How to think about Quantum Mechanics—Part 3: The pointer and Schmidt bases

[Other parts in this series: 1,2,3,4,5,6,7,8.]

A common mistake made by folks newly exposed to the concept of decoherence is to conflate the Schmidt basis with the pointer basis induced by decoherence.

[Show refresher on Schmidt decompsition]

Given any two quantum systems $\mathcal{S}$ and $\mathcal{E}$ and a pure joint state $\vert \psi (t) \rangle \in \mathcal{H} = \mathcal{S} \otimes \mathcal{E}$ , there always exists a Schmidt decomposition of the form

(1) $\begin{align*} \vert \psi (t) \rangle = \sum_k c_k \vert S_k (t) \rangle \vert E_k (t) \rangle \end{align*}$

where $\vert S_k (t) \rangle$ and $\vert E_k (t) \rangle$ are local orthonormal Schmidt bases on $\mathcal{S}$ and $\mathcal{E}$ , respectively.

Now, any state in such a joint Hilbert space can be expressed as $\vert \psi \rangle = \sum_{i,j} d_{i,j} \vert S_i \rangle \vert E_j \rangle$ for arbitrary fixed orthonormal bases $\vert S_i \rangle$ and $\vert E_j \rangle$ . What makes the Schmidt decomposition non-trivial is that it has only a single index $k$ rather than two indices $i$ and $j$ . (In particular, this means that the Schmidt decomposition constains at most $\mathrm{min}(\mathrm{dim}\,\mathcal{S},\mathrm{dim}\,\mathcal{E})$ non-vanishing terms, even if $\mathrm{dim}\,\mathcal{E} \gg \mathrm{dim}\,\mathcal{S}$ .) The price paid is that the Schmidt bases, $\vert S_k \rangle$ and $\vert E_k \rangle$ , depend on the state $\vert \psi \rangle$ .

When the values $\vert c_i \vert$ in the Schmidt decomposition are non-degenerate, the local bases are unique up to a phase. As $\vert \psi (t) \rangle$ evolves in time, this decomposition is defined for each time $t$ . The bases $\vert S_i (t) \rangle$ and $\vert E_i (t) \rangle$ evolve along with it, and can be considered to be a property of the state $\vert \psi (t) \rangle$ . In fact, they correspond to the eigenvectors of the respective reduced density matrices of $\mathcal{S}$ and $\mathcal{E}$ .

In the ideal case of so-called pure decoherence, the environment $\mathcal{E}$ begins in an initial state $\vert E_0 \rangle$ and is coupled to the system $\mathcal{S}$ through a unitary of the form

(2) $\begin{align*} U(t) = \sum_k \vert S_k \rangle \langle S_k \vert \otimes U^{\mathcal{E}}_k(t) \end{align*}$

with $\langle E_k(t) \vert E_l(t) \rangle \to \delta_{k,l}$ as $t \to \infty$ , where $U^{\mathcal{E}}_k(t)$ is a conditional unitary on $\mathcal{E}$ and $\vert E_k(t) \rangle \equiv U^{\mathcal{E}}_k(t) \vert E_0 \rangle$ . The elements of the density matrix $\rho$ of the system evolve as $\rho_{k,l}(t) = \langle E_k(t) \vert E_l(t) \rangle \rho_{k,l}(0)$ , i.e. $\rho$ approaches a diagonal form in the basis $\vert S_k \rangle$ , and this basis is the pointer basis.

The Schmidt basis approaches the pointer basis asymptotically but the bases do not coincide for any finite time $t$ . In other words, what makes $\vert S_k \rangle$ the pointer basis isn’t the instantaneous value of the density matrix of $\mathcal{S}$ , it’s the form in which $\vert S_k \rangle$ appears in the dynamics given by eq. (2).

