1. IntroductionMajorana fermion (MF) refers to a kind of exotic particle that is identical to its own antiparticle. MF has been proposed[1] in particle physics as a possible elementary building block. In the past few years, due to possibly being realized in condensed matter systems using quasiparticle excitations[2] and the potential applications in topological quantum computations,[3,4] MFs have been widely investigated from both theoretical and experimental sides. These quasiparticles have been proposed to exist in different solid-state systems, e.g., the 5/2 fractional quantum Hall systems,[5] the chiral p-wave superconductors,[6–9] and the surface state of topological insulators with superconducting proximity effect.[10] Experimentally,[11–14] the end states of a semiconductor nanowire[15,16] with strong spin–orbit coupling[17] are demonstrated to be a pair of MFs under proper experimental conditions, i.e., the nanowire has been put on the top of an s-wave superconductor and subject to a proper Zeeman field. As a quasiparticle, MF in solid-state systems is a superposition of particle and hole states. The MF can not emerge separately in solid-state systems, a proper combination of MF-pair can form a nonlocal electronic level. This unique property makes the non-locality to be an intrinsic feature of MFs even if the two MFs are separated far away from each other and the quantum entanglement[18,19] between these two MFs is vanishing.
Several approaches[20–25] have been proposed to detect the quantum non-locality of MFs.[26–31] Wang et al. proposed[20] an experiment to detect the MFs by measuring the quantum entanglement between two QDs. However, it is difficult to discriminate the MFs and the other intermediaries, i.e., the topological trivial superconductors and regular fermions, in their proposal. In Ref. [21], Wang et al. studied the spectral density of the cross-correlation of two QDs mediated by a pair of MFs. They found that the spectral density mediated by MFs is antisymmetric with respect to the simultaneous modulation of the QD energy levels, while the spectrum is symmetric if the current is mediated by the regular fermion. Li et al. studied[22] the quantum correlation between two QDs mediated by a pair of MFs. As an evidence of the non-locality of MFs, they have shown that, being different from the quantum entanglement, the quantum discord is not vanishing when the MFs are completely separated from each other.
Recently, quantum coherence has been proposed as a physical resource to measure the quantum correlation.[32] Napoli et al. proposed[33] an operational and observable measure of the quantum coherence, the robustness of coherence (ROC), under the general abstract notion of robustness of asymmetry.[34] The quantum coherence as a resource to identify the non-locality of MFs has not been investigated rigorously as far as we know. In this paper, we study the ROC between two QDs mediated by a pair of MFs.
2. Model HamiltonianThe model Hamiltonian is set up as follows (see Fig. 1 for schematic setup). A quantum wire with strong spin–orbit coupling is put on the top of an s-wave superconductor, the proximity effect drives the quantum wire into a superconducting phase. In the regime of topological superconductor phase, a pair of zero-energy MFs are associated to the two ends of the quantum wire. The model Hamiltonian used to describe the MFs is given by
where
εMF∼e
−L/ξ and
L is the length of the quantum wire,
ξ is the superconducting coherent length.
γ1 and
γ2 are the two Majorana operators which satisfy the anticommutation relation {
γi,
γj} =
δij and the Majorana condition
(
i,
j = 1,2). Two QDs are put close to the ends of the quantum wire, i.e., QD
1 and QD
2. We use the following Hamiltonian to describe the QDs:
where
εd1 and
εd2 are the on-site energies of the two QDs, and
and
dj are the creation and annihilation operators on QD
j. The tunneling coupling between the QDs and MFs can be written as
[9,15,21,35]
where
λj is the tunneling coupling constant between QD
j and MFs. The total Hamiltonian
Under the following transformation:
we find that the total Hamiltonian can be rewritten as
where
nf =
f†f and
are the electron occupation number operators of the paired MFs and the on-site states at QD
j, and h.c. refers to the Hermitian conjugate. On the electron occupation number basis, the model Hamiltonian can be written as an 8 × 8 matrix.
In order to find the distinguishable features of MFs, here we consider two other setups for comparison. The first case is QDs mediated by regular fermions as shown in Ref. [22], where the Hamiltonian is written as
for one and two non-interacting regular fermions between the QDs, respectively. Here
εc and
εcj are the energies of the regular fermions,
c† (
c) and
(
cj) are the creation (annihilation) operators of the regular fermions,
λj (
j = 1,2) is the tunneling coupling constant between QD
j and regular fermions. The second case is QDs mediated by a pair of superconducting electrons, the Hamiltonian can be written as
The time evaluation of the density matrix for these model Hamiltonians (
6)–(
9) can be solved analytically for given initial states when the decoherence effect is not considered.
Moreover, in order to investigate the decoherence effect on quantum correlations, we assume that the two electrode leads are weakly coupled to the two QDs. This coupling is described by
where
tj is the coupling strength between the
j-th electrode lead and the corresponding QD,
ckj and
.
are annihilation and creation operators of a quantum state with wave vector
k on the
j-th electrode lead. The two electrode leads are described as
, where
εkj is the energy level of the quantum state with wave vector
k on the
j-th electrode lead. The chemical potential
μ of the electrode leads is assumed to be much lower than the QD on-site energy,
μ ≪
ε1,
ε2, such that the electrons can only jump outwards from the QDs to the leads. The leads are regarded as an environment and the quantum state in the central regime is described by a reduced density matrix
ρ(
t). The time evaluation of this reduced density matrix can be described by the standard Born–Markov master equation
[36] in the weak coupling limit
where
Htot is the model Hamiltonian of the whole system as shown in Eqs. (
6)–(
9)for different situations.
.
is the dissipator of the quantum master equation at QD
j,
Γj is the corresponding tunneling rate, it is associated with both the coupling strength
tj and the density of states on the
j-th lead. For simplicity, we assume
Γ1 =
Γ2 =
Γ. Generally, one can express this reduced density matrix
ρ(
t) as a 8 × 8 or 16 × 16 matrix for different situations in the electron occupation number representation. By tracing over the degrees of freedom of the intermediate MFs, regular fermions, or superconducting states in
ρ(
t), we can obtain the further reduced 4 × 4 density matrix of the two QDs,
ρd(
t), and study the quantum correlation between these QDs.
In this work, we focus on the ROC as a resource to measure the quantum correlation between the two QDs mediated by the MFs. Here we give a brief introduction to ROC. The ROC of a state ρ is given by the minimum weight of another state τ such that its convex mixture with ρ yields an incoherent state δ.[33] More specifically, let
be the convex set of density operators acting on a d-dimensional Hilbert space,
be the subset of incoherent states in
. The ROC of a state
is given by
where
τ is another state in
and
δ is an incoherent state. Numerically, it is proved that the ROC can be recast via the semidefinite program as
[37]
for any given witness operator
W. The witness operator
W is defined as the Hermitian operator which satisfies
Π(
W) ≥ 0 if and only if Tr(
δW) = Tr(
δΠ(
W)) ≥ 0 for all incoherent states
. Here
Π is the full dephasing operator, which maps the witness operator
W into its diagonal part
in a computational reference basis
. The MATLAB
[38] codes used to evaluate the semidefinite program of ROC can be found in the supplemental materials of Refs. [
33] and [
34]. The open-source modeling system for convex optimization, CVX,
[39] is required in running the MATLAB codes.
Two other measurements, the concurrence[40,41] and the quantum discord,[42,43] which are used to identify the quantum correlations between the two QDs, are calculated as well, they are widely studied in different systems in quantum information field. For the sake of completeness, here we give a brief introduction to the definitions of these measurements. We refer to Ref. [22] and references therein for the detailed discussion of these quantities for the model Hamiltonian investigated in this work. The concurrence, as a measure of the quantum entanglement, can be obtained as follows for the two-qubit states:
where
Λ1–
Λ4 are the four eigenvalues of the following matrix in decreasing order:
where
ρ* is the complex conjugate of the density matrix
ρ.
The definition of quantum discord is accomplished by using two other important quantities in quantum information theory, the quantum mutual information and the classical correlation. The quantum mutual information
is a quantum substitution of the mutual information in classical information theory,
, where
(
) is the reduced density matrix for the subsystem
(
) of the composite bipartite system
, and ρ refers to the total density matrix of the
system. S(ρ) = −tr(ρ log2ρ) is the von Neumann entropy. The other quantity, classical correlation
, is defined by using the von Neumann-type measurements,
. The inferior limit is taken over different sets of orthogonal projection operators,
, acting on the subsystem
only. For a given set of orthogonal projection operators
, one can calculate the conditional density matrix ρj associated with the measurement result j, and the corresponding probability pj to obtain this measurement result j. Their expressions are given by
, where
is the identity matrix acting on the subsystem
,
. Finally, the quantum discord is the difference of these two quantities
 | |
3. ResultsFirstly, we consider the quantum correlations for the model Hamiltonian without decoherence, Eq. (6). In the electron occupation number representation |nd1,nd2,nf⟩, equation (6) can be reexpressed as a 8 × 8 matrix, it is easy to check that the parity of the electron number is conserved, which makes the model Hamiltonian be expressed as
, where Hu and Hg are both 4 × 4 matrices in the parity-even and parity-odd subspaces of the Hilbert space, respectively. Here we consider the product state |ψ(0)⟩ = |1,1,0⟩ in the parity-even subspace as an initial state, the corresponding Hamiltonian Hu can be written as
on the bases |0,0,0⟩, |0,1,1⟩, |1,0,1⟩, and |1,1,0⟩. The time evaluation of the initial state is solved to be
where the time-dependent coefficients are given by
with
ω =
εM/2 and
. By tracing over the MF states in the density matrix
ρ(
t) = |
ψ(
t)⟩⟨
ψ(
t)|, one can obtain the reduced density matrix
ρd(
t). Expressed in the electron occupation number representation of the QDs, |
nd1,
nd2⟩ = |0,0⟩, |0,1⟩, |1,0⟩, and |1,1⟩,
ρd(
t) has a X-type shape. All of the 8 non-zero elements in
ρd(
t) can be obtained by the joint measurement experiment proposed in Ref. [
22].
Figures 2(a) and 2(b) show the time evaluation of the quantum correlations calculated from ρd(t). When the direct coupling between the two MFs, εM, is not vanishing, there are two frequencies ω/2π and Ω/2π (see Fig. 2(a) for εM = 0.5λ). Both quantum entanglement (red solid line with the area bellow the line filled) and quantum discord (dashed blue line) have the similar tendency, i.e., the position of the fast oscillation peak of both quantum entanglement and quantum discord is located at t = 2Nπ/Ω, the position of the slow oscillation peak is located at t = 2Nπ/ω, both quantum entanglement and quantum discord achieve their maxima at these points, concurrence = discord = 1. Near the nodes of the slow oscillation, quantum entanglement and quantum discord are a bit different from each other. However, when we consider the quantum coherence (the solid black line), it is quite different from the quantum entanglement and quantum discord. Firstly, the ROC is greater than (at least equals to) the quantum entanglement in the whole time domain. Secondly, the maximum quantum coherence in each fast oscillation period can approach one. Thirdly, the location of the fast oscillation peak (valley) of the quantum coherence is associated to the oscillation valley (peak) of the quantum entanglement. Their tendencies are opposite to each other.
Figure 2(b) shows the quantum correlations of two QDs mediated by a pair of MFs without direct coupling between the two MFs, εM = 0. One can find that, similar to the quantum discord, the quantum coherence can also be used to identify the quantum correlation between two qubits without quantum entanglement. For comparison, we show the quantum correlations between two QDs mediated by a regular fermion, Figs. 2(c) and 2(d) for different parameters. Both figures 2(c) and 2(d) show that the quantum entanglement and quantum coherence are exact the same. Figure 2(e) shows that the quantum correlation between the two QDs is vanishing if these two QDs are mediated by a pair of regular fermions without interaction. Furthermore, figures 2(f) and 2(g) show the quantum correlations between two QDs mediated by a pair of normal superconducting fermions with a superconducting energy gap Δ = 5λ and different parameters given in the caption. The trembles on the lines are induced by the finite superconducting energy gap. We find that the quantum entanglement, quantum discord, and quantum coherence have some differences, they are not coincide to each other exactly, however, the general trends are the same. These results indicate that the differences between the quantum entanglement and quantum coherence can be considered as an important identity for MFs in the one-dimensional quantum wire system.
Now we consider the decoherence induced by the leads. By solving the Born–Markov master equation in Eq. (10), we can obtain the time-dependent density matrices. Figure 3 shows the quantum correlations between the two QDs mediated by a pair of MFs. In Fig. 3(a), the direct coupling between the two MFs, εM = 0.5λ. There are three different time scales, the fast oscillation, the slow oscillation, and the decoherence time induced by the environment. Similar to the case without decoherence, one can find that the ROC is great than the quantum entanglement in the whole time domain. More importantly, the oscillation peak of concurrence and quantum discord is smaller than the valley of ROC in each fast oscillation period, there is a gap between the ROC and the other two measurements, which demonstrates that the quantum coherence is more robust against decoherence from the environment. Figure 3(b) shows the quantum correlations between the two QDs mediated by a pair of MFs without direct coupling between the two MFs, εM = 0. One can find that, similar to the quantum discord, the ROC can also be used to identify the quantum correlations between two QDs without quantum entanglement, and numerically, the ROC is more significant than the quantum discord. The insert in Fig. 3(b) shows the details of quantum discord in the regime t ∈ [0,8] in units of 1/λ, where one can find that the maximum quantum discord is about 0.1.
