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$2M$ Majorana zero modes define $M$ complex fermionic modes. Fixing total fermion parity leaves a $2^{M-1}$-dimensional ground space, encoding $M-1$ qubits.

The Kitaev chain is a one-dimensional spinless p-wave superconductor. Its topological phase has a Majorana zero mode at each end of an open chain.

When Majorana zero modes have disjoint exponentially localized supports, every local parity-even observable acts as a scalar on the fixed-parity ground space. Finite overlap changes this by terms exponentially small in separation divided by localization length.

A Majorana fermion operator is a self-adjoint operator in a fermionic algebra satisfying $\{{\gamma }_i,{\gamma }_j\}=2{\delta }_{ij}I$. A single complex fermionic mode $c$ can be written using two such operators:The canonical anticommutation relations verify their Hermiticity and normalization, and the occupation is $c^{†}c=(I+i{\gamma }_1{\gamma }_2)/2$.

This algebraic rewriting is always possible for a complex mode. A spatially separated Majorana zero mode requires additional physical properties, including localization and near-zero commutator with the Hamiltonian. The existence of two Majorana operators in a formula alone therefore does not establish protected boundary states or a topological phase.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

Two counterpropagating Majorana fields coupled by a real mass $m(x)$ bind a Majorana zero mode when $m$ changes sign. Its envelope is proportional to $\exp [-{\int }^{x}m(s)ds/(\hslash v)]$ up to orientation conventions, so a constant asymptotic mass gives localization length $\hslash v/|m|$.

A Majorana zero mode is a localized Hermitian quasiparticle operator ${\gamma }_j={\gamma }_j^{†}$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relationsAn exchange is adiabatic when its duration is long compared with $\hslash$ divided by the bulk excitation gap, so the adiabatic theorem keeps the state in the ground-state manifold. It should also be short compared with the inverse exponentially small zero-mode splitting if that splitting is not neglected.

Locality and conservation of fermion parity imply that exchanging modes $a$ and $b$ mapswith all other well-separated Majoranas unchanged. Since $({\gamma }_a{\gamma }_b{)}^2=-1$, exponentiation givesDirect conjugation using the Clifford relations produces exactly the displayed quarter-turn of $({\gamma }_a,{\gamma }_b)$. Thus, up to an overall physically irrelevant phase,The two signs are inverse Majorana braiding operators and correspond to the two braid orientations, clockwise and counterclockwise.

Each of the $M$ vortices contributes one bulk Majorana zero mode. A finite fermionic system must have an even total number of Majorana zero modes. The boundary Chiral Majorana edge mode has no zero momentum in the antiperiodic sector but has one $k=0$ Majorana mode in the periodic sector. Thereforeor compactly $c(x+L)=(-1{)}^{M+1}c(x)$. This boundary condition of a chiral Majorana edge mode supplies the extra boundary zero mode exactly when the vortex count is odd.

The Bogoliubov--de Gennes Hamiltonian has particle-hole symmetry. A locally nondegenerate zero mode can therefore be chosen particle-hole invariant. If its Nambu wavefunction is $(u_j,u_j^{\ast })$, defineThen ${\gamma }_j^{†}={\gamma }_j$. Exponential localization and $\ell /\xi \to \infty$ make distinct zero-mode wavefunctions orthogonal. Normalizing each one and using the fermionic canonical anticommutation relations gives

Pair the $2M$ Majorana zero modes into ordinary zero-energy fermionsTheir occupations produce $2^M$ states. Physical operations preserve total fermion parity, so choosing one parity sector imposes one binary constraint and leaves $2^{M-1}$ states. Thus Dense encoding with Majorana zero modes gives

A Majorana zero mode is a normalized self-adjoint fermion operator localized near a boundary and commuting with the Hamiltonian. With open boundaries,Neither $c_1$ nor $c_{2M}$ appears. Each anticommutes with both factors in every displayed bilinear and consequently commutes with their product. Thusand $c_1,c_{2M}$ are Majorana zero modes localized exactly at the two end sites.

Let $U$ be the quasi-local unitary carrying the ground space of $H_0$ to that of a topologically equivalent $H$. DefineUnitary conjugation preserves self-adjointness and the Majorana algebra. Quasi-locality spreads each endpoint operator only into an exponentially decaying tail, and the two operators preserve the ground space because $c_1,c_{2M}$ preserve the ground space of $H_0$.

Diagonalize the quadratic Hamiltonian asThe hint gives a real expansion $\gamma ={\sum }_j{g}_j{\chi }_j$. Any coefficient along a pair with ${ϵ}_j>0$ would create or remove a positive-energy quasiparticle and send some ground state outside the two-lowest-state subspace. Ground-space preservation therefore forces ${\gamma }_1$ and ${\gamma }_{2M}$ to have support only in the zero-energy Majorana subspace. Hencein the ideal infinite-chain limit, with only exponentially small finite-size corrections. They are the exponentially localized endpoint Majorana zero modes throughout the topological phase.

Use $c_j=({\gamma }_j^A+i{\gamma }_j^B)/2$ and $c_j^{†}=({\gamma }_j^A-i{\gamma }_j^B)/2$. Moving operators from different modes past one another introduces a minus sign. Direct expansion givesThus, for real hopping and pairing parameters,At $\Delta =t\ne 0$ this becomesUse $\tau$ for time to distinguish it from the hopping amplitude $t$. Each unused Majorana operator anticommutes with both factors in $H$, so commuting it through their product gives two minus signs. Hence $[H,{\gamma }_1^A]=[H,{\gamma }_2^B]=0$. The Heisenberg picture equation $dO/d\tau =(i/\hslash )[H,O]$ therefore givesThese are the two uncoupled Majorana zero modes.