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.

The central lesson is:

$$\text{coherence is representation dependent,} \qquad \text{preferred structure is dynamical.}$$

This page explains the distinction. The basic environmental mechanism is developed in What Is Decoherence?, and dynamically robust record states are treated in Pointer States.

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.

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:

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

When $|D_{ab}|$ becomes small for $a\ne b$, interference between those alternatives is suppressed. The canonical finite-dimensional model is the Dephasing Channel.

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

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

If the system is initially in $\lvert a\rangle$, the interaction correlates that label with an environmental state:

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

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 environmental states become distinguishable, the corresponding off-diagonal terms are suppressed. The preferred basis is therefore selected by the system-environment coupling, not by a purely formal choice of coordinates. The environmental-record mechanism and its rate estimates are developed in Environment-Induced Decoherence.

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

For a damped harmonic oscillator or optical mode, coherent states can remain comparatively stable under linear loss. 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,

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

These quantities are useful only after the reference basis and allowed operations have been specified. A basis-free statement such as “this state has coherence” hides the operational context.

Resource Theories owns the general free-set, free-transformation, and conversion ledger; this page owns basis dependence and preferred-structure physics.

Relation to Measurement

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.

Common Mistakes

• Saying “coherence disappeared” without naming the basis or decomposition.
• Treating the eigenbasis of $\rho(t)$ as the physical preferred basis.
• Assuming decoherence suppresses all possible off-diagonal matrix elements.
• Forgetting that preferred structure may be approximate, coarse-grained, or overcomplete.
• Confusing a diagonal reduced density matrix with a proper ignorance mixture.
• Calling a computational basis “preferred” just because it is convenient in a calculation.
• Ignoring the system Hamiltonian, which may rotate states away from the basis selected by the environment.
• Applying a resource-theory coherence measure without specifying the reference basis.

References

• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715-775 (2003).
• M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer (2007).
• E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, Springer, 2nd ed. (2003).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Physical Review Letters 113, 140401 (2014).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 10th anniversary ed. (2010).

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 operator $Z$ has eigenstates $\lvert0\rangle$ and $\lvert1\rangle$ in the usual convention. The interaction distinguishes those eigenvalues, so the environment monitors the $Z$ basis. Superpositions of $\lvert0\rangle$ and $\lvert1\rangle$ become entangled with different environmental states and dephase in that basis.

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.