Coherence and Preferred Bases

Coherence is not an absolute label attached to a density matrix. It is coherence relative to a chosen basis, decomposition, observable, or set of operational alternatives.

A state can be coherent in one basis and diagonal in another. Decoherence, correspondingly, does not mean that every possible off-diagonal matrix element disappears. It means that interference between the alternatives monitored by an apparatus or environment becomes locally inaccessible.

Helpful background. Density operators describe populations and interference terms. What Is Decoherence? derives their suppression by environmental overlap.

Coherence Relative to a Basis

Let $\{\lvert a\rangle\}$ be an orthonormal basis. A density operator has matrix elements

$$\rho_{ab} = \langle a\rvert\rho\lvert b\rangle.$$

The diagonal entries $\rho_{aa}$ are populations in that basis. The off-diagonal entries

$$\rho_{ab}, \qquad a\ne b,$$

are coherences between the alternatives $\lvert a\rangle$ and $\lvert b\rangle$.

Those statements are basis-relative. If one changes basis,

$$\lvert \tilde a\rangle = \sum_b U_{ba}\lvert b\rangle,$$

then the transformed matrix elements are

$$\tilde\rho = U^\dagger\rho U.$$

A matrix diagonal in the original basis need not be diagonal in the new basis. This coordinate change does not physically disturb the state: observable probabilities are unchanged when the observables are transformed with it. Applying a unitary to the state while keeping the apparatus fixed is a different, physical operation.

Qubit Example

In the $Z$ basis, the state

$$\lvert +\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}$$

has density matrix

$$\lvert+\rangle\langle+\rvert = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

The off-diagonal entries are nonzero, so the state is coherent relative to the $Z$ basis.

In the $X$ basis,

$$\{\lvert+\rangle,\lvert-\rangle\}, \qquad \lvert-\rangle = \frac{\lvert0\rangle-\lvert1\rangle}{\sqrt2},$$

the same state is

$$\begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}.$$

It is diagonal in that basis. Thus the sentence “the state has coherence” is incomplete unless the alternatives have been specified.

The maximally mixed state

$$\rho=\frac12 I$$

is special: it is diagonal in every basis and has no basis-relative coherence.

Dephasing as Removing Relative Coherence

Given a preferred orthonormal basis $\{\lvert a\rangle\}$, the ideal dephasing map is

$$\Delta(\rho) = \sum_a \lvert a\rangle\langle a\rvert \rho \lvert a\rangle\langle a\rvert.$$

In matrix elements,

$$\Delta(\rho)_{ab} = \delta_{ab}\rho_{aa}.$$

The map keeps populations and removes off-diagonal coherence in that basis. It is a completely positive trace-preserving channel, and it is idempotent:

$$\Delta^2=\Delta.$$

Partial dephasing multiplies off-diagonal terms by decoherence factors:

$$\rho_{ab} \longmapsto D_{ab}\rho_{ab}, \qquad D_{aa}=1.$$

The matrix $D$ must be positive semidefinite with unit diagonal for this to define a channel. When $|D_{ab}|$ becomes small, interference between those alternatives is suppressed. Dephasing Channel explains the CP condition and why bounds on individual entries are not enough.

Coherence Relative to a Decomposition

The alternatives need not be one-dimensional. Often the environment distinguishes coarse labels $\alpha$ while leaving coherence inside each sector:

$$\mathcal H = \bigoplus_\alpha \mathcal H_\alpha.$$

Let $P_\alpha$ project onto $\mathcal H_\alpha$. The corresponding block-dephasing map is

$$\Delta_P(\rho) = \sum_\alpha P_\alpha\rho P_\alpha.$$

The off-block terms

$$P_\alpha\rho P_\beta, \qquad \alpha\ne\beta,$$

are the coherences removed by this coarse-grained monitoring. Coherence within a block $P_\alpha\rho P_\alpha$ may remain.

This is why preferred structure can be a basis, a family of subspaces, or an approximate set of wavepackets rather than a single exact orthonormal basis.

What Selects a Preferred Basis?

A basis becomes preferred when the dynamics treats its alternatives differently in a stable, record-producing way.

The simplest monitoring Hamiltonian has the form

$$H_{\mathrm{int}} = \sum_a \lvert a\rangle\langle a\rvert \otimes B_a.$$

Assume an initially independent pure environment, and that the system Hamiltonian preserves these alternatives or is negligible during the interaction. For an initial $\lvert a\rangle$, conditional environmental evolution gives:

$$\lvert a\rangle\lvert E_0\rangle \longmapsto \lvert a\rangle\lvert E_a(t)\rangle.$$

For a superposition,

$$\sum_a c_a\lvert a\rangle\lvert E_0\rangle \longmapsto \sum_a c_a\lvert a\rangle\lvert E_a(t)\rangle.$$

The reduced density matrix then contains overlap factors:

$$\rho_{ab}(t) = \langle E_b(t)\vert E_a(t)\rangle \rho_{ab}(0).$$

When the conditional environment vectors become physically distinguishable, the corresponding off-diagonal magnitudes are suppressed. Distinct $B_a$ are not sufficient by themselves: if they act only by different scalar phases on the initial environment vector, the system undergoes a phase rotation without losing coherence. The coupling and initial environment state determine this behavior. See Environment-Induced Decoherence for rate estimates.

Eigenbasis Is Not Automatically Preferred

Every density matrix can be diagonalized:

$$\rho = \sum_k p_k\lvert k\rangle\langle k\rvert.$$

This spectral decomposition is mathematically useful, but it does not by itself identify the physically preferred alternatives. The eigenbasis can change with time, can be unstable under small perturbations when eigenvalues are nearly degenerate, and may have no direct relation to environmental records.

A pointer basis or pointer structure is different. It is selected by:

• the system Hamiltonian;
• the system-environment interaction;
• measurement and amplification channels;
• timescale and coarse graining;
• robustness of records under later dynamics.

This is why diagonalizing $\rho(t)$ at one instant is not a solution to the preferred-basis problem. It finds a basis in which that matrix has no off-diagonal entries; it does not explain why those alternatives are the ones recorded, stable, or classical-looking.

Common Physical Examples

Pure dephasing

If a qubit environment monitors $Z$, then the preferred alternatives are $\lvert0\rangle$ and $\lvert1\rangle$. Superpositions in the equatorial plane lose relative phase coherence, while the $Z$ populations are preserved.

Energy relaxation

For weakly coupled systems where the secular approximation is valid, coherences between different energy eigenstates often decouple from populations and decay. The energy basis is then dynamically important. But relaxation also changes populations, so this is not merely dephasing.

Position decoherence

For macroscopic objects, environmental scattering often distinguishes different positions or localized wavepackets. The preferred structure is then approximately position-localized, not exactly a discrete basis of normalizable position eigenstates.

Coherent states under loss

Under quantum-limited linear loss into vacuum, coherent states remain pure coherent states with a reduced amplitude. At finite bath temperature they generally become mixed; their robustness must be stated relative to the chosen dynamics. They form an overcomplete set rather than an orthonormal pointer basis.

Collective noise

If the environment couples only to a collective observable, it may distinguish total labels while leaving coherence inside degenerate sectors. Preferred structure is then a block decomposition, and some subspace coherence may be protected.

Coherence as a Resource

In quantum information, one often fixes a reference basis and treats coherence relative to that basis as a resource. In that setting, incoherent states are diagonal in the chosen basis:

$$\rho_{\mathrm{inc}} = \sum_a p_a\lvert a\rangle\langle a\rvert.$$

Two common diagnostics are:

$$C_{\ell_1}(\rho) = \sum_{a\ne b}|\rho_{ab}|,$$

and the relative entropy of coherence,

$$C_{\mathrm{rel}}(\rho) = S(\Delta(\rho))-S(\rho).$$

The measures require a reference basis; interpreting them as resources also requires specifying the allowed operations. Use base-two logarithms for $S$ when quoting coherence in bits.

For the partially dephased qubit

$$\rho=\frac12\begin{pmatrix}1&0.6\\0.6&1\end{pmatrix},$$

$C_{\ell_1}=0.6$ in the $Z$ basis. The eigenvalues are $0.8$ and $0.2$, so

$$C_{\mathrm{rel}}=1-h_2(0.8)\approx0.2781\ \text{bits},$$

where $h_2(p)=-p\log_2p-(1-p)\log_2(1-p)$. In the $X$ basis the same state is diagonal and both measures vanish. This does not remove the $Z$-basis interference experiment; it changes the alternatives relative to which coherence is counted. Resource Theories explains the role of free states and allowed transformations.