A crude analogy can be made with the case of a rock falling in the atmosphere under the pull of gravity. The instantaneous velocity of the rock is well-defined and unambiguous at any given time. On the other hand, the terminal velocity is not so much a property of the rock as it is a property of the model containing the rock, the air, and various mathematical idealizations. Insofar as the model is a good approximation for reality, the instantaneous velocity will approach arbitrarily close to the terminal velocity for large times. However, it never exactly reaches the terminal velocity and, more importantly, at some point the separation between the two velocities becomes small enough that deficiencies in the model begin to dominate. Likewise, the Schmidt basis is an instantaneous property defined for any physical system, while the pointer basis is an asymptotic property within an idealized model.

Unfortunately, this distinction is not well appreciated for a number of reasons. First, the simple and ubiquitous nature of the Schmidt decomposition is enticing, while dynamical descriptions of decoherence are fairly complicated except in a few simplified models. (Pure decoherence being almost the simplest possible.) Second, there are modal interpretations of quantum mechanics that assign “ontic” status to some eigenvectors of the reduced density matrix of systems. Likewise, a simple reading of Hugh Everett’s original interpretation^a would suggest that one could simply read off the “relative states” coinciding with the Schmidt basis in a von Neumann measurement.

This isn’t just pedantry. All sorts of things can happen to the system that depart from its idealized dynamics, and you can become confused if you equate the pointer basis and the Schmidt basis. After all, decoherence-induced “branching” events like the one described above do not usually happen in isolation in physical systems. They are chained together, sometimes in rapid succession for a system that is constantly being monitored. Often, the self-dynamics of the system (and of the environment) cannot be easily disentangled from the decoherence interaction.

This is especially well known in the case of collisional decoherence. Collisional decoherence is when the wavefunction of a small particle, like a dust grain, is kept spatially localized by repeated microscopic scattering events with a multipartite environment, like photons or a gas. The decoherence competes with the natural spatial dispersion of the particle’s wavefunction in isolation. For a widely separated initial spatial superposition of the particle, dynamics of the form in eq. (2) are a good approximation for short times (with $\vert S_k \rangle$ the position basis), but in general things are much more complicated.

Collisional decoherence of a C70 fullerene (blue) by a methane gas environment (red).

This is usually studied with a very general class of Caldeira-Leggett models where the system and the parts of the environment are harmonic oscillators in Gaussian states, and are coupled by linear terms in the Hamiltonian. Although the important distinction between the Schmidt and pointer bases in such models has been long appreciated by practitioners in the field^b, the most striking example of how this mistake can lead one astray is found in the relatively recent work of Page^c:

It may be true that with interactions that are local in space, the density matrix in a basis that each has an appropriate single macroscopic state (e.g., an appropriate superposition of quantum microstates that each have the same unique macroscopic values) is often approximately diagonal, but the basis in which the density matrix really is precisely diagonal is, as I shall show for a wide class of simple examples, far from each having definite macroscopic states. In particular, I shall show that for many simple examples the mean uncertainty of the position variables in each of the Schmidt basis states is just as great as the full uncertainty that they have in the complete quantum density matrix of the subsystem.

In other words: if you look at the mathematically exact Schmidt basis for a particle decohered in the position basis, you’d find states that aren’t any more localized than the undecohered system!

As a corollary, any attempts to define the “real” branches in a Many Worlds interpretation can’t just use the Schmidt basis; more is needed. (Edit: Kent and McElwaine have a great paper^d cataloging the issues with these attempts.)

Footnotes

(↵ returns to text)

Hugh Everett (1957). “‘Relative state’ formulation of quantum mechanics”. Reviews of Modern Physics 29 (3): 454–462. [Text (HTML)].↵
W. G. Unruh and W. H. Zurek, “Reduction of a Wave Packet in Quantum Brownian Motion,” Phys. Rev. D 40, 1071-1094 (1989).↵
D. Page, “Quantum Uncertainties in the Schmidt Basis Given by Decoherence” [arXiv:1108.2709].↵
A. Kent and J. McElwaine, “Quantum prