NitreneGlog

POVM -- Generalized Quantum Measurement

Thu, 24 Sep 2026 00:00:00 GMT

As a fundamental concept in quantum information, a positive operator-valued measure (POVM) extends the traditional projective description of quantum measurement. Its simple but beautiful mathematical structure provides a general language for describing measurement statistics and information extraction. In this article, we try to give a brief introduction to POVMs, and gain some insights into the modern measurement theory.

Measure#

A POVM is an operator-valued analogue of an ordinary probability measure. A positive Measure μ\muμ is a map defined on a measurable space

(Ω,Σ)\begin{aligned} (\Omega, \Sigma) \end{aligned}(Ω,Σ)

where Ω\OmegaΩ is the sample space and Σ\SigmaΣ is the σ\sigmaσ-algebra of measurable subsets of Ω\OmegaΩ.

μ:Σ→[0,+∞]\begin{aligned} \mu: \Sigma \to [0, +\infty] \end{aligned}μ:Σ→[0,+∞]

satisfying the following properties:

μ(∅)=0\mu(\emptyset) = 0μ(∅)=0
For any sequence of disjoint sets {An}n=1∞⊂Σ\{A_n\}_{n=1}^{\infty} \subset \Sigma{An}n=1∞⊂Σ, we have μ(⋃n=1∞An)=∑n=1∞μ(An)\mu\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mu(A_n)μ(n=1⋃∞An)=n=1∑∞μ(An)

A POVM generalizes an ordinary measure by replacing nonnegative real values with positive semidefinite operators.

E:Σ→B(H)+\begin{aligned} E : \Sigma \to \mathcal{B}(\mathcal{H})_+ \end{aligned}E:Σ→B(H)+

where B(H)+\mathcal{B}(\mathcal{H})_+B(H)+ is the set of positive semi-definite bounded operators, {Em}\{E_m\}{Em} is the set of positive semi-definite operators on a Hilbert space H\mathcal{H}H, satisfying the following properties:

E(⋃mAm)=∑mE(Am)E(\bigcup_{m} A_m) = \sum_{m} E(A_m)E(⋃mAm)=∑mE(Am) for any disjoint sets {Am}m=1∞⊂Σ\{A_m\}_{m=1}^{\infty} \subset \Sigma{Am}m=1∞⊂Σ
E(Ω)=IE(\Omega) = IE(Ω)=I

Given a quantum state ρ\rhoρ, the POVM induces an ordinary probability measure, specifies the statistics of the measurement outcomes

p(m)=Tr(Emρ)p(m) = Tr(E_m \rho)p(m)=Tr(Emρ)

where ρ\rhoρ is the density matrix of any quantum state in H\mathcal{H}H

Measurement#

With the above definition, a POVM may initially look like nothing more than a set of operators used to calculate outcome probabilities. But traditional quantum mechanics already describes measurement using Hermitian observables and projectors. What, then, does a POVM add? Now let’s talk about the Measurement to see the difference.

Classical to Quantum Measurement#

Built on classical mechanics and probability theory, one of the main topics of classical measurement is “determinism”. In practice, a system is described by a physical theory in these 3 aspects

State (what is it)
Dynamics (how it evolves)
Measurement (how to observe)

Classical Mechanics tells us that Newton’s laws give the deterministic path. Measurement is just revealing pre-existing properties. An ideal measurement reads out a property without necessarily changing it.

Quantum Mechanics does not only change the way we describe the state and dynamics, but also add a new bond to the measurement. The outcome of a measurement is an intrinsically probabilistic event. A quantum measurement can also change the state, and the state update depends on how the measurement is physically implemented.

The observable can be described as the sum of projectors PPP with the eigenvalue omo_mom (spectral decomposition):

O^=∑momPm\begin{aligned} \hat{O} = \sum_{m} o_m P_m \end{aligned}O^=m∑omPm

for a state expanded in the eigenbasis ϕ=∑mcmψm\phi = \sum_{m} c_m \psi_mϕ=∑mcmψm, where

O^ψm=omψm\begin{aligned} \hat{O} \psi_m = o_m \psi_m \end{aligned}O^ψm=omψm

Suppose we perform a non-degenerate projective measurement and obtain the outcome omo_mom. Conditioned on this outcome, the state is updated to the corresponding eigenstate ∣ψm⟩\ket{\psi_m}∣ψm⟩. This is the usual measurement “collapse”.

It is useful to distinguish this conditioned state update from decoherence. If the measurement is performed but its outcome is ignored, the state is instead described by

ρ⟶∑iPiρPi,\rho \longrightarrow \sum_i P_i \rho P_i,ρ⟶i∑PiρPi,

which removes the coherence between different measurement eigenspaces. The decoherence changes the way we detect the state and decode the Quantum furnished information.

General Quantum Measurement#

A projective measurement is described by mutually orthogonal projectors. A POVM relaxes this requirement: its effects only need to be positive and complete, and they need not be orthogonal or idempotent.

It is allowed that

non-orthogonal
EmEn≠0,m≠n\begin{aligned} E_m E_n \neq 0, \quad m \neq n \end{aligned}EmEn=0,m=n
non-projective
Em2≠Em\begin{aligned} E_m^2 \neq E_m \end{aligned}Em2=Em

but it is still

Complete ∑mEm=I\begin{aligned} \sum_{m} E_m = I \end{aligned}m∑Em=I

for example we have an inefficient detector, it cannot always detect the “occurrence” ∣1⟩\ket1∣1⟩ when it really happens.

E1=η∣1⟩⟨1∣E0=∣0⟩⟨0∣+(1−η)∣1⟩⟨1∣\begin{aligned} E_1 &= \eta |1\rangle\langle 1| \\ E_0 &= |0\rangle\langle 0| + (1-\eta) |1\rangle\langle 1| \\ \end{aligned}E1E0=η∣1⟩⟨1∣=∣0⟩⟨0∣+(1−η)∣1⟩⟨1∣

So we can construct any number of POVMs to describe the measurement, not limited by the dimension of the Hilbert space. A POVM on a ddd-dimensional system may have more than ddd outcomes.

Detection#

Having defined POVMs mathematically, a natural question remains: can every valid POVM be physically realized? Naimark’s dilation theorem answers this question affirmatively. In fact, any POVMs can be realized by a projective measurement with an auxiliary system. Naimark’s Dilation Theorem states the following:

For every POVM {Em}\{E_m\}{Em} on a Hilbert space HS\mathcal{H}_SHS, satisfying

∑mEm=IsEm≥0\begin{aligned} \sum_{m} E_m = I_s \\ E_m \geq 0 \end{aligned}m∑Em=IsEm≥0

there exists an ancilla HA\mathcal{H}_AHA, an initial ancilla state ∣0⟩A\ket{0}_A∣0⟩A, a unitary operator UUU on HS⊗HA\mathcal{H}_S \otimes \mathcal{H}_AHS⊗HA, and a set of projectors {Πm}\{\Pi_m\}{Πm}

such that

Pr(m∣ρ)=Tr(Emρ)=Tr(ΠmU(ρ⊗∣0⟩A⟨0∣)U†)\begin{aligned} Pr(m|\rho) = Tr(E_m \rho) = Tr(\Pi_m U (\rho \otimes |0\rangle_A \langle 0|) U^\dagger) \end{aligned}Pr(m∣ρ)=Tr(Emρ)=Tr(ΠmU(ρ⊗∣0⟩A⟨0∣)U†)

This looks like:

we have our Em∈Hom(HS)E_m \in Hom(\mathcal{H}_S)Em∈Hom(HS)
the embedding operator J:HS→HS⊗HAJ: \mathcal{H}_S \to \mathcal{H}_S \otimes \mathcal{H}_AJ:HS→HS⊗HA appends an ancilla in the state ∣0⟩A|0\rangle_A∣0⟩A.
We take an unitary U∈Hom(HS⊗HA)U \in Hom(\mathcal{H}_S \otimes \mathcal{H}_A)U∈Hom(HS⊗HA) to transform the state. So V=UJV = UJV=UJ is an isometry, and we have Em=V†ΠmVE_m = V^\dagger \Pi_m VEm=V†ΠmV.
We perform a projective measurement on the larger tensor-product space, and get the same outcome statistics.

Every POVM can be regarded as a projective measurement after embedding the system into a larger Hilbert space.

Example: the trine POVM#

There are 3 nonzero effects E0,E1,E2E_0, E_1, E_2E0,E1,E2 given by

Ek=13(I+nk⋅σ)=23∣ψk⟩⟨ψk∣,k=0,1,2\begin{aligned} E_k = \frac{1}{3} (I + \mathbf{n}_k \cdot \boldsymbol{\sigma})= \frac{2}{3} |\psi_k\rangle\langle\psi_k|, \quad k = 0, 1, 2 \end{aligned}Ek=31(I+nk⋅σ)=32∣ψk⟩⟨ψk∣,k=0,1,2

and the quantum states are given by

∣ψk⟩=12(∣0⟩+eiθk∣1⟩),θk=2π3k\begin{aligned} |\psi_k\rangle = \frac{1}{\sqrt{2}} (|0\rangle + e^{i \theta_k} |1\rangle), \quad \theta_k = \frac{2\pi}{3} k \end{aligned}∣ψk⟩=21(∣0⟩+eiθk∣1⟩),θk=32πk

which are not orthogonal to each other. With an input state ∣ψ⟩\ket{\psi}∣ψ⟩, we have the probability of the outcome kkk as

Pr(k∣ψ)=Tr(Ek∣ψ⟩⟨ψ∣)=23∣⟨ψk∣ψ⟩∣2\begin{aligned} Pr(k|\psi) = Tr(E_k |\psi\rangle\langle\psi|) = \frac{2}{3} |\langle\psi_k|\psi\rangle|^2 \end{aligned}Pr(k∣ψ)=Tr(Ek∣ψ⟩⟨ψ∣)=32∣⟨ψk∣ψ⟩∣2

If outcome kkk is interpreted as the guess that the input was ∣ψk⟩|\psi_k\rangle∣ψk⟩, the measurement does not identify the state perfectly because the trine states are nonorthogonal.

Naimark’s dilation works when we take an ancilla HA=span{∣0⟩A,∣1⟩A,∣2⟩A}\mathcal{H}_A = \text{span}\{|0\rangle_A, |1\rangle_A, |2\rangle_A\}HA=span{∣0⟩A,∣1⟩A,∣2⟩A} where A⟨j∣k⟩A=δjk{}_A\langle j|k\rangle_A=\delta_{jk}A⟨j∣k⟩A=δjk

we choose the measurement operators (not unique) to build the isometry

Mk=Ek=23∣ψk⟩⟨ψk∣\begin{aligned} M_k = \sqrt{E_k} = \sqrt{\frac{2}{3}} |\psi_k\rangle\langle\psi_k| \end{aligned}Mk=Ek=32∣ψk⟩⟨ψk∣

and the isometry V:HS→HS⊗HAV: \mathcal{H}_S \to \mathcal{H}_S \otimes \mathcal{H}_AV:HS→HS⊗HA is given by

V∣ψ⟩S=∑k=02Mk∣ψ⟩S⊗∣k⟩A=23∑k=02∣ψk⟩⟨ψk∣ψ⟩S⊗∣k⟩A\begin{aligned} V \ket{\psi}_S &= \sum_{k=0}^{2} M_k \ket{\psi}_S \otimes |k\rangle_A \\ &= \sqrt{\frac{2}{3}} \sum_{k=0}^{2} |\psi_k\rangle\langle\psi_k|\psi\rangle_S \otimes |k\rangle_A \end{aligned}V∣ψ⟩S=k=0∑2Mk∣ψ⟩S⊗∣k⟩A=32k=0∑2∣ψk⟩⟨ψk∣ψ⟩S⊗∣k⟩A

easy to verify that

V†V=∑kMk†Mk=∑kEk=IS\begin{aligned} V^\dagger V &= \sum_k M_k^\dagger M_k \\ &= \sum_k E_k = I_S \end{aligned}V†V=k∑Mk†Mk=k∑Ek=IS

via the isometry, the projective measurement can be performed on the larger space

Πk=IS⊗∣k⟩⟨k∣AEk=V†ΠkV=Mk†Mk\begin{aligned} \Pi_k = I_S \otimes |k\rangle \langle k|_A \\ E_k = V^\dagger \Pi_k V = M_k^\dagger M_k \end{aligned}Πk=IS⊗∣k⟩⟨k∣AEk=V†ΠkV=Mk†Mk

With the orthogonal outcome branches, we can measure the ancilla projectively

p(k∣ψ)=∣∣(IS⊗⟨k∣A)V∣ψ⟩S∣∣2=⟨ψ∣Mk†Mk∣ψ⟩=⟨ψ∣Ek∣ψ⟩\begin{aligned} p(k|\psi) = ||(I_S \otimes \langle k|_A) V \ket{\psi}_S ||^2 = \bra{\psi} M_k^\dagger M_k \ket{\psi} = \bra{\psi} E_k \ket{\psi} \end{aligned}p(k∣ψ)=∣∣(IS⊗⟨k∣A)V∣ψ⟩S∣∣2=⟨ψ∣Mk†Mk∣ψ⟩=⟨ψ∣Ek∣ψ⟩

POVMs are the Natural language for optimal quantum measurement. If we have prepared one of the several possible states, it’s necessary to choose the best measurement to distinguish them.

Discrimination —I want/I don’t know?#

Uncertainty lies at the heart of quantum information.You prepare a quantum state to store information, but you don’t exactly know which one it is. You receive the state, and you perform a projective measurement through the Naimark’s dilation, but you may still be not sure if it is a “0” or “1”. Quantum Mechanically, two non-orthogonal states cannot be perfectly distinguished.

0<∣⟨ψ1∣ψ2⟩∣<1\begin{aligned} 0< |\langle\psi_1|\psi_2\rangle| < 1 \end{aligned}0<∣⟨ψ1∣ψ2⟩∣<1

We therefore face a choice. We may always make a guess and minimize the error probability, or we may forbid erroneous conclusive answers while allowing an inconclusive result.

Minimum Error Discrimination#

Always make a guess. — I want an answer, even if you are lying.

Binary minimum error discrimination#

Sample: S={p0,ρ0;p1,ρ1}S=\{p_0, \rho_0; p_1, \rho_1\}S={p0,ρ0;p1,ρ1}, with POVMs {E0,E1}\{E_0, E_1\}{E0,E1} the success/error probability

Psucc=p0Tr(E0ρ0)+p1Tr(E1ρ1)Perr=1−Psuccp0+p1=1,E0+E1=I\begin{aligned} P_{succ}&= p_0 Tr(E_0 \rho_0) + p_1 Tr(E_1 \rho_1) \\ P_{err}&= 1- P_{succ} \\ p_0 + p_1 &= 1, \quad E_0 + E_1 = I \end{aligned}PsuccPerrp0+p1=p0Tr(E0ρ0)+p1Tr(E1ρ1)=1−Psucc=1,E0+E1=I

the Helstrom operator Δ=p0ρ0−p1ρ1\Delta = p_0 \rho_0 - p_1 \rho_1Δ=p0ρ0−p1ρ1 then,

Psucc=p0Tr(E0ρ0)+p1Tr((I−E0)ρ1)=p1+Tr(E0Δ)\begin{aligned} P_{succ}&= p_0 \text{Tr}(E_0 \rho_0) + p_1 \text{Tr}((I-E_0) \rho_1) \\ &= p_1 + Tr(E_0 \Delta) \\ \end{aligned}Psucc=p0Tr(E0ρ0)+p1Tr((I−E0)ρ1)=p1+Tr(E0Δ)

what we need to do is to maximize Tr(E0Δ)Tr(E_0 \Delta)Tr(E0Δ) by choosing the optimal E0E_0E0. We can spectral decompose the Helstrom operator.

Δ=∑kλk∣k⟩⟨k∣\begin{aligned} \Delta = \sum_{k} \lambda_k | k\rangle\langle k| \end{aligned}Δ=k∑λk∣k⟩⟨k∣

take a direct sum decomposition of the eigenspaces of Δ\DeltaΔ into positive and negative subspaces

Δ=Δ+−Δ−=∑λk>0λk∣k⟩⟨k∣+∑λk<0λk∣k⟩⟨k∣∣Δ∣=Δ++Δ−whereΔ+=∑λk>0λk∣k⟩⟨k∣Δ−=−∑λk<0λk∣k⟩⟨k∣\begin{aligned} \Delta &= \Delta_+ - \Delta_- \\ &= \sum_{\lambda_k > 0} \lambda_k |k\rangle\langle k| + \sum_{\lambda_k < 0} \lambda_k |k\rangle\langle k| \\ |\Delta| &= \Delta_+ + \Delta_- \\ \text{where} \quad \Delta_+ &= \sum_{\lambda_k > 0} \lambda_k |k\rangle\langle k| \\ \Delta_- &= -\sum_{\lambda_k < 0} \lambda_k |k\rangle\langle k| \\ \end{aligned}Δ∣Δ∣whereΔ+Δ−=Δ+−Δ−=λk>0∑λk∣k⟩⟨k∣+λk<0∑λk∣k⟩⟨k∣=Δ++Δ−=λk>0∑λk∣k⟩⟨k∣=−λk<0∑λk∣k⟩⟨k∣

Since 0≤E0≤I0\leq E_0\leq I0≤E0≤I, the maximum is obtained by assigning eigenvalue one to the positive eigenspace of Δ\DeltaΔ and zero to its negative eigenspace.

E0=Π+(Δ)E1=I−E0\begin{aligned} E_0 &= \Pi_{+}(\Delta) \\ E_1 &= I - E_0 \end{aligned}E0E1=Π+(Δ)=I−E0

So we have the Helstrom measurement

max⁡0≤E0≤ITr(E0Δ)=Tr(Δ+)\begin{aligned} \max_{0\leq E_0 \leq I} Tr(E_0 \Delta) = Tr(\Delta_+) \end{aligned}0≤E0≤ImaxTr(E0Δ)=Tr(Δ+)

So the optimal success probability is

Psuccopt=p1+Tr(Δ+)=12(1+∣∣Δ∣∣1)=12(1+∑i∣λi∣)Perropt=1−Psuccopt=12(1−∣∣Δ∣∣1)=12(1−∑i∣λi∣)\begin{aligned} P_{succ}^{opt} = p_1 + Tr(\Delta_+) = \frac{1}{2} (1 + ||\Delta||_1) = \frac{1}{2} (1 + \sum_i |\lambda_i| ) \\ P_{err}^{opt} = 1 - P_{succ}^{opt} = \frac{1}{2} (1 - ||\Delta||_1) = \frac{1}{2} (1 - \sum_i |\lambda_i| ) \end{aligned}Psuccopt=p1+Tr(Δ+)=21(1+∣∣Δ∣∣1)=21(1+i∑∣λi∣)Perropt=1−Psuccopt=21(1−∣∣Δ∣∣1)=21(1−i∑∣λi∣)

Helstrom Bounds

for the equal prior p0=p1=12p_0 = p_1 = \frac{1}{2}p0=p1=21

Perropt=12(1−D(ρ0,ρ1))\begin{aligned} P_{err}^{opt} = \frac{1}{2} (1 - D(\rho_0, \rho_1)) \end{aligned}Perropt=21(1−D(ρ0,ρ1))

where DDD is the trace distance defined as

D(ρ,σ)=12∣∣ρ−σ∣∣1\begin{aligned} D(\rho, \sigma) = \frac{1}{2} ||\rho - \sigma||_1 \end{aligned}D(ρ,σ)=21∣∣ρ−σ∣∣1

Suppose we have 2 pure states ∣ψ0⟩\ket{\psi_0}∣ψ0⟩ and ∣ψ1⟩\ket{\psi_1}∣ψ1⟩, the error rate becomes

Perropt=12(1−1−∣⟨ψ0∣ψ1⟩∣2)\begin{aligned} P_{err}^{opt} = \frac{1}{2} (1 - \sqrt{1 - |\langle\psi_0|\psi_1\rangle|^2}) \end{aligned}Perropt=21(1−1−∣⟨ψ0∣ψ1⟩∣2)

For binary minimum-error discrimination, optimizing over all POVMs yields an optimal projective measurement (PVM): The Helstrom measurement.

Unambiguous discrimination#

Never make a wrong conclusive guess. — I don’t know.

E0+E1+E?=IE_0+E_1+E_?=IE0+E1+E?=I we allow an inconclusive outcome E?E_?E? to avoid the error. We are sure that if we get the outcome E0E_0E0 or E1E_1E1

Tr(E0ρ1)=Tr(E1ρ0)=0\begin{aligned} Tr(E_0 \rho_1) = Tr(E_1 \rho_0) = 0 \end{aligned}Tr(E0ρ1)=Tr(E1ρ0)=0

The success probability to be optimized is

Psucc=η0Tr(E0ρ0)+η1Tr(E1ρ1)\begin{aligned} P_{succ} = \eta_0 Tr(E_0 \rho_0) + \eta_1 Tr(E_1 \rho_1) \end{aligned}Psucc=η0Tr(E0ρ0)+η1Tr(E1ρ1)

The Unambiguous State Discrimination (USD) requires:

⟨ψ0∣E1∣ψ0⟩=0,⟨ψ1∣E0∣ψ1⟩=0\begin{aligned} \bra{\psi_0} E_1 \ket{\psi_0} = 0, \quad \bra{\psi_1} E_0 \ket{\psi_1} = 0 \end{aligned}⟨ψ0∣E1∣ψ0⟩=0,⟨ψ1∣E0∣ψ1⟩=0

Suppose η0=η1=12\eta_0 = \eta_1 = \frac{1}{2}η0=η1=21, for 2 pure states ∣ψ0⟩\ket{\psi_0}∣ψ0⟩ and ∣ψ1⟩\ket{\psi_1}∣ψ1⟩, which are not orthogonal to each other. we can choose the basis ∣0⟩,∣1⟩\ket{0}, \ket{1}∣0⟩,∣1⟩ such that

∣ψ0⟩=∣0⟩,∣ψ1⟩=c∣0⟩+s∣1⟩\begin{aligned} \ket{\psi_0} = \ket{0}, \quad \ket{\psi_1} = c\ket{0} + s \ket{1} \end{aligned}∣ψ0⟩=∣0⟩,∣ψ1⟩=c∣0⟩+s∣1⟩

choose their orthogonal complements

∣ψ0⊥⟩=∣1⟩,∣ψ1⊥⟩=s∣0⟩−c∣1⟩\begin{aligned} \ket{\psi_0^\perp} = \ket{1}, \quad \ket{\psi_1^\perp} = s\ket{0} - c\ket{1} \end{aligned}∣ψ0⊥⟩=∣1⟩,∣ψ1⊥⟩=s∣0⟩−c∣1⟩

then we can construct the POVMs as

E0=a∣ψ1⊥⟩⟨ψ1⊥∣E1=a∣ψ0⊥⟩⟨ψ0⊥∣E?=I−E0−E1=(1−as2ascasc1−a(1+c2))\begin{aligned} E_0 &= a \ket{\psi_1^\perp}\bra{\psi_1^\perp} \\ E_1 &= a \ket{\psi_0^\perp}\bra{\psi_0^\perp} \\ E_? &= I - E_0 - E_1 = \begin{pmatrix} 1 - as^2 & asc\\ asc & 1 - a(1 + c^2) \end{pmatrix} \end{aligned}E0E1E?=a∣ψ1⊥⟩⟨ψ1⊥∣=a∣ψ0⊥⟩⟨ψ0⊥∣=I−E0−E1=(1−as2ascasc1−a(1+c2))

For E?E_?E? to be positive semidefinite, its principal minors must be non-negative. Together with a≥0a\geq0a≥0, the limiting determinant condition gives

det⁡(E?)=(1−as2)(1−a(1+c2))−(asc)2≥0a≤11+c\begin{aligned} \det(E_?) &= (1 - as^2)(1 - a(1 + c^2)) - (asc)^2 \geq 0 \\ a &\leq \frac{1}{1 + c} \end{aligned}det(E?)a=(1−as2)(1−a(1+c2))−(asc)2≥0≤1+c1

So we have the Ivanovic-Dieks-Peres (IDP) limit

PsuccUSD=amax(1−c2)=1−c=1−∣⟨ψ0∣ψ1⟩∣\begin{aligned} P_{succ}^{USD} = a_{max}(1-c^2) = 1 - c = 1 - |\langle\psi_0|\psi_1\rangle| \end{aligned}PsuccUSD=amax(1−c2)=1−c=1−∣⟨ψ0∣ψ1⟩∣

Conclusion#

POVMs extend projective measurements by allowing outcome effects that need not be orthogonal projectors. Naimark dilation shows that these generalized measurement statistics can be realized by coupling the system to an ancilla and projectively reading orthogonal pointer states. In state discrimination, the choice of POVM determines how we trade errors against inconclusive results. Quantum measurement is therefore not merely the reading of an eigenvalue, but a physically constrained way of extracting classical information from a quantum state.

Quantum information is about what can be learned from quantum states, and the laws of quantum measurement impose fundamental limits on information extraction.

References:

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010).

2. C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press (1976).

3. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland (1982).

4. A. Chefles, “Quantum State Discrimination,” Contemporary Physics 41, 401–424 (2000), DOI: 10.1080/00107510010002599.

5. S. M. Barnett and S. Croke, “Quantum State Discrimination,” Advances in Optics and Photonics 1, 238–278 (2009), DOI: 10.1364/AOP.1.000238.

6. Quantum Information Lecture, Prof. J.Y.Fan, Department of Physics, SUSTech.

ILLIT_Minju

What is Wannier? --the wisdom of simplification and localization

Mon, 10 Aug 2026 00:00:00 GMT

This article is a simple introduction to the Wannier function. It won’t go into more technical details, but will give a general idea of what it is and what it does in the practical calculation with my own understanding and thinking framework. It’s not a formal article and may contain some mistakes, feel free to contact me if you have any questions or suggestions.

Where does it come from?#

The modern condensed matter physics aims to figure out the electron structures and behaviours in the crystal-like, i.e. the periodic potential field. As a fundamental theory, Bloch’s theorem tells the most important property of the wave function while it is noticed that the crystal momentum kkk is a good quantum number in such periodic potential. [1]

ψnk(r+R)=eik⋅Rψnk(r).\begin{align} \psi_{n\mathbf k}(\mathbf r+\mathbf R) &= e^{i\mathbf k\cdot\mathbf R} \psi_{n\mathbf k}(\mathbf r). \end{align}ψnk(r+R)=eik⋅Rψnk(r).

However, Bloch’s picture describe the electron as a “running wave” in the infinite potential field, which is not naturally consistent with the “atomic” picture. Early quantum mechanics provided a clear orbital picture for isolated atoms, most notably through the exact solution of the hydrogen atom and approximate treatments of many-electron atoms. It is simpler and more intuitive to treat the electron states in the periodic potential as the linear combination of the localized atomic-orbital-like real space functions.

Gregory Wannier gave his answer in 1937 when studying the electron-hole excitation in the crystal. He constructed the Wannier Functions (WFs) to bridge the above gap. [2]

Wannier functions are real-space representatives of a chosen Bloch-band subspace. For a finite crystal containing NNN unit cells with periodic boundary conditions, the Wannier states associated with an isolated band are defined by

∣wnR⟩=1N∑ke−ik⋅R∣ψnk⟩.\begin{align} \left|w_{n\mathbf R}\right\rangle &= \frac{1}{\sqrt{N}} \sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R} \left|\psi_{n\mathbf k}\right\rangle. \end{align}∣wnR⟩=N1k∑e−ik⋅R∣ψnk⟩.

For a composite subspace containing JJJ bands, the Bloch states at each k\mathbf kk can also mix with each other:

∣wnR⟩=1N∑ke−ik⋅R∑m=1J∣ψmk⟩Umn(k).\begin{align} |w_{n\mathbf R}\rangle &= \frac{1}{\sqrt N} \sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R} \sum_{m=1}^{J} |\psi_{m\mathbf k}\rangle U_{mn}(\mathbf k). \end{align}∣wnR⟩=N1k∑e−ik⋅Rm=1∑J∣ψmk⟩Umn(k).

Here U(k)∈U(J)U(\mathbf k)\in U(J)U(k)∈U(J) is a k\mathbf kk-dependent unitary matrix. It represents the gauge freedom inside the selected band subspace. Different choices of U(k)U(\mathbf k)U(k) keep the same subspace, but generally give different Wannier functions and different degrees of localization.

Their position-space representations are complex-valued functions:

wnR(r)=⟨r|wnR⟩,\begin{align} w_{n\mathbf R}(\mathbf r) &= \left\langle\mathbf r\middle|w_{n\mathbf R}\right\rangle, \end{align}wnR(r)=⟨r∣wnR⟩,

Wannier functions have the following important properties.

Lattice-translation covariance

Wannier functions in different unit cells are related by lattice translations:
wnR(r)=wn0(r−R).\begin{align} w_{n\mathbf R}(\mathbf r) &= w_{n\mathbf 0}(\mathbf r-\mathbf R). \end{align}wnR(r)=wn0(r−R).
Orthonormality
⟨wnR|wmR′⟩=δnmδRR′\begin{align} \left\langle w_{n\mathbf R} \middle| w_{m\mathbf R'} \right\rangle &= \delta_{nm}\delta_{\mathbf R\mathbf R'} \end{align}⟨wnR∣wmR′⟩=δnmδRR′
Completeness within the selected band subspace

The Wannier states form an orthonormal basis for the range of a band projector P^\hat PP^:
HP:=Ran⁡P^=span⁡‾{∣wnR⟩}n,R\begin{align} \mathcal H_P &:= \operatorname{Ran}\hat P = \overline{\operatorname{span}} \left\{ \left|w_{n\mathbf R}\right\rangle \right\}_{n,\mathbf R} \end{align}HP:=RanP^=span{∣wnR⟩}n,R
The projector satisfies
P^2=P^, P^†=P^P^=∑n,R∣wnR⟩⟨wnR∣\begin{aligned} &\hat P^2= \hat P, \ \hat P^\dagger = \hat P \\ &\hat P = \sum_{n,\mathbf R} \left|w_{n\mathbf R}\right\rangle \left\langle w_{n\mathbf R}\right| \end{aligned}P^2=P^, P^†=P^P^=n,R∑∣wnR⟩⟨wnR∣
Real-space localization

When a sufficiently smooth gauge exists, the Wannier functions can be chosen to decay rapidly away from the centers. For exponentially localized Wannier functions,
∣wnR(r)∣≤Cne−αn∣r−R∣,Cn,αn>0.\begin{align} \left| w_{n\mathbf R}(\mathbf r) \right| &\leq C_n e^{-\alpha_n \left|\mathbf r-\mathbf R\right|}, \qquad C_n,\alpha_n>0. \end{align}∣wnR(r)∣≤Cne−αn∣r−R∣,Cn,αn>0.

Therefore, Wannier functions do not satisfy the Bloch condition. For an isolated band, the Bloch state is obtained as a momentum-labelled superposition of Wannier functions distributed over all lattice cells:

∣ψnk⟩=1N∑Reik⋅R∣wnR⟩.\begin{align} \left|\psi_{n\mathbf k}\right\rangle &= \frac{1}{\sqrt N} \sum_{\mathbf R} e^{i\mathbf k\cdot\mathbf R} \left|w_{n\mathbf R}\right\rangle. \end{align}∣ψnk⟩=N1R∑eik⋅R∣wnR⟩.

For composite bands, the inverse transformation instead reconstructs a Bloch-like state in the Wannier gauge,

∣ψ~nk⟩=∑m=1J∣ψmk⟩Umn(k)=1N∑Reik⋅R∣wnR⟩.\begin{align} \left|\widetilde\psi_{n\mathbf k}\right\rangle &= \sum_{m=1}^{J} \left|\psi_{m\mathbf k}\right\rangle U_{mn}(\mathbf k) \notag \\ &= \frac{1}{\sqrt N} \sum_{\mathbf R} e^{i\mathbf k\cdot\mathbf R} \left|w_{n\mathbf R}\right\rangle. \end{align}ψnk⟩=m=1∑J∣ψmk⟩Umn(k)=N1R∑eik⋅R∣wnR⟩.

The two representations, the Bloch ∣ψnk⟩\ket{\psi_{n\mathbf k}}∣ψnk⟩ and the Wannier ∣wnR⟩\ket{w_{n\mathbf R}}∣wnR⟩ both describe the correct band dispersion. Bloch states are extended throughout the crystal, whereas Wannier states sacrifice a definite momentum label in exchange for real-space localization.

What does it do?#

In the condensed matter calculation, Wannierization is most commonly used as a post-processing step following a first-principles electronic structure calculation, mostly Density Functional Theory (DFT), even though it is not the initial purpose of Wannier’s original work.

In 1997, Marzari and Vanderbilt introduced the maximally localized Wannier functions (MLWFs) for composite bands [3]. In 2001, Souza, Marzari, and Vanderbilt extended this method to entangled bands through a disentanglement procedure [4]. In 2008, the Wannier90 package was published [5].

Wannier90 does not determine the electronic structure by itself. Instead, it takes the eigenvalues, Bloch-state overlaps, and trial-orbital projections supplied by a first-principles calculation and constructs a localized representation of a selected band subspace. Within a fixed isolated subspace, Wannierization is a unitary change of basis. Approximations enter when an effective subspace is disentangled and when long-ranged real-space matrix entries are truncated.

In a first-principles workflow, DFT supplies the Kohn-Sham Hamiltonian, eigenvalues, and Bloch eigenstates:

H^KS[ρ]∣ψnk⟩=εnk∣ψnk⟩.\begin{align} \hat H_{\mathrm{KS}}[\rho] \left|\psi_{n\mathbf k}\right\rangle &= \varepsilon_{n\mathbf k} \left|\psi_{n\mathbf k}\right\rangle. \end{align}H^KS[ρ]∣ψnk⟩=εnk∣ψnk⟩.

Wannierization gives the Real Space Hamiltonian under the Real Space Wannier basis

Hmn(R)=⟨wm0∣H^∣wnR⟩\begin{align} H_{mn}(\mathbf R) = \bra{w_{m\mathbf 0}}\hat{H}\ket{w_{n\mathbf R}} \end{align}Hmn(R)=⟨wm0∣H^∣wnR⟩

And the Fourier transformation to momentum reciprocal space

Hmn(k)=∑Reik⋅RHmn(R).\begin{align} H_{mn}(\mathbf k) &= \sum_{\mathbf R} e^{i\mathbf k\cdot\mathbf R} H_{mn}(\mathbf R). \end{align}Hmn(k)=R∑eik⋅RHmn(R).

So why do we wannierize it instead of using the original DFT result directly?

First-principles DFT calculations provide an accurate description of the electronic structure. In principle, most physical quantities can be evaluated directly from the Kohn-Sham eigenstates. In practice, the difficulty lies in repeating such calculations over the extremely dense momentum meshes required for Brillouin-zone integration. Such large plane-wave diagonalizations on the mesh is too expensive to be practical, especially for large systems.

Wannierization replaces large-scale calculation with a compact interpolation problem. Dense-mesh calculations require only the diagonalization of a much smaller Wannier Hamiltonian. Topological invariants, Berry curvature, transport coefficients, and dynamical response functions can therefore be evaluated efficiently without solving the original large plane-wave problem.

For example, the primitive bcc Fe unit cell contains only 1 magnetic atom, but the DFT calculation requires more than 600 plane waves to converge the Kohn-Sham eigenstates. In contrast, a Wannier Hamiltonian with 18 basis functions per unit cell is sufficient to reproduce the correct band structure over a wide energy range centered on the Fermi level. The Wannier Hamiltonian is therefore more than an order of magnitude smaller than the original DFT Hamiltonian, and its diagonalization is correspondingly faster.

The bcc Fe band structure, The grey lines represent the vasp results, and the red lines represent the Wannier90 results

1-dimensional ssh-like example#

Suppose we have a 1-dimensional SSH-like model with two identical atoms in a unit cell.

periodic potential

lattice constant aaa
relative position ddd

The potential is given by

V(x)=−V0∑R∈Λ[g(x−R)+g(x−R−d)]\begin{align} V(x) &= -V_0 \sum_{R\in \Lambda} [g(x-R) + g(x-R-d)] \end{align}V(x)=−V0R∈Λ∑[g(x−R)+g(x−R−d)]

where g(x)g(x)g(x) is the Gaussian function defined as

g(x)=e−x22σ2.\begin{align} g(x) &= e^{-\frac{x^2}{2\sigma^2}}. \end{align}g(x)=e−2σ2x2.

and the lattice set Λ\LambdaΛ is defined as

Λ={ma∣m∈Z}.\begin{align} \Lambda &= \{ma| m\in \mathbb Z\}. \end{align}Λ={ma∣m∈Z}.

Plane wave solution#

The Hamiltonian is given by

H^=−ℏ22md2dx2+V(x)\begin{align} \hat H &= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \end{align}H^=−2mℏ2dx2d2+V(x)

from the Bloch theorem, the Bloch wave function is the eigenfunction of the Hamiltonian.

ψnk(x)=eikxunk(x)\begin{align} \psi_{n k}(x) = e^{ikx}u_{nk}(x) \end{align}ψnk(x)=eikxunk(x)

expand the periodic function unk(x)u_{nk}(x)unk(x)

ψnk(x)=∑GCnG(k)ei(k+G)x.\begin{align} \psi_{nk}(x) &= \sum_G C_{nG}(k)e^{i(k+G)x}. \end{align}ψnk(x)=G∑CnG(k)ei(k+G)x.

We can define the normalized plane wave basis in the real space as

⟨x∣k+G⟩=1Lei(k+G)x.\begin{align} \braket{x|k+G} &= \frac{1}{\sqrt L}e^{i(k+G)x}. \end{align}⟨x∣k+G⟩=L1ei(k+G)x.

Then we have the Bloch states in the plane wave basis as

∣ψnk⟩=∑GCnG(k)∣k+G⟩.\begin{align} \ket{\psi_{nk}} &= \sum_G C_{nG}(k)\ket{k+G}. \end{align}∣ψnk⟩=G∑CnG(k)∣k+G⟩.

Then the eigenvalue problem can be written as

∑G′⟨k+G∣H^∣k+G′⟩CnG′(k)=EnkCnG(k).\begin{align} \sum_{G'}\bra{k+G}\hat H\ket{k+G'}C_{nG'}(k) &= E_{nk}C_{nG}(k). \end{align}G′∑⟨k+G∣H^∣k+G′⟩CnG′(k)=EnkCnG(k).

Kinetic energy#

⟨k+G∣T^∣k+G′⟩=ℏ22m(k+G)2δGG′\begin{align} \bra{k+G}\hat T\ket{k+G'} &= \frac{\hbar^2}{2m}(k+G)^2\delta_{GG'} \end{align}⟨k+G∣T^∣k+G′⟩=2mℏ2(k+G)2δGG′

which is the diagonal term

Potential energy#

⟨k+G∣V^∣k+G′⟩=VG−G′\begin{align} \bra{k+G}\hat V\ket{k+G'} &= V_{G-G'} \end{align}⟨k+G∣V^∣k+G′⟩=VG−G′

In order to find the off-diagonal term, we can use the Fourier transformation to get the potential in the reciprocal space

VG=1a∫0aV(x)e−iGxdx\begin{align} V_G &= \frac{1}{a}\int_0^a V(x)e^{-iGx} dx \end{align} VG=a1∫0aV(x)e−iGxdxVG=−V0a∑R∈Λ∫0a[g(x−R)+g(x−R−d)]e−iGxdx\begin{aligned} V_G= -\frac{V_0}{a}\sum_{R\in \Lambda} \int_0^a [g(x-R) + g(x-R-d)] e^{-iGx} dx \end{aligned}VG=−aV0R∈Λ∑∫0a[g(x−R)+g(x−R−d)]e−iGxdx

here we only show the contribution from the first term, the second term is similar

VG(1)=−V0a∫0a∑R∈Λg(x−R)e−iGxdx=−V0a∑m∈Z∫0ae−(x−ma)22σ2e−iGxdx=−V0a∑m∈Z∫−maa−mae−x22σ2e−iG(x+ma)dx\begin{aligned} V_G^{(1)} &= -\frac{V_0}{a}\int_0^a \sum_{R\in \Lambda} g(x-R) e^{-iGx} dx \\ &= -\frac{V_0}{a} \sum_{m \in \mathbb{Z}} \int_{0}^{a} e^{-\frac{(x-ma)^2}{2\sigma^2}} e^{-iGx} dx \\ &= -\frac{V_0}{a} \sum_{m \in \mathbb{Z}} \int_{-ma}^{a-ma} e^{-\frac{x^2}{2\sigma^2}} e^{-iG(x+m a)} dx \\ \end{aligned}VG(1)=−aV0∫0aR∈Λ∑g(x−R)e−iGxdx=−aV0m∈Z∑∫0ae−2σ2(x−ma)2e−iGxdx=−aV0m∈Z∑∫−maa−mae−2σ2x2e−iG(x+ma)dx

mmm enumerates all the integers

e−iGma=e−i2πajma=1e^{-iGma} = e^{-i\frac{2\pi}{a}jm a} = 1e−iGma=e−ia2πjma=1VG(1)=−V0a∫−∞∞e−x22σ2e−iGxdx=−V0a2πσe−G2σ22V_G^{(1)} = -\frac{V_0}{a} \int_{-\infty}^{\infty} e^{-\frac{x^2}{2\sigma^2}} e^{-iGx} dx = -\frac{V_0}{a} \sqrt{2\pi}\sigma e^{-\frac{G^2\sigma^2}{2}}VG(1)=−aV0∫−∞∞e−2σ2x2e−iGxdx=−aV02πσe−2G2σ2

The second term is similar, we can get the final result

VG=−V0a2πσe−G2σ22(1+e−iGd)\begin{align} V_G &= -\frac{V_0}{a} \sqrt{2\pi}\sigma e^{-\frac{G^2\sigma^2}{2}}(1+e^{-iGd}) \end{align}VG=−aV02πσe−2G2σ2(1+e−iGd)

so the potential entry is

VG−G′=−V0a2πσe−(G−G′)2σ22(1+e−i(G−G′)d)\begin{align} V_{G-G'} &= -\frac{V_0}{a} \sqrt{2\pi}\sigma e^{-\frac{(G-G')^2\sigma^2}{2}}(1+e^{-i(G-G')d}) \end{align}VG−G′=−aV02πσe−2(G−G′)2σ2(1+e−i(G−G′)d)

with the Hamiltonian entry

⟨k+G∣H^∣k+G′⟩=−V0a2πσe−(G−G′)2σ22(1+e−i(G−G′)d)+ℏ22m(k+G)2δGG′\begin{align} \bra{k+G}\hat H\ket{k+G'} &= -\frac{V_0}{a} \sqrt{2\pi}\sigma e^{-\frac{(G-G')^2\sigma^2}{2}}(1+e^{-i(G-G')d}) + \frac{\hbar^2}{2m}(k+G)^2\delta_{GG'} \end{align}⟨k+G∣H^∣k+G′⟩=−aV02πσe−2(G−G′)2σ2(1+e−i(G−G′)d)+2mℏ2(k+G)2δGG′

we can construct the Hamiltonian matrix and diagonalize it at any given kkk point to get the band structure. We take total 111 dimension with G=−552πa,−542πa,⋯ ,552πaG = -55\frac{2\pi}{a}, -54\frac{2\pi}{a}, \cdots, 55\frac{2\pi}{a}G=−55a2π,−54a2π,⋯,55a2π, the proper parameters are selected. We get the lowest band structure as shown in the figure below.

band structure

Tight-binding approximation#

In practical calculation, we only care about the specific bands of the system, which is usually near the Fermi surface, or with nontrivial topological structures by selecting an energy window to construct a subspace. The lowest two plane-wave bands are separated from the higher-energy bands by a finite gap and therefore form a well-defined two-dimensional invariant subspace. We would like to describe this subspace using two localized orbitals per unit cell.

Suppose we have 2 orbitals in the 2 positions of the unit cell as ∣A,R⟩\ket{A, R}∣A,R⟩ and ∣B,R⟩\ket{B, R}∣B,R⟩, we can write Hopping term

Onsite ⟨A,R∣H^∣A,R⟩=⟨B,R∣H^∣B,R⟩=ϵ0\begin{align} \bra{A, R}\hat H\ket{A, R} &= \bra{B, R}\hat H\ket{B, R} = \epsilon_0 \end{align}⟨A,R∣H^∣A,R⟩=⟨B,R∣H^∣B,R⟩=ϵ0
Inter-cell same site ⟨A,R∣H^∣A,R+a⟩=⟨B,R∣H^∣B,R+a⟩=t0\begin{align} \bra{A, R}\hat H\ket{A, R+a} &= \bra{B, R}\hat H\ket{B, R+a} = t_0 \end{align}⟨A,R∣H^∣A,R+a⟩=⟨B,R∣H^∣B,R+a⟩=t0
Intra-cell ⟨A,R∣H^∣B,R⟩=t1\begin{align} \bra{A, R}\hat H\ket{B, R} &= t_1 \end{align}⟨A,R∣H^∣B,R⟩=t1
Inter-cell different site ⟨A,R+a∣H^∣B,R⟩=t2\begin{align} \bra{A, R+a}\hat H\ket{B, R} &= t_2 \end{align}⟨A,R+a∣H^∣B,R⟩=t2

Second quantization form

H^ext=∑Rϵ0(c^A,R†c^A,R+c^B,R†c^B,R)+∑R[t0(c^A,R+a†c^A,R+c^B,R+a†c^B,R)+h.c.]+∑R[t1c^A,R†c^B,R+t2c^A,R+a†c^B,R+h.c.].\begin{align} \hat H_{\mathrm{ext}} ={}& \sum_R \epsilon_0 \left( \hat c_{A,R}^\dagger\hat c_{A,R} + \hat c_{B,R}^\dagger\hat c_{B,R} \right) + \sum_R \left[ t_0 \left( \hat c_{A,R+a}^\dagger\hat c_{A,R} + \hat c_{B,R+a}^\dagger\hat c_{B,R} \right) +\mathrm{h.c.} \right] \notag \\ &+ \sum_R \left[ t_1\hat c_{A,R}^\dagger\hat c_{B,R} + t_2\hat c_{A,R+a}^\dagger\hat c_{B,R} + \mathrm{h.c.} \right]. \end{align}H^ext=R∑ϵ0(c^A,R†c^A,R+c^B,R†c^B,R)+R∑[t0(c^A,R+a†c^A,R+c^B,R+a†c^B,R)+h.c.]+R∑[t1c^A,R†c^B,R+t2c^A,R+a†c^B,R+h.c.].

Fourier transforming the annihilation and creation operators

c^A,R=1N∑keikRc^A,kc^B,R=1N∑keikRc^B,k\begin{align} \hat c_{A,R} &= \frac{1}{\sqrt N} \sum_k e^{ikR} \hat c_{A,k} \notag \\ \hat c_{B,R} &= \frac{1}{\sqrt N} \sum_k e^{ikR} \hat c_{B,k} \end{align}c^A,Rc^B,R=N1k∑eikRc^A,k=N1k∑eikRc^B,k

We can write the Hamiltonian in the momentum space by

H^=∑kΨk†H(k)Ψk\begin{align} \hat H = \sum_k \Psi_k^\dagger H(k) \Psi_k \end{align}H^=k∑Ψk†H(k)Ψk

where Ψk=[c^A,k,c^B,k]T\Psi_k = [\hat c_{A,k}, \hat c_{B,k}]^TΨk=[c^A,k,c^B,k]T and the Hamiltonian matrix is

H(k)=[ϵ0+2t0cos⁡(ka)t1+t2e−ikat1+t2eikaϵ0+2t0cos⁡(ka)]\begin{align} H(k) &= \begin{bmatrix} \epsilon_0 + 2t_0\cos(ka) & t_1 + t_2 e^{-ika} \\ t_1 + t_2 e^{ika} & \epsilon_0 + 2t_0\cos(ka) \end{bmatrix} \end{align}H(k)=[ϵ0+2t0cos(ka)t1+t2eikat1+t2e−ikaϵ0+2t0cos(ka)]

(ϵ0,t0,t1,t2∈R\epsilon_0, t_0, t_1, t_2 \in \mathbb{R}ϵ0,t0,t1,t2∈R)

With the proper parameters, we can get the band structure as shown below.

TB band structure

Where Wannier Enters#

The two calculations above reveal the trade-off between the plane-wave and tight-binding descriptions. The plane-wave method solves the continuum Hamiltonian in a large and delocalized basis. However, the resulting Hamiltonian is large and its real-space interpretation is not immediately transparent.

The extended SSH model takes the opposite approach. It assumes two localized orbitals, ∣A,R⟩\ket{A,R}∣A,R⟩ and ∣B,R⟩\ket{B,R}∣B,R⟩, and retains only a small number of hopping parameters. The resulting 2×22\times22×2 Hamiltonian is simple and physically intuitive. However, the model itself does not tell us what the two orbital wave functions actually are, nor does it determine the values of ϵ0,t0,t1,\epsilon_0,t_0,t_1,ϵ0,t0,t1, and t2t_2t2. In the example above, these parameters were fitted manually to the plane-wave band structure.

Wannierization provides the missing connection between these two descriptions. Starting from the isolated two-band subspace obtained in the plane-wave calculation, it constructs two localized Wannier functions per unit cell. In the MLWF method, the gauge freedom of the Bloch states is chosen by minimizing the real-space spread of these functions. The states ∣A,R⟩\ket{A,R}∣A,R⟩ and ∣B,R⟩\ket{B,R}∣B,R⟩ in the tight-binding model can therefore be replaced by explicitly constructed Wannier functions rather than assumed atomic-like orbitals.

Once the Wannier basis has been constructed, the original Hamiltonian is represented by the real-space matrix elements as shown in Eq. (12).

The onsite energies and hopping parameters are then identified from these matrix elements. For example,

ϵ0∼H11(0),t1∼H12(0),t0∼H11(a),t2∼H12(−a).\begin{align} \epsilon_0 &\sim H_{11}(0), & t_1 &\sim H_{12}(0), \notag \\ t_0 &\sim H_{11}(a), & t_2 &\sim H_{12}(-a). \end{align}ϵ0t0∼H11(0),∼H11(a),t1t2∼H12(0),∼H12(−a).

Unlike the SSH model above, the Wannier Hamiltonian also reveals longer-ranged hopping processes whenever their matrix elements are non-negligible. A practical tight-binding model is obtained by choosing a real-space cutoff and retaining the relevant matrix elements:

Hmn(k)=∑R∈cutoff seteik⋅RHmn(R).\begin{align} H_{mn}(k) &= \sum_{R \in \text{cutoff set}} e^{ik \cdot R} H_{mn}(R). \end{align}Hmn(k)=R∈cutoff set∑eik⋅RHmn(R).

Wannierization does more than provide another tight-binding model. It systematically derives the localized basis and its hopping parameters from the original plane-wave calculation, while allowing the accuracy and complexity of the effective model to be controlled through the real-space cutoff. With the real-space Wannier Hamiltonian, position operator, and other physical operators represented in the Wannier basis, we gain efficient access to these physical quantities. Wannierization does not change the physics; it changes the language in which the physics becomes transparent.

References:

1. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929), DOI: 10.1007/BF01339455.

2. G. H. Wannier, “The Structure of Electronic Excitation Levels in Insulating Crystals,” Phys. Rev. 52, 191–197 (1937), DOI: 10.1103/PhysRev.52.191.

3. N. Marzari and D. Vanderbilt, “Maximally Localized Generalized Wannier Functions for Composite Energy Bands,” Phys. Rev. B 56, 12847–12865 (1997), DOI: 10.1103/PhysRevB.56.12847.

4. I. Souza, N. Marzari, and D. Vanderbilt, “Maximally Localized Wannier Functions for Entangled Energy Bands,” Phys. Rev. B 65, 035109 (2001), DOI: 10.1103/PhysRevB.65.035109.

5. A. A. Mostofi et al., “wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions,” Comput. Phys. Commun. 178, 685–699 (2008), DOI: 10.1016/j.cpc.2007.11.016.

6. N. Marzari et al., “Maximally Localized Wannier Functions: Theory and Applications,” Rev. Mod. Phys. 84, 1419–1475 (2012), DOI: 10.1103/RevModPhys.84.1419.

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The First Post for NitreneGlog!!!

Thu, 23 Jul 2026 00:00:00 GMT

Congratulations on the First Post of NitreneGlog!#

Born on June 30, 2026, NitreneGlog has once been built as a static site and then with halo system, and now, it has been rebuilt with Astro framework with Firefly theme. It is finally ready to share the first post with the world at 03:06 a.m. (UTC+8) on July 23, 2026!!! Congratulations!!!