be many others that I cannot even begin to list. These five years passed very quickly, but thanks to all of you they have been among the best of my life.
Five years are quite a long time, so in these few rows I would like to mention some of all those friends from home that have not forgotten my face in these years: Daniele, Riccardo, Giovanna, Daniele, Valentina, Alice, Nicola, Matteo. Your friendship means a lot to me, even if sometimes my absence seems to prove otherwise: thank you.
Last, but not least, I want to thank my family that has never stopped encouraging and supporting me in all my choices. Among all the relatives, the most important thanks go to my parents and to my sister Ilaria: this dissertation stands as proof of your unconditional support.
Abstract i
Acknowledgements ii
Contents iii
1 Correlations in Quantum Information Theory 1
1.1 Introduction . . . 1
1.2 Some entanglement measures . . . 6
1.3 Quantum Discord . . . 9
1.3.1 Entropic formalism . . . 9
1.3.2 Definition and properties . . . 12
1.3.3 Symmetric Quantum Discord . . . 14
2 Measures of Quantum Correlations 17 2.1 Geometric measures . . . 18
2.1.1 Zero and One Way Quantum Deficit . . . 20
2.1.2 Geometric Discord . . . 21
2.1.3 Bures Geometric Discord . . . 22
2.1.4 Trace Distance Discord . . . 23
Explicit evaluation on two qubit states . . . 24
Maximum on separable states . . . 27
2.2 Non-geometric measures . . . 31
2.2.1 Global action of local unitary maps . . . 31
Impact Power Gap . . . 32
Discord of Response . . . 32
2.2.2 Quantumness of Correlation . . . 33
2.2.3 Quantum metrology - related measures. . . 35
Interferometric Power. . . 35
Local Quantum Uncertainty . . . 37
3 Discriminating Strength 39 3.1 State discrimination scheme . . . 40
3.2 Properties . . . 43 iii
3.2.1 DS as a measure of non-classical correlations . . . 43
3.2.2 A formal connection between DS and LQU measures . . . 45
3.2.3 Discriminating Strength for pure states. . . 46
Hamiltonians with harmonic spectrum . . . 48
3.2.4 Discriminating Strength for qubit-qudit systems. . . 48
3.3 Dependence upon the spectrum . . . 49
3.3.1 Harmonic spectrum optimality for low-dimensional pure states 50 3.3.2 Majorization-related ordering cannot be found . . . 53
3.4 Discriminating Strength maximization over the set of separable states 55 3.4.1 pure-QC optimality: subsystem B infinite dimensional . . . . 56
3.4.2 pure-QC optimality: subsystem B at least three dimensional. 59 3.4.3 pure-QC optimality: subsystem B two dimensional . . . 60
Maximum DS over QC states . . . 60
Separable qubit-qubit states: numerical results . . . . 62
4 Enhancing Quantum Correlations through Local Channels 63 4.1 Quantum operations . . . 64
4.2 Amplitude-damping channel and state evolution. . . 66
4.3 Correlation measures: evaluation . . . 68
4.3.1 Quantum Discord and One Way Deficit . . . 69
4.3.2 Geometric Discord and Trace Distance Discord . . . 73
4.3.3 LQU and Interferometric Power . . . 74
4.3.4 Bures Geometric Discord and Discord of Response . . . 75
4.4 Overall plot and comments . . . 77
5 Discriminating Strength Generalization to Gaussian States 79 5.1 Gaussian states and symplectic formalism . . . 80
5.1.1 Definition and standard form . . . 80
5.1.2 Characteristic function in phase space . . . 83
5.1.3 Gaussian-preserving unitary maps . . . 84
5.1.4 Entanglement in Gaussian states . . . 88
5.2 Generalized Discriminating Strength . . . 89
5.2.1 Maximization set . . . 90
5.2.2 DS for Gaussian states . . . 93
5.2.3 Explicit evaluation for some two-mode states . . . 95
Two-mode squeezed thermal states . . . 96
6 Conclusions 100 6.1 Original results . . . 100
6.2 Open problems and possible future developments . . . 101
B Maximum distance on the phase space 104
B.1 Hamiltonian with harmonic spectrum. . . 105
B.2 Bipartite system with A qutrit . . . 106
C DS for qubit-qubit separable states - Numerical Analysis 109
D Useful inequality for Bures-related measures 112
E Generalized DS: minimum in “s” for a two-mode squeezed thermal
state 115
Correlations in Quantum
Information Theory
1.1
Introduction
Quantum information is the study of the information processing tasks that can be achieved using physical systems described by the laws of quantum mechanics. This development of classical information theory can be understood under at least two points of view. First, from a theoretical perspective, the digitalization can results quite “unnatural” in a world described by continuously varying quantities. A quan-tum mechanical world is the natural environment where to search for systems that can exist only in certain fixed states. A more practical explanation, on the other side, would state that the reduction in size of the used hardware will lead us in a few years to atomic-scaled devices, where quantum mechanical features cannot be ignored anymore.
One can safely assume the state of a physical system as completely described by a normalized vector|ψi in a separable Hilbert space H, that is often considered to be only finite dimensional. However, during the manipulations of a quantum system, one can lose information about its actual state that is no more fully known. This general situation, in which a physical state can be only statistically described, is represented by an hermitian semi-positive definite matrixρ, called density matrix of the system. More precisely, if the considered system is in one of the states {|ψji}j
with probability distribution{pj}j, its density matrix will be
ρ =X
j
pj|ψjihψj|, (1.1)
where |ψjihψj|, in the usual Dirac notation, represents the projector on the vector |ψji. Let us now assume to perform a measurement on such a state. It is well
known that generally in quantum mechanics there is an uncertainty in its outcome, even if the initial state is completely known. In the general situation described by the density matrix in Equation1.1an additional classical uncertainty must be taken into account. The amount of classical mixing of a state can be measured by its purity. Definition 1.1. Given a quantum stateρ, its purity is defined to be Tr ρ2 .
According to this definition, we say that a quantum state is completely known, i.e. described by a density matrix given by a projector|ψihψ|, if and only if it has purity 1. Such state is said to be pure.
In classical information theory the basic object is the so called “bit”, whose value can assume only one of the two states0 and 1. When we move to the quantum world, it is natural to consider its generalization: the “qubit”, which is just a quantum system whose Hilbert space is spanned by two orthonormal vectors |0i and |1i. With this formalism a qubit can assume many more states with respect to a normal bit, actu-ally an infinite number, since all superpositions of the basis vectors are acceptable. Although higher dimensional Hibert spaces could be considered, often quantum in-formation and computation protocols are described in terms of qubits because they are more easy to manipulate, both formally and in a laboratory. This simplicity can be seen for example in the description of the more general mixed qubit state, whose density matrix can be expanded in the complete operator basis, composed by the identity and by the Pauli matrices,{1, ~σ}:
ρ = 1 2(1 + ~n· ~σ) , (1.2) where σ1 = 0 1 1 0 ! , σ2 = 0 −i i 0 ! , σ3 = 1 0 0 −1 ! . (1.3)
The vector n, called Bloch Vector of the state, has to be real and it is constrainedˆ to be in the sphere|~n| ≤ 1, with equality saturated on pure states. In the opposite condition~n = 0, the state is completely mixed and we do not have any information on it: the outcome of an orthogonal measure will have indeed an uniform probability distribution.
In order to fix the notation the standard, or computational, basis indicated with the vectors|0i, |1i will be the one composed respectively by the +1 and −1 eigenstates of σ3, while the states indicated as|±i will be the eigenstates of σ1:
σ1|+i = |+i, σ1|−i = −|−i =⇒ |±i = |0i ± |1i√
2 . (1.5)
Throughout this thesis we will be mostly interested in studying the correlations among the parts A and B of a bipartite system, whose components can be thought to be in two different laboratories controlled by two distant parties Alice and Bob. If the global state is described byρAB, its local componentA is completely characterized
by the reduced density matrix [1]
ρA= TrB[ρAB] , (1.6)
which can be used to evaluate probability outcomes and expectation values of any local measure. Usually the two componentsA and B are not completely independent, and they can be related in some way. When this happens the two subsystem are said to be correlated. Some bipartite states have correlations that can be interpreted classically; for example let us consider a couple of qubits labelled with A and B, whose global state is described by the density matrix
ρcl =
1
2|0iAh0| ⊗ |0iBh0| + 1
2|1iAh1| ⊗ |1iBh1|. (1.7) This simply describes a situation where A and B are both |0i or both |1i, but we do not know which one is the actual state of the system. A completely different situation appears considering the so called “Bell State”:
|ψBelli = |0, 0iAB +|1, 1iAB √ 2 = |+, +iAB+|−, −iAB √ 2 . (1.8)
Here we have as much information as possible, since we are dealing with a globally pure state. However, if we measure subsystemA in the computational basis we could find |0i or |1i, both with 12 probability. Let us consider for example the first case, where the output would be |ψ0i = |0, 0i: if the same measure is carried out in B at this stage, the exact same output of A is found. Analogously, in the second case, the state will be projected in |ψ1i = |1, 1i and Bob would find an answer identical
to Alice’s one. Interestingly enough, the same perfect correlations will occur even if Alice and Bob should decide to measure their qubits on the basis {|±i}. Therefore subsystemsA and B are strictly correlated, in a way completely different with respect to what was going on in (1.7), where the outcomes are surely identical only if the measure is performed along the same basis used to define the state. The Bell state (1.8) is only an example of a whole class of states, with no classical analogous. Their key property is called entanglement.
Definition 1.2. A quantum state of a composite system is said to be entangled with respect to the bipartition A and B if it is not separable, i.e. if it cannot be written as a finite convex combination of pure product states as in
ρsep = n
X
j=1
pj|ψjiAhψj| ⊗ |φjiBhφj|. (1.9)
For quite a long time, separable states have been considered as the density matrices that fully capture the notion of “classicality” in quantum mechanics. Their structure is simple indeed, so much that they can be created just by means of Local Operations and Classical Communication (LOCC)1. Here for operation we mean any generalized measurement (see Sec. 3.1) or any trace-preserving evolution (see Sec. 4.1). Suppose that Alice and Bob live in two different laboratories and are interested in sharing a separable state. Starting with a default global state |0iA|0iB, they can reach every state of the form (1.9) just performing local operations in a cooperative way, property that is assured by the possibility of exchanging classical bits of information, for example through an ordinary phone. This simple setup cannot create strongly correlated states as the one in (1.8), for which some global operations, involving A and B at the same time, are needed. Entanglement is one of the main features of quantum mechanics, which deeply characterizes several of its aspects.
From a theoretical point of view, entangled states can violate Bell inequalities [1], which are relations limiting the expectation values of some operators when com-puted on separable states or assuming theories alternative to the standard quantum mechanics, like for example those involving hidden variables. From a quantum infor-mation and communication perspective, entanglement is seen as a resource and it is this the aspect that will mostly interest us. Indeed a shared entangled state between the parts allows the realization of very powerful protocols as quantum teleportation [1], where an unknown state can be destroyed in Alice’s laboratory and created in Bob’s one, no matter how far away, at the price of losing the initial entangled state and transmitting some classical bits. Furthermore, from a computational perspec-tive, the use of entangled states in quantum algorithms yields important speedups with respect to the best classical known solution. The most known example is Shor’s algorithm, which thanks to entangled states solves the factoring problem with a number of steps scaling polynomially in the input size[1]. Classically such problem is thought to be very hard to solve (exponentially scaling solution), so much that the most common cryptographic protocols are based upon this assumption.
1
Considering all these facts, we are led to believe that entanglement marks the bound-ary between classical and quantum behaviour. However, this seems to be not exactly the case. A flavour of this statement can be caught by considering, for example, the bipartite state2: ρB92= 1 2|0iAh0| ⊗ |0iBh0| + 1 2|+iAh+| ⊗ |1iBh1|. (1.10) Even if ρB92 is separable, it shows a richer structure with respect to ρcl in (1.7).
Indeed while in both cases subsystemB is described by two orthonormal states (i.e. |0i and |1i), the situation for subsystem A is different: Ah+|0iA = √12 6= 0 and
this two local states cannot be distinguished with a single measurement. It results that the quantum state ρB92 cannot be exactly mapped in a classical probability
distribution pab, as can be done for (1.7). This can be seen because classically it is
always possible to distinguish the content of a register from another just measuring it, fact that is now no longer true. This idea suggests that when mixed states are involved, non-classicality can arise even in separable states.
Returning to the computational aspect, we could ask ourselves whether entanglement is really needed to achieve remarkably faster algorithms. If the computation involves only pure states Josza and Linden [3] have shown that entanglement, whose amount has to increase with the input size, is a necessary resource to achieve an exponential speedup over a classical counterpart. However, the situation seems to be different if mixed states are used in the computation. The Deterministic Quantum Computation with one qubit (DQC1) [4] is a computational model which usesn maximally mixed qubits (with null Bloch vectors) and a single qubit with non minimal initial purity:
ρ1 =
1− α
2 1+ α|0ih0|, α ∈]0, 1]. (1.11)
Although being probably less powerful than a pure-state quantum computer, this model can still perform some tasks exponentially more efficiently than the best known classical algorithm. In particular the DQC1 model with non-zero α can be used to evaluate the normalized trace, Tr(Un)/2n, of a n-sized unitary matrix with a
number of operations scaling polynomially with n [5]. This problem seems to be very hard to solve classically, with the best known algorithm scaling exponentially with the parameter n [6]. The amount of entanglement that can be found during the computation is analysed by Datta et al. in [5]. In the paper the entanglement between any bipartition is measured using the negativity N , that we will formally
2
The index B92 is used here because this particular state is the key resource in the Bennett-92 protocol for quantum cryptography. [2]
introduce in Section1.2, and is upper bounded by a function of α: N ≤ 1
2 p
1 + α2− 1. (1.12)
Notice that it goes to zero for α→ 0. This fact shows how an exponential speedup can be achieved even with a negligible amount of entanglement in the system. These considerations suggest that classical correlations can be found also in non-entangled states. Their study will be the aim of this thesis, with particular interest in their identification and their quantification. Recently several papers addressed this topic, from both a theoretical and an experimental point of view. A thorough summary of theoretical results on the subject up to the end of 2012 can be found in the review by Modi et al. [7], while references to more recent papers can be found along the discussion of Chapter 2. Among the related experiments, we can mention the ones described in references [8,9][10]. The first two papers provide an experimental proof for the operational interpretation of two measures of correlations that will be discussed in the following: the Negativity of Quantumness (see Section 2.2.2) and the Interferometric Power (see Section 2.2.3). The experimental implementations exploit the nuclear magnetic resonance and several optical devices to manipulate qubits in the form, respectively, of nuclear spin (first case) and photon polarization or photon path (second case). Optical devices are also used in the experimental setup described in the last paper, where correlations in continuous variables Gaussian states are modified, exploiting different noise additions.
In next section, some measures of entanglement that will be useful in the following are described, while in Sec. 1.3 the Quantum Discord is presented, which is the first measure of non-classical correlation historically introduced. In Chapter 2 the discussion is extended considering a list of other measures found in the literature along with their main properties, while in Chapter 3a new quantifier, based on the Quantum Chernoff Bound, is introduced and studied. In Chapter 4 a comparison among different measures is performed, considering their values on a locally modified state. Chapter5will be devoted to continuous variable bosonic systems, with a short introduction on Gaussian states and a generalization of the Discriminating Strength defined in Chapter 3 to these infinite-dimensional systems. Conclusions are left for Chapter6.
1.2
Some entanglement measures
Detecting and measuring the entanglement of a state is not a trivial task, and a detailed discussion of the topic goes beyond the purpose of this thesis. Here we
present just some of the measures that will be mentioned in the following. The interested reader can find more details in [11].
Definition 1.2provides the formal characterization of what an entangled state is. It turns out, however, that for a generic mixed state just deciding if it is entangled or not is not an easy task. We would have to prove that a decomposition similar to the one in (1.9) is impossible. The situation is much more simpler if we are dealing with pure states, because the Schmidt decomposition [1] helps us.
Proposition 1.1 (Schmidt Decomposition). Given a pure state |ψi ∈ HA⊗ HB,
there exist two orthonormal basis {|jiA}dA
j=1 in A, {|jiB}dj=1B in B and a probability
distribution qj for which the state can be written as
|ψi = n X j=1 √q j|jiA|jiB. (1.13)
The coefficient n ≤ min{dA, dB} depends only on |ψi and it is called the Schmidt
rank of the state, while theqj are its Schmidt coefficients.
Corollary 1.2. A pure state |ψi of a composite system is entangled if and only if the number of its Schmidt coefficients is strictly greater than 1.
An entanglement measureE ∈ R should satisfy some reasonable requirements:
1. E(ρAB)≥ 0 with equality iff ρAB is pure;
2. E(ρAB) has to be invariant under local unitary operations, they just correspond to a change of basis;
3. E(ρAB) should decrease under LOCC.
Corollary 1.2 suggests a way to measure the entanglement of a pure state, that intuitively has to be zero if only one Schmidt coefficient is non null (and thus equal to 1), and maximum for the uniform distribution qj ≡ 1d, whered = min{dA, dB}.
A good entanglement measure for pure states is thus the von Neumann entropy S of its reduced density matrix, that satisfies such requirements:
S(ρA) =−Tr [ρAlog ρA] =−
X
i
qilog qi= S(ρB), (1.14)
where in such definition we meanq log q≡ 0 when q = 0. This entanglement measure is called Entropy of Entanglement. One generalization to mixed states can be built in the following way:
Definition 1.3 (Entanglement of Formation). Let us consider a generic unravelling in pure states of ρ:
ρ =X
i
pi|ψiihψi|.
The Entanglement of Formation is defined to be EF(ρ) = inf {pi,|ψii} X i piS
ρ(i)A, where ρ(i)A = TrB[|ψiiABhψi|] . (1.15)
It can be proven that this is a good entanglement measure, satisfying all properties 1-3 [12].
Despite its theoretical interest, the Entanglement of Formation is still hardly com-putable, involving a difficult minimization. An easier approach would be searching for sufficient criteria of entanglement, in the hope of showing that they are also nec-essary at least in some simple situations. The most known among these criteria is the one that goes under the name of Positive Partial Transpose Criterion [1]. Definition 1.4. Chosen an orthonormal basis {|ii}i in a Hilbert space H, we can
define the transposition operator acting on it as T (|iihj|) = |jihi|, ∀ i, j. Proposition 1.3. T ⊗ 1n(ρ) is semi-positive definite for each ρ separable.
Proposition 1.3 assures us that if a state is mapped to a non semi-positive definite operator, than it is entangled. Unfortunately, it is known that there exists some entangled states that by chance are sent to an acceptable matrix density under par-tial transposition. For generic systems the PPT criterion is hence only a sufficient criterion for entanglement, the only exceptions being qubit-qubit, qubit-qutrit and 1-n mode Gaussian states, where PPT has been shown to be also necessary [13–15]. Exploiting Proposition1.3, a measure of entanglement can be constructed by taking the absolute value of the sum of negative output eigenvalues, which is equivalent to the following definition involving the Trace Norm:
||O||1 = Tr
h√
O†Oi. (1.16)
Definition 1.5 (Negativity of Entanglement). LetρABbe a bipartite quantum state.
The Negativity of Entanglement of ρAB is defined as
N (ρAB) = ||T ⊗ 1(ρAB
)||1− 1
2 . (1.17)
It can be easily seen that this quantity is well defined, being independent on the basis chosen forT and on the system in which the transposition is applied. Property
2 is trivial with the definition of the trace norm, while property 3 is more involved to prove [16]. The great advantage of this measure3 is that it can be very easily computed, not involving any minimization.
1.3
Quantum Discord
The first functional which allows one to identify the peculiar correlations of a state like 1.10 was originally introduced by Ollivier and Zurek [17] and it is known as Quantum Discord. In the following paragraph we will review this quantity. Since it is an entropic functional we start by recalling some basic entropic notions.
1.3.1 Entropic formalism
In classical information theory, the key quantity to measure the amount of informa-tion contained in a stochastic variable is the Shannon Entropy :
H(pj) =−
X
j
pjlog pj, (1.18)
which depends only on the events probability distribution4. If we are dealing with two variables X and Y , described by the joint probability distribution pxy, we could
search for the amount of information shared by them. The quantity of interest in this situation is the mutual information:
I(X : Y ) = H(X) + H(Y )− H(X, Y ), (1.19)
where probability distributions px =Pypxy, py = Pxpxy and pxy are respectively
used. The idea sub-standing this definition can be grasped watching at figure 1.1, where the amount of information contained in a variable is represented with a Venn diagram. The intuitive meaning of definition 1.19 is that, to measure the shared information, we should consider the superposition area. A different approach to the problem involves a measurement process. In this second scenario one aim to measure one subsystem, say X, trying to get informations about Y . The important quantity here is the conditional entropy
H(Y|X) =X
x
pxH(Y|x), (1.20)
3It is rigorously a measure only for the simple situation where the partial transposition criterion is
necessary and sufficient. In a generic situation there could be entangled states with zero negativity.
4
H ( X )
H( X | Y ) I ( X : Y ) H( Y | X ) H( Y )
Figure 1.1: Graphical representation of Shannon entropies and mutual informa-tion.
where H(Y|x) depends on the probability distribution of Y knowing that X = x, which is
py|x =
pxy
px
. (1.21)
The conditional entropy describes the average amount of information still inY after the X measurement. With this in mind, a second possible definition of mutual information could be
J(Y|X) = H(Y ) − H(Y |X). (1.22)
The notation J(Y|X) is used to show the asymmetry between the two variables in the definition. We can now ask ourselves whetherI and J are different quantities or not.
Proposition 1.4. If X and Y are two classical stochastic variables described by the joint probability pxy, the mutual information does not change if measured with
I(X : Y ) or with J(Y|X).
Proof. To see this we just need to prove that
H(Y|X) = H(X, Y ) − H(X), (1.23)
which follows from Bayes rule (1.21): H(Y|X) = −X
x
px
X
y
py|xlog py|x=−X
x,y
pxylog
pxy
px
= H(X, Y )− H(X) (1.24)
If we move to a quantum scenario density matrices take the place of stochastic vari-ables. The quantum situation is less trivial, since all classical probability distributions can be thought as purely diagonal density matrices. The generalization of (1.18) is
the von Neumann entropy, already introduced in (1.14):
S(ρ) =−Tr [ρ log ρ] , (1.25)
which is exactly the Shannon entropy evaluated on ρ eigenvalues. Some facts about von Neumann entropy which will be useful are:
• Given an unravelling ρ =P ipiρi [1]: X i piS(ρi)≤ S(ρ) ≤ H(pi) + X i piS(ρi), (1.26)
where the second inequality is saturated iffρi are orthogonal.
• Given two quantum states, we can define the relative entropy of ρ to σ as S(ρ||σ) = Tr [ρ log ρ] − Tr [ρ log σ] . (1.27) Some of its properties are:
– Klein’s Inequality [1]:
S(ρ||σ) ≥ 0, with equality iff ρ = σ; (1.28) – not increasing under local operations5 Φ [18]:
SΦ(ρ)||Φ(σ)≤ S(ρ||σ). (1.29)
We want now try to generalize the concept of mutual information to the quantum scenario, where the amount of information shared by the two subsystems A and B has to be evaluated. The generalization of (1.19) can be easily done using the reduced density matrices of the problem:
I(A : B) = S(ρA) + S(ρB)− S(ρAB). (1.30)
Such quantity is known to be positive, with equality only for completely uncorrelated states, i.e. that can be factorized in a tensor product ρ = ρA⊗ ρB [1]. Non-zero
values of the mutual information are hence due to the presence of general correlations within the considered state, being them entanglement-like or just classical. On the other hand, the analogous of (1.22) is less trivial to write, since a measurement on subsystem A is involved. Considering only projective measurements, we can write a J for each choice of the basis in which the measure is carried out. Let us call
5
Πi(A) = |eiiAhei| the projectors in this basis. The output B states with respective probabilities are ρB|j = 1 pj TrA h Πj(A)ρABΠj(A) i , pj = TrAB h Πj(A)ρABΠj(A) i , (1.31)
so that the average global final state is ρ0AB =X
j
pjΠj(A)⊗ ρB|j. (1.32)
The generalization of the conditional entropy (1.20) now depends on the chosen basis SB|{Πj(A)}
=X
j
pjS ρB|j , (1.33)
as well as the quantum version of J: J(B|A){Π(A) j } = S(ρB)− S B|{Πj(A)}= S(ρB)− X j pjS ρB|j . (1.34)
Our goal was to find the amount of information shared between the two subsystems, but to do so we had to chose arbitrarily a basis in A. To avoid unlucky choices the reasonable definition for the quantum version ofJ(B|A) is
J(B|A) = max
{Πj(A)}
J(B|A){Π(A) j }
, (1.35)
where the best measurement is chosen. However, even after this maximization, the two quantum versions of mutual information (i.e. (1.30) and (1.35)) are different. Their difference is what is called Quantum Discord.
1.3.2 Definition and properties
Formally the Quantum Discord is the difference between the two quantum mutual information expressions (1.30) and (1.35):
D(B|A) = I(A : B) − J(B|A) = S(ρA)− S(ρAB) + min {Πj(A)}
X
j
pjS ρB|j . (1.36)
As already pointed out in the previous section, being inequalityI ≥ 0 saturated only on tensor product states, theI mutual information can be considered a measure of the global amount of correlations between the parts of the system. This interpretation is confirmed by the intuitive idea that the evaluation of such quantity does not alter
the considered state, while the projective measurement involved in the definition ofJ disturbs and modifies the quantum system, destroying some of the initial correlations that in this way cannot be measured. Indeed, J captures only the classical fraction of such correlations [19], that can be acquired via a measurement. This fact leads to an interpretation for Quantum Discord as a measure of purely quantum correlations. This intuition is confirmed by the discord positivity. The formal statement is the following (a rigorous proof can be found in Datta’s doctoral thesis [20]):
Proposition 1.5. D(B|A) ≥ 0, with equality if and only if there exists a basis for A such that the global state is left in average unmodified by the measurement in that basis: ρAB = ρDAB, with ρDAB = X j Πj(A)ρABΠj(A) = X j pjΠj(A)⊗ ρB|j. (1.37)
The key result of this section is the condition obtained for zero discord: equality (1.37) shows that to detect some non-classicality by measuring subsystem A, the state has to be written as a superposition of some orthogonal projectors in A. Due to the asymmetry in the definition of Quantum Discord, however, the states ρB|j
left in B at the end of the measuring process could still present quantum features, e.g. non commutativity. For this reason, having a zero D(B|A), gives informations only about classicality in the measured subsystem. This class of states is called Classical-Quantum (CQ):
ρCQ=
X
j
pj |ejiAhej| ⊗ ρB|j, Ahej|ekiA= δjk. (1.38)
Analogously, if a measurement on subsystem B is considered, a state with null D(A|B) is called Quantum-Classical (QC) and has the typical structure:
ρQC =
X
j
pj ρA|j ⊗ |fjiBhfj|, Bhfj|fkiB = δjk. (1.39)
In order to have a completely classical state, we have hence to require at the same time D(B|A)(ρ) = D(A|B)(ρ) = 0. This class of states will be called Classical-Classical (CC) and its elements can be written as
ρCC =
X
j
pj|ejiAhej| ⊗ |fjiBhfj|, Ahej|ekiA=Bhfj|fkiB = δjk. (1.40)
With these definitions, the B92 state considered in (1.10) is QC, being classical on subsystemB, but with non zero D(B|A). Its actual evaluation would give a value of D(B|A) ∼ 0.8, that can be compared for example with the discord of the Bell state
(1.8), which is1. Having explained quantitatively the non-classicality of such a state, we can now address again the discussion about the DQC1 computational model, that was the second hint to the presence of non-classicality beyond entanglement. In his paper [5], Datta showed how, measuring the first qubit, a non-zero Quantum Discord D(n|1) could be found in the DQC1 system, at least for a simple class of unitary matrices. Such non-classical correlations could be the key resource to achieve a speedup in that situation.
The condition (1.37) for having a zero discord implies separability if the state con-sidered is pure. Actually, the discord itself reduces to a measure of entanglement for pure states: the entropy of entanglementS(ρA). This can be easily seen considering
the measurement on the Schmidt basis of the state, which yieldsS(ρB|j) = 0. This result confirms that to find quantum features that cannot be described by the mere entanglement, mixed states have to be taken into account.
1.3.3 Symmetric Quantum Discord
To conclude the discussion on Quantum Discord let us present a symmetrized version [7] that keeps under consideration, at the same time, both subsystems. The key observation is thatJ(B|A){Π(A)
j }
can be written as the mutual informationI(A : B) evaluated on the average state after measuring A:
ρ0 = ΠA⊗ 1B(ρ)≡ X j Πj(A)ρΠj(A) =X j pjΠj(A)⊗ ρB|j. (1.41)
This can be easily seen with the help of property (1.26): S(ρ0AB) = H(pj) + X j pjS(ρB|j) = S(ρ0A) + S(ρB)− J(B|A){Π(A) j } , (1.42)
and noticing thatρB = ρ0B. An equivalent formula for asymmetric Quantum Discord
is then
D(B|A) = I(ρAB)− max {Πj(A)}
I(ρ0AB). (1.43)
A straightforward generalization that can be adopted to have a symmetric measure [7] is
D(A : B) = I(ρAB)− max {Πj(A),Πk(B)} I(ρ00AB), (1.44) where ρ00AB = ΠA⊗ ΠB(ρ)≡ X j,k Πj(A)Πk(B)ρΠk(B)Πj(A) (1.45)
is the average state after two local projective measurements on bothA and B. It is reasonable to expect that if a state is classical in one of the two subsystem (i.e. it is QC or CQ), then the symmetric Quantum Discord (1.44) does not add anything to the usual asymmetric version which measures the non-classical component of the system. This fact is proven in the following proposition, that concludes this chapter. Proposition 1.6. If the state is QC, the symmetric Quantum Discord D(A : B) reduces to the Quantum DiscordD(B|A) in which only the A subsystem is measured:
ρQC =
X
i
piρ(A)i ⊗ |iiBhi|. (1.46)
Proof. Let us indicate with ρ0
i = ΠA(ρ(A)i ) = P jΠ (A) j ρ (A) i Π (A)
j the decohered local
state ρ(A)i along the basis{|kiA}, so that after the A measurement the state is ρ0; if
also B is measured the state will be indicated with ρ00:
ρ0 =X i piρ0i⊗ |iiBhi|; ρ00 = X i,j piqijρ0i⊗ |jiBhj|, (1.47)
whereqij =|Bhi|jiB|2. Since both measurements are local, considering the definition
of D(A : B) (1.44) we can perform the maximization in ΠB first, showing that for
eachΠAsuch maximum is reached when the projection is carried out on the classical
base|iiBhi|. Since ρ00A= ρ0
A, the symmetric discord can be rewritten as
D(A : B) = I(ρ)− max
ΠA S(ρ0A)− min ΠB S(ρ00) − S(ρ00B) . (1.48)
The first equality in Equation (1.42), can be rewritten as S ρ00 − S ρ00B =X
j
pjS
ρ0A|j, (1.49)
whereρ0A|j is the state ofA after the measurement, assuming that the measurement of B has returned the jth outcome, with probabilityp
j. From the concavity of the
von Neumann entropy (1.26): X j pjS(ρA|j) = X i,j piqijS X α paqαj ρ0 α P βpβqβj ! ≥X αj pαqαjS(ρ0α) = X α pαS(ρ0α), (1.50)
which is the result obtained when qij = δij, i.e. when the measurement is done on
the|iiB basis. The proof is now concluded because
min ΠB S(ρ00 )− S(ρ00B) = S(ρ 0 )− S(ρ0B),
Measures of Quantum Correlations
After the definition of Quantum Discord, several other measures of purely quantum correlations have been introduced, in an effort to gain a better operational under-standing on the subject (a process which closely remind what happened in the study and characterization of entanglement). Often for each one of these new measures there is a version in which only the classicality of one subsystem, say A, is tested and another slightly different version that considersA and B symmetrically, exactly as was done in chapter 1 with Quantum Discord and its symmetrical formulation. In this chapter a list is presented, with the measures discussed in Modi et al. review [7], along with others recently introduced. Their definition and main properties are stated together with references of interest. The measures are regrouped to emphasize their similarities, with the exception of the Discriminating Strength, that will be the argument of Chapter 3.
Particular attention will be dedicated to the Trace Distance Discord (TDD) in Section
2.1.4, for which we extended the classes of qubit-qubit states for which a closed formula is known. Entangled states always contains discord-like correlations, but the viceversa is not true. An interesting problem is to find the maximum value that a measure of correlation can reach over the set of separable states. Addressing this problem for the TDD at the end of Section 2.1.4, we found an upper bound that can be reached with a limit procedure when the unmeasured subsystemB is infinite-dimensional (e.g. an harmonic oscillator), but that seems unlikely to be reached for dB<∞.
Like the Quantum Discord, a functional of the density matrix M (ρ) should satisfy the following properties to be considered a valid measure of quantum correlations:
1. M (ρ) = 0 iff ρ∈ CQ, CC depending on whether classicality has to be evaluated for a single subsystem (A throughout this thesis) or for both of them;
2. M (ρ) = MUAUBρ UA†UB†
has to be invariant under local unitary opera-tions;
3. if ρ is pure M (ρ) has to be an entanglement monotone, i.e. if|ψi is sent to |φi by means of a LOCC, inequalityM (|φihφ|) ≤ M(|ψihψ|) must hold.
If only the classicality of subsystem A is being tested, a final desirable property is:
4. M should be non-increasing under any local operation (see Sec. 4.1) on the unmeasured subsystemB.
While the first and the second conditions are self-explanatory, it is worth to spend a few words on the last two. The third property is required because on pure states each measure should quantify the entanglement, decreasingly monotone under LOCC, because there cannot be correlations of a different kind without classical mixing. The fourth condition must hold because a local operation on the unmeasured subsystem B should not increase the quantumness of A, which is being tested. Notice that a local operation onA can in general increase its quantumness, since orthogonal states can be mapped to non-orthogonal ones. This is explicitly shown in Chapter4. The chapter is divided into two big sections, describing the measures which have or not a geometrical interpretation.
2.1
Geometric measures
Proposition 1.5 (in Section 1.3.2) states that a quantum state has non-zero quan-tum correlations if it cannot be written as a Classical-Quanquan-tum (1.38) or Classical-Classical state (1.40). The idea underlying the geometric measures is to quantify the amount of correlations in a state by computing its distance from those set of states, as measured by some functionald(ρ, σ):
Dd(ρ) = min
σ∈CQd(ρ, σ); D (S)
d (ρ) = minσ∈CCd(ρ, σ). (2.1)
Such operation is graphically represented in Figure 2.1, where the set of classical states is explicitly shown to be non convex. This is because the property of being a QC, CQ or CC state is not preserved by a convex combination, since the involved
ρ
Classical
Separable
Physical States
Figure 2.1: Graphical representation for the geometric distance of a given state ρ from the set of classical states. Depending on the measured subsystem these could
be QC, CQ or CC states.
reduced density matrices for the classical subsystem could be diagonal in different basis.
Usually also the distance of ρ from the separable states is a good measure of entan-glementE. The relations CC ⊂ CQ ⊂ SEP lead to the inequalities
Dd(S)(ρ)≥ Dd(ρ)≥ E(ρ), (2.2)
which are an interesting feature of the geometric approach at the problem.
The minimisation in (2.1) is usually not easy to carry out. Often there is a sim-plification if the minimum is reached on a state σmin obtained projecting ρ in some
orthogonal basis:
σmin = Π(ρ) =
X
j
ΠjρΠj, (2.3)
where projectors Πj are chosen on A, or on both subsystem, depending on the
symmetry required to the measure. If this is the case, it is possible to minimize only on the projection basis, evaluating how much a local measurement disturbs the state:
D0d(ρ) = min Πa d(ρ, ΠA(ρ)); D 0(S) d (ρ) = minΠ aΠb d(ρ, ΠAΠB(ρ)). (2.4)
This simpler approach is usually called “Measurement Induced Disturbance”, but even when Equation (2.3) holds and this second approach is equivalent to the first one of Equation (2.1), the minimization can still be very hard.
There are many available choices for the functionald, and each one of them leads to a different measure of correlations. The most common are:
• Squared Hilbert-Schmidt distance −→ Geometric Discord; • Squared Bures distance −→ Bures Geometric Discord; • Trace Distance −→ Trace Distance Discord;
which are discussed in the following.
2.1.1 Zero and One Way Quantum Deficit
Also known as “Thermal Discord” [21], the One and the Zero Way Deficit are defined respectively as ∆→A(ρ) = min Πa S ρ0AB − S(ρAB), and ∆0(ρ) = min Πa⊗Πb S ρ0AB − S(ρAB), (2.5)
whereρ0AB = Π(ρAB) is the state dephased in the local basis shown in the
minimiza-tion. These quantities are shown to be exactly the minimum relative entropy ofρ to the set ofCQ or CC states [22]. It is important to point out that for its asymmetry, relative entropy is not a real distance. Its positivity (1.28), however, allows us to consider this measure as a geometric one. Modi et al. [23] showed that simplification (2.4) is possible, since the minimum is reached dephasing the state.
Properties 1 and 2 are satisfied thanks to the properties of the relative entropy, like Equation (1.28) and the invariance under unitary operations. We want now to prove the third one. Let us consider the Relative Entropy of Entanglement [11]
ES(ρ) = min
σ∈SEPS(ρ||σ), (2.6)
which is well known to reduce to the entropy of entanglementS(ρB) for a pure state.
Since CQ⊂ SEP the general inequality is min
σ∈CQS(ρ||σ) ≥ minσ∈SEPS(ρ||σ) = S(ρB).
If we find a σ0 ∈ CQ such that S(ρ||σ0) = S(ρB), then the two minimums are the
same and the One Way Deficit reduces to entropy of entanglement on pure states, which is a measure of entanglement and then monotone under LOCC. Given a pure state ρ =|ψihψ|, it can be expanded in its Schmidt basis
ρ =X ij √q iqj|iiAhj| ⊗ |iiBhj| =⇒ ρB = X i qi|iiBhi|, (2.7)
and the searchedσ0 can be built as
σ0 =
X
i
qi|iiAhi| ⊗ |iiBhi|, (2.8)
as can be easily checked. The last requirement for being a good correlation measure is satisfied for the One Way Deficit thanks to relation (1.29).
At the end of Section 1.3.3 we proved that the symmetric Quantum Discord actu-ally reduces to its usual asymmetric version when the state is QC (or CQ), if the measurement is performed on the non-classical subsystem. The same fact holds also when the One and Zero Way Deficit are involved.
Proposition 2.1. The Zero Way Deficit ∆0 reduces to the One Way deficit ∆→ A in
a QC state.
Proof. From the definitions (2.5), the thesis is proven if we show that for each mea-surement inA the minimum among the measurement in B is reached for the projec-tors |iiBhi|, that leave the state unchanged. Using the same notation of (1.47), that denotes byρ0 andρ00the states after the measurementsΠAandΠAΠB respectively,
and the concavity of the von Neumann entropy (first inequality in (1.26)), one has: S(ρ00)≥ −X ij piqijlog (piqij) + X ij piqijS(ρ0i) =X i piS(ρ0i) + H({pi}) − X ij piqijlog(qij) = S(ρ0) + X i piH(qij)≥ S(ρ0). (2.9) SinceS(ρ0) = S(ρ00) when the B measurement is performed along the base|iiBwhere
ρB is diagonal, the proof is concluded.
2.1.2 Geometric Discord
This measure is based upon the Hilbert-Schmidt squared distance, obtained from the operator norm||O||2 =pT r [O†O]. Firstly introduced in [24] as
DG= min σ∈CQ||ρ − σ|| 2 2 and D (S) G = minσ∈CC||ρ − σ|| 2 2, (2.10)
it has been shown in [25] that the formulation with projectors, like in Equation (2.4), is equivalent to it. Properties 1-3 apply to this definition, only the third being non trivial. For pure states it has been stated in [26] that Geometric Discord (GD)
reduces to the linear entropy: DG(|ψi) = 1 − Tr ρ2A = 1 − X j q2 j, (2.11)
where {qj} are the Schmidt coefficients. A known measure of entanglement, called
Concurrence, can be written in terms of such quantity [11]: Cpure(ρ) =
q
2 1− T r(ρ2 A),
so that the GD is an entanglement monotone for pure states.
The great advantage of this definition is that such quantity is very easy to compute, at least in the two-qubit case where each state can be represented with Pauli Matrices σµ={1, σ1, σ2, σ3} as ρAB = 1 4 3 X µ=0 3 X ν=0 ΓµνσAµ⊗ σBν. (2.12)
The minimization involved in the definition can be performed explicitly, so that the Geometric Discord of ρAB equals to:
DG= 1 4 3 X k=1 3 X ν0 Tkν2 − λmax, (2.13)
whereλmax is the largest eigenvalue of the matrix L = ~xA~x|A+ T T|, built from the
local Bloch vectorxAi≡ Ti0and the 3-dimensional correlation matrixTkl≡ Γkl.
However the GD has a major drawback: it does not satisfy the fourth property, as pointed out by M. Piani in [27]. The argument is simple and consists in adding to the unmeasured subsystemB an ancillary system C in a state σC. Since||ρ ⊗ σ||2 =
||ρ||2·||σ||2, this would reduce the GD of a factor||σC||22 = Tr(σC2)≤ 1. The problem
emerges if we start with a mixed ancillary state already added to B. We can freely and reversibly removeC from the problem, increasing the quantumness of A. This fact has led the community to search for different geometrical approaches, like the following two.
2.1.3 Bures Geometric Discord
This measure is based on the Bures Distance dB(ρ, σ) = h 21−pF (ρ, σ)i 1 2 , (2.14)
whereF is the Uhlmann’s Fidelity [1] F (ρ, σ) =||√ρ√σ||2 1= Tr q√ σρ√σ 2 . (2.15)
The Bures Geometric Discord [28] is defined as DB(ρ) = min σ∈CQd 2 B(ρ, σ), D (S) B (ρ) = minσ∈CCd 2 B(ρ, σ), (2.16)
the square being introduced to simplify the expression. Notice that hereB stands for “Bures” and not for the single subsystem whose classicality is being tested. The first two requirements and the fourth are satisfied from the properties of the Uhlmann’s Fidelity [1], while the third is true because for pure states the expressions above reduce to the corresponding geometric measure of entanglement EB, based on the
squared Bures distance from the set of separable states (see [28,29]).
IfA is a two-dimensional system, a simpler expression for DB has been computed in
[29]. Indeed, in such a case we have
FA(ρ)≡ max σ∈CQF (ρ, σ) = 1 2max|ˆu|=1 " 1− TrΛ(ˆu) + 2 dB X l=1 λl(ˆu) # , (2.17)
whereλ(ˆu) are the decreasing ordered eigenvalues of Λ(ˆu)≡√ρ (~σA· ˆu ⊗ 1)√ρ. In
[28] is shown that if A is a qubit the maximum value of DB is 2−
√
2, so it is convenient normalize this measure as:
˜ DB= DB 2−√2 = 1−pFA(ρ) 1−√1 2 . (2.18)
2.1.4 Trace Distance Discord
In order to use distances more similar to the well known Hilbert-Schmidt one, is possible to define
Dp(ρ)∝ min
σ∈CQ||ρ − σ|| p p,
where the Schatten p-norm||O||p = Tr
h
(O†O)p2
i1
p
was used. Forp = 2 this quantity reduces exactly the Geometric Discord. It has been shown by Paula et al. [30] that the only value of p which does not present the same problem as the GD is p = 1, corresponding to the so called “Trace Distance”. Indeed, it is contractive under trace-preserving quantum operations on the state (see Section4.1for their definition). The
resulting one-sided measure is called Trace Distance Discord (TDD): T DD(ρ) = 1 2σ∈CQmin ||ρ − σ||1 = 1 2σ∈CQmin Tr hp (ρ− σ)2i, (2.19)
where the global coefficient1/2 is just the standard normalization for the trace dis-tance. While the first two requirements to be a good measure of correlation are easily proven using the main properties of the trace norm [1], for the third one there is not yet a formal proof that we are aware of. Only very recently the same problem has been solved for its analogous quantifier based on the measurement induced distur-bance approach (2.4), showing that it is indeed an entanglement monotone on pure states [31]. These two approaches are equivalent whenA is a qubit [32], but unfor-tunately there are numerical counterexamples for mixed states in higher dimensions. However, the situation is still unclear when pure states are involved, and it is very likely that also the geometric measure (2.19) behaves monotonically under LOCC on such class of states.
Differently from the 2-norm geometric discord discussed in Section 2.1.2, the mini-mization in Equation (2.19) is not easily resolved analytically. When A is a qubit, the equivalence with the measurement induced disturbance approach allows the min-imization to be carried out only considering as σ those CQ states which can be obtained from ρ via the action of a dephasing channel, that projects A into an or-thonormal basis characterized by the projectors {Πj(A)}j. Accordingly, in this case
we can write
T DD(ρ) = 1 2{ΠminA}
||ρ − (ΠA⊗ 1B) (ρ)||1, (2.20)
where ΠA stands for the considered qubit dephasing channel, so that the output
state is locally diagonal in the chosen basis:
ΠA(ρ) = 2 X j=1 Tr h Πj(A)ρi Πj(A). (2.21)
Even in this way, however, the minimization is not trivial. In the first following paragraph it is shown how to approach the problem for a two-qubit state, while in the second we present a bound on the maximum value that the TDD can reach on the set of separable states in the qubit-qudit case.
Explicit evaluation of the TDD for some class of two qubit states A two-qubit state can be represented by the matrix Γµν introduced in (2.12). To
simplify the expressions involved we can set xAi = Γi0 and xBi = Γ0i, which are
Tufarelli and Giovannetti in [33]. Using the singular value decomposition on the 3× 3 correlation matrix Tkl≡ Γkl, withk = 1, 2, 3, we can rewrite it as
T = O| γ1 0 0 0 γ2 0 0 0 γ3 Ω,
where T is real and O and Ω can be chosen in SO(3). It derives that {γi} are real
(not necessarily non-negative) quantities whose moduli correspond to the singular eigenvalues ofT . In the paper it is shown that, defining±ˆe the Bloch vectors of the measurement projectors, for a generic qubit-qubit state the TDD can be evaluated as T DD(ρ) = √ 2 4 q min ˆ e h(ˆe), h(ˆe) = a + p a2− b; (2.22) with a = x2A⊥+X i γi2ω2i⊥, b = 4χ2+ g2 , (2.23) χ = γ1(~xA⊥· ~w1⊥) γ2(~xA⊥· ~w2⊥) γ3(~xA⊥· ~w3⊥) , g = γ2γ3(ˆe· ˆω1) γ1γ3(ˆe· ˆω2) γ1γ2(ˆe· ˆω3) . (2.24)
In these formulas we have used the definitions wˆk =
P
jOkjxˆj, ˆvk =
P
jΩkjxˆj,
where xˆj are the unit Cartesian vectors. In addition, the convention of writing
~n⊥≡ ~n − (~n · ˆe)ˆe to represent the orthogonal component of a vector ~n with respect
to e, has been adopted.ˆ
In the paper [33] the minimization in Equation (2.22) has been explicitly performed in some particular cases, like with homogeneous |γi|, when ~xA = 0 or with a single
γj 6= 0. In the following will be presented the results of the minimization in a
symmetric case with two non-zeroγi.
γ3 = 0, ~xA on the plane spanned by ˆw1 and ˆw2:
Let θ and φ be the polar angles of ~e respect to the basis ˆwi. If λ =|~xA| and θA is
the angle between~xAand w1, the relevant quantities are:
x2A⊥ = λ21 − sin2θ cos2(φ− θA) ,
w21⊥ = 1− sin2θ cos2φ, ω2⊥2 = 1− sin2θ sin2φ, ( ~w1⊥· ~xA⊥)2 = λ2cos θA− sin2θ cos φ cos(φ− θA)
2 , ( ~w2⊥· ~xA⊥)2 = λ2sin θA− sin2θ sin φ cos(φ− θA)
2 ,
which yield
a = λ21 − sin2θ cos2(φ− θ
A) + γ12 1− sin2θ cos2φ + γ22 1− sin2θ sin2φ ,
b = 4γ12γ22cos2θ + 4λ2γ12cos θA− sin2θ cos φ cos(φ− θA)
2 + 4λ2γ22sin θA− sin2θ sin φ cos(φ− θA)
2 .
Numerically the minimum ofa +√a2− b seems to be reached when ˆe is in the plane
spanned bywˆ1 and wˆ2. Indeed writing
ˆ
e = [~e− (ˆe · ˆw3) ˆw3] + ˆw3,
with = (ˆe· ˆw3), the function h(ˆe) can be written as a function of φ and 2. Fixing
φ, together with all the other state’s parameters (γ1, γ2, λ, θA), such function seems
to reach the minimum in 2 for 2 = 0. This statement has been numerically verified
randomly generating ∼ 106 set of parameters. For the complexity of the quantities involved, an analytical proof has not been found, but with this hypothesis the mini-mum inφ can be calculated. Indeed with this assumption a2− b can be shown to
be a perfect square: a = λ2sin2(φ− θA) + ¯ γ−1 2∆ cos(2φ) , b = ∆
2 cos(2θA) + cos(2φ) cos
2(φ− θ A)− 2 cos(φ − θA) cos(φ + θA) + ¯γ sin2(φ− θA), a2− b = λ2sin2(φ− θA)− ¯γ + ∆ 2 cos(2φ) 2 ,
where we have defined¯γ ≡ γ12+γ22
2 and∆≡ γ 2
1−γ22. At this point we have to minimize
inφ the function h = λ2sin2(φ− θA) + ¯γ− ∆ 2 cos(2φ) + λ2sin2(φ− θA)− ¯γ + ∆ 2 cos(2φ) . (2.25) It can be noted that:
• if λ2sin2(φ− θ
A)≤ ¯γ − ∆2 cos(2φ) =⇒ h = 2¯γ2−∆2 cos(2φ) ,
• if λ2sin2(φ− θ
A)≥ ¯γ − ∆2 cos(2φ) =⇒ h = 2λ2sin2(φ− θA) ,
so considering the two sides of the inequality we should always take the minimum of the bigger one, multiplied by a factor of 2. The actual presence of intersections, between the graphs of the two inequality sides, leads to3 cases:
1. Exists aφ0 where the two functions have the same value, so that: min φ h = h(φ0) for γ¯ 2 −∆2 cos(2φ0) = λ2sin2(φ0− θA); (2.26) 2. λ2sin2(φ− θ
A) < ¯γ−∆2 cos(2φ) = γ12sin2φ + γ22cos2φ ∀φ. In this case:
min φ h = 2 minφ ¯ γ− ∆ 2 cos(2φ) = 2 min{γ2 1, γ22}
Notice that a sufficient condition to be in this case isλ < min{|γ1|, |γ2|}.
3. λ2sin2(φ− θ
A) > ¯γ − ∆2 cos(2φ) = γ12sin2φ + γ22cos2φ ∀φ. This condition
with the strict inequality cannot be satisfied, because for φ = θA there is an
absurd.
To sum up, in the numerically based hypothesis that the minimum is reached for ˆe in the plane ofwˆ1 and wˆ2, there is an easy answer ifλ < min{|γ1|, |γ2|}:
T DD = 1 2
q
min{γ2 1, γ22}.
If this is not the case, we will have to determine in which case we are among 1 and 2, find φ0 from (2.26) if needed, and eventually use Equation (2.22).
Maximum on separable states In this section we address the problem of finding the maximum TDD over the set of separable states when A is a qubit. It can be compared for example with its value on the maximally entangled state (1.8), which is shown to be 12 in [33]. First of all we find the maximum for a system withdB=∞,
where the problem can be simplified considering the maximum over the subset of pure-QC (or pQC) states, defined as those quantum-classical states (generally not pure) whose components on the “quantum” system (the first) are taken to be pure. Accordingly, their elements can be written as
ρ(pQC) =X
i
pi|ψiiAhψi| ⊗ |iiBhi|, (2.27)
with {|iiB} orthonormal in B and |ψiiAhψi| pure, but not necessarily orthogonal1.