Relation to Measurement

An ideal Lüders implementation of a projective measurement in basis $\{\lvert a\rangle\}$ has nonselective state update

$$\rho \longmapsto \sum_a \lvert a\rangle\langle a\rvert \rho \lvert a\rangle\langle a\rvert.$$

This is the same ideal dephasing map $\Delta$. If the outcome is ignored, coherence between different outcomes is removed. If the outcome is recorded and conditioned on, the state update is selective rather than merely dephasing.

Environmental decoherence is measurement-like because the environment stores records. The difference is that the record may be uncontrolled, distributed through many fragments, and inaccessible to the observer who assigns the reduced state.

For the operational distinction among selective, nonselective, and conditioned updates, see Projective Measurements and POVMs.

Exercises

1. Same state, different basis. Write $\lvert+\rangle\langle+\rvert$ in the $Z$ basis and in the $X$ basis. Identify which matrix elements are coherences in each basis.

Solution

In the $Z$ basis,

$$\lvert+\rangle\langle+\rvert = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

The off-diagonal entries are nonzero, so the state is coherent relative to the $Z$ basis.

In the $X$ basis $\{\lvert+\rangle,\lvert-\rangle\}$,

$$\lvert+\rangle\langle+\rvert = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}.$$

It is diagonal in that basis. The state did not change; the reference alternatives changed.

2. Dephasing is idempotent. Show that $\Delta(\rho)=\sum_a\lvert a\rangle\langle a\rvert\rho\lvert a\rangle\langle a\rvert$ satisfies $\Delta^2=\Delta$.

Solution

Let $P_a=\lvert a\rangle\langle a\rvert$. The projectors obey $P_aP_b=\delta_{ab}P_a$. Then

$$\begin{aligned} \Delta(\Delta(\rho)) &= \sum_{a,b} P_aP_b\rho P_bP_a\\ &= \sum_a P_a\rho P_a\\ &= \Delta(\rho). \end{aligned}$$

After one dephasing step, all off-diagonal terms in the chosen basis are already removed.

3. Diagonal is basis-relative. Let $\rho=p\lvert0\rangle\langle0\rvert+(1-p)\lvert1\rangle\langle1\rvert$. Compute $\langle+\rvert\rho\lvert-\rangle$. For which $p$ is the state diagonal in both the $Z$ and $X$ bases?

Solution

Using

$$\lvert\pm\rangle = \frac{\lvert0\rangle\pm\lvert1\rangle}{\sqrt2},$$

one finds

$$\langle+\rvert\rho\lvert-\rangle = \frac12 \left( p-(1-p) \right) = \frac{2p-1}{2}.$$

This vanishes only when $p=1/2$. In that case $\rho=I/2$, which is diagonal in every basis.

4. Preferred basis from a coupling. For $H_{\mathrm{int}}=Z\otimes B$, which qubit basis is monitored by the environment?

Solution

The eigenstates $\lvert0\rangle$ and $\lvert1\rangle$ are the candidate stable alternatives of this interaction. For an initial pure environment $|E\rangle$ and negligible competing dynamics, the conditional vectors are $e^{-iBt/\hbar}|E\rangle$ and $e^{iBt/\hbar}|E\rangle$. Decoherence occurs only if their overlap magnitude becomes less than one. If $|E\rangle$ is an eigenvector of $B$, they differ only by a phase: the system rotates without dephasing. A noncommuting system Hamiltonian also changes the robustness question.

5. Block coherence. Suppose $P_1$ and $P_2$ are two orthogonal projectors with $P_1+P_2=I$. What parts of $\rho$ are removed by $\Delta_P(\rho)=P_1\rho P_1+P_2\rho P_2$?

Solution

Decompose

$$\rho = P_1\rho P_1 + P_1\rho P_2 + P_2\rho P_1 + P_2\rho P_2.$$

The map $\Delta_P$ keeps the within-block terms $P_1\rho P_1$ and $P_2\rho P_2$. It removes the cross-block coherences $P_1\rho P_2$ and $P_2\rho P_1$. Coherence inside each block is not removed.

References

• T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Physical Review Letters 113, 140401 (2014). DOI.
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007). DOI.
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715–775 (2003). DOI.