Abstract
Entanglement and nonlocal correlations are fundamental resources in quantum information.
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Notes
- 1.
Concurrence is defined as follows: Given the density matrix describing a pair of qubits, denoted \(\rho _{AB}\), one takes the so-called spin-flipped density matrix \(\tilde{\rho }_{AB}:=(\sigma _y \otimes \sigma _y)\rho _{AB}^*(\sigma _y \otimes \sigma _y)\), where \({}^*\) denotes complex conjugation and \(\sigma _y\) is the Pauli matrix. Both \(\rho _{AB}\) and its spin-flipped counterpart are positive operators and its product \(\rho _{AB}\tilde{\rho _{AB}}\) has real and non-negative eigenvalues (although it is non-Hermitian). By denoting the square root of them \(\lambda _1, \ldots , \lambda _4\), sorted in decreasing order, the concurrence of \(\rho _{AB}\) is defined as \(C_{AB}:= \max \{\lambda _1-\lambda _2-\lambda _3-\lambda _4,0\}\).
- 2.
This is by normalization in the case \(d=2\). In the general case, one further needs to take into consideration that there may be a fixed offset k between the outcomes of X and C.
- 3.
Note that the observables of \(A^{(2)}\) have already been relabelled according to the rule \(\alpha _{1}\mapsto \alpha _{1}+\alpha _{2}-1\).
- 4.
In order to see that explicitly, it is convenient to rename some of the indices. We can assume, without loss of generality, that \(k=1\) and that we rename \(A\leftrightarrow A^{(1)}\), \(B \leftrightarrow A^{(2)}\) and \(C \leftrightarrow A^{(3)}\) in Eq. (6.9). In addition, we set \(\alpha = \alpha _1\) for Alice and \(\alpha = \alpha _1+\alpha _2-1\) for Bob. One has to take into account that those observables \(A^{(2)}_{\alpha _1+\alpha _2-1}\) for which \(\alpha _1+\alpha _2-1\ge m\) have to be handled with especial care: Actually, one needs to apply the rule that, for all \(\gamma \) and i, \(X_{i\cdot m + \gamma }=[X_\gamma + i]\) so that the terms \([A^{(2)}_\gamma + i]\) that appear (for some \(\gamma \) and i) can be replaced by another variable, which we call \(\tilde{A}^{(2)}_{\gamma }\). Observe that the latter variable is the former with the outcomes shifted by a constant amount. The case \(k=2\) follows from Eq. (6.11). This proves the equivalence between Eqs. (6.16) and (6.2).
- 5.
The other case follows from exchanging B and C.
- 6.
In this section it is convenient to distrust that external observer, so we call it Eve, which is a shortening of a placeholder name like Alice or Bob, which here stands for Eavesdropper.
- 7.
The fact that we consider Eve to be a supra-quantum eavesdropper makes the proof even stronger, as \(\mathbf {Q}\) \(\subsetneq \) \(\mathbf {P}_{NS}\). Hence, even in the extreme scenario that QT were incomplete, the proof would hold as long as one could not transmit information instantaneously. Note that, although we make the eavesdropper stronger, this allows us to have simpler the proofs, as \(\mathbf {P}_{NS}\) is easier to characterize than \(\mathbf {Q}\), a fact that is reflected in the monogamy relations (compare (6.35) with (6.37)).
- 8.
This claim depends on the interpretation that we take of QT. It is possible to have a deterministic QT, at the expense of dropping the NS principle (often referred to as Bohmian mechanics [Boh52]), but this theory is in direct contradiction with Einstein’s Relativity theory, as it allows for instantaneous signalling.
- 9.
Measurement independence (the fact that \(p(\mathbf {x}|\lambda ) = p(\mathbf {x})\), where \(\mathbf {x}\) are the settings that are chosen in the Bell experiment when the state of the system (the hidden variable) is \(\lambda \)) is needed in a Bell test. Absence of it is often referred to as denial of free will, as the measurement settings could, in principle, be chosen by human beings (although in practice one uses a computer to choose randomly a large number of measurement choices) whose free will would be maimed.
- 10.
In particular, Eve can give them local correlations from time to time, as generally the NS bound is higher than the QT bound; otherwise, an observation of a supra-quantum violation would look very suspicious to the parties in \(\mathbf {A}\).
- 11.
The reason for this assumption is that Eve does not want Alice to notice her action from \(p(\mathbf {x})\).
- 12.
We need this NS violation to be maximal; otherwise Eve could mix such correlations with local ones, for which she has full knowledge.
- 13.
The problem of Randomness Expansion is typically stated as follows: Given a finite sequence of perfectly random bits, one has to generate longer sequence of perfectly random bits. In our case, the parameter that we increase is the dimension of the dits that we are using: from 2 to d. We start, however, with imperfect bits, as they are given by a SV source. As we do not use an infinite amount of \(\varepsilon \)-free bits, we can achieve at the same time Randomness Amplification and Randomness Expansion.
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Tura i Brugués, J. (2017). Atomic Monogamies of Correlations. In: Characterizing Entanglement and Quantum Correlations Constrained by Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-49571-2_6
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