With a limiting procedure the maximum will be found to be π/8, a value that
1
Notice that a pure-QC state is different from a CQ state, since the set {|ψiiA} is not required
to be a basis for A. Indeed, they could be linearly dependent and if dB> dAthey even outnumber
each basis of subsystem A. Even the role played by these two class of states is very different: while the TDD is identically zero on the CQ states, we will see that in certain conditions (e.g. dB= ∞)
it reaches values asymptotically near the maximum over the set of separable states exactly on particular pure-QC states.
hardly seems achievable considering only a finite sum in (2.27), because of the general formula for the TDD on such a state (see Equation (2.37)). Considering a finite-dimensional subsystem B, all separable states should be taken into account, but such an optimization is highly non trivial since an efficient parametrization of a separable state is still lacking. It is reasonable to think that the maximum can be always reached on a pure-QC state, but there is no formal proof of this fact. It is surely true that the maximum reached for dB =∞ is an upper bound for all dB, as
can be seen embedding the second space in the first one, so the problem would be solved if a state with T DD = π/8 could be found for each choice of dB. For other
measures of discord-like correlations (like the Discriminating Strength discussed in Chapter 3), the maximum found for dB = ∞ is already reached for dB = 3 on the
generalizedB92 state ρGB92=
1 3 h
|0iAh0| ⊗ |αiBhα| + |+iAh+| ⊗ |βiBhβ| + |×iAh×| ⊗ |γiBhγ|
i
, (2.28) with|×i the +1 eigenvector of σ2and|αi, |βi, |γi describing an orthonormal basis of
B. Unfortunately, for the TDD the situation is more difficult: its value on this state is only1/3 and even the proof of its optimality among the pure-QC states is missing. To conclude this discussion we can talk about the situation for dB = 2, which is
only slightly better because the set of generic QC states has been analysed in [33]. The maximum on such class is found to be 1/4, reached on the B92 state, which is pure-QC. Even in this case, however, there is no proof of its actual optimality among all the separable states. Even a numerical analysis is non trivial, for the difficulties in performing the minimization involved in the definition. The technical details of this discussion will be now presented, starting with the casedB=∞. In the end the
calculation for the GB92 state (2.28) is carried out.
• dB = ∞
The key property to see that the maximum is actually achieved on the pure-QC set, is the trace distance joint convexity [1]:
1 2 X i piρi− X i piσi 1 ≤X i pi 1 2 ρi− σi 1. (2.29)
With this statement it can be shown that for every separable state, that can always be written as
ρ(sep)=X
i
a pure-QC state with greater TDD can be found just substituting |φiiB with |iiB. Notice that this operation is legitimate only when dB = ∞. Being A a
qubit, we are allowed to use the simpler expression in (2.20): T DD(ρ(sep)) = min ΠA 1 2 X i piρ(A)i ⊗ ρ (B) i − X i piΠA h ρ(A)i i⊗ ρ(B)i 1 ≤ min ΠA X i pi 1 2 ρ(A)i − ΠA h ρ(A)i i 1 · ||ρ(B)i ||1,
so unravellingρ in pure states, non necessarily orthogonal, like in (2.30) we get TDD(ρ(sep))≤ min ΠA X i pi 1 2 |ψiihψi| − ΠA[|ψiihψi|] 1. (2.31)
It can be seen that for a pure-QC state, like the one in Equation (2.27), that have orthogonal |iiB such inequality is actually an equality. Hence:
T DDρ(sep)|ψ iiA|φiiB ≤ T DDρ(pQC)|ψ iiA|iiB . (2.32)
Using TDD invariance under local unitary operations, we can now find a state with greater Trace Distance Discord than a givenρAB ∈ pure-QC:
ρUA = 1 2 ρAB⊗ |0iCh0| + UA[ρAB]⊗ |1iCh1| ; UA[ρ]≡ UAρUA†. (2.33)
In (2.33), orthogonal states in C actually label orthogonal subspaces of B, which can always be found if dB = ∞. The correlations on ρUA are bigger
because: TDD(ρUA) = 1 2minΠA 1 2 ρAB− ΠA[ρAB] 1+ 1 2 UA[ρAB]− ΠAUA[ρAB] 1 ≥ 1 2 min ΠA 1 2 ρAB− ΠA[ρAB] 1+ minΠA 1 2 UA[ρAB]− ΠAUA[ρAB] 1 = 1 2TDD(ρAB) + 1 2TDD (UA[ρAB]) = TDD(ρAB), (2.34)
where it can be noted that equality holds if the minimum of the sum is reached for the same measurement that minimize each term separately. Considering a pure-QC state, we can think at this unitary transformation as a rotation of the Bloch vector describing each |ψiiA with a matrix R ∈ SO(3). In particular,
N of these rotations generated by the unitaries Uk can be considered, such
interested in the pure-QC state with such pure vectors inA: ρ(pQC)N =X k Uk h |0iAh0| i ⊗ |kiBhk|. (2.35)
In the limit N → ∞, ρ(pQC)N tends to a uniform state ρ∞ without privileged
direction. The minimization defining the TDD is therefore trivial, because the basis are all equivalent to one another.
Analogously to what done forρUA in (2.34), where a single rotation was
intro-duced, we can at this point consider a state in which all the rotations Uk are
involved, obtaining a more correlated state than the initial ρ∈ pure-QC:
TDD N X k 1 NUk[ρ]⊗ |kiChk| ! ≥ N1 X k TDD(ρ) = TDD(ρ). (2.36)
On the other hand such a state can be arranged differently:
N X k 1 NUk(ρ)⊗ |kiChk| = X i pi X k 1 NUk h |ψiiAhψi| i ⊗ |kiChk| ! ⊗ |iiBhi|,
where now the state between brackets is just the a rotation of ρ(pQC)N which can be labelled as ρ(pQC)N,i . When N → ∞ each of that states is minimized by every projective measurement, because all of them haveρ∞ as limit. For this
reason: TDD X i piρ(pQC)N,i ⊗ |iihi| ! →X i piTDD(ρ∞) = TDD(ρ∞).
With this statement, and using Equation (2.36), it can be seen that the TDD superior extreme on pure-QC states, and then on separable ones, is reached on the asymptotic uniform state ρ∞.
In Appendix A is shown that for a pure-QC state (2.27) the TDD can be obtained as T DDρ(pQC)= 1 2minRˆ X i pi q 1− (ˆn(i)· ˆR)2, (2.37)
where nˆ(i) are the Bloch vectors of the pure states |ψ
iiAhψi| and ˆR is a unit
vector related to the projection basis chosen. This allows us to evaluate the TDD for the limit stateρ∞, measuringA for example with projectors described
ˆ R = ˆz (see AppendixA): TDD (ρ∞) = 1 2 1 4π Z dΩ q 1− (ˆr(Ω) · ˆz)2 = 1 4 Z π 0 dθ sin2θ = π 8. (2.38) • dB = 3: generalized B-92 state
The TDD of (2.28) can be computed using the general formula for a pure-QC state (2.37): TDD(ρGB92) = 1 6minRˆ p 1− R2 z+p1 − R2x+ q 1− R2 y = 1 6x2min+y2≤1 p x2+ y2+p1− x2+p1− y2= 1 3, where we used the following inequality, saturated for x = y = 0:
p
x2+ y2+p1− x2+p1− y2 ≥ 2 ⇐⇒p1− x2+p1− y2≥ 2 −px2+ y2
⇐⇒p(1 − x2)(1− y2)≥ −2px2+ y22−px2+ y2,
which holds because the right hand side is negative in the domain of minimiza-tion.
2.2
Non-geometric measures
These discord-like measures can be divided into three groups. The idea substanding the first group of quantifiers is to measure the change induced on the global state by a local unitary map, while the second set considers the entanglement created in a measurement process. Finally, the third one is composed by measures with a direct operational interpretation as figures of merit in a quantum metrology task.
2.2.1 Global action of local unitary maps
In the previous section we have seen how consider the disturbance induced in a state by a local measurement (see (2.4)) can lead to a well defined measure of non-classicality, often equivalent to a geometric approach (2.1). This suggests the idea that the non-classical character of a state can be linked to its susceptibility with respect to local perturbations, non necessarily in the form of measurements. In the quantifiers discussed in this section, the basic idea is to evaluate the global change which is induced on the state by a local unitary operation. Indeed, a non-CQ state is necessarily modified by all local unitary transformations, acting on the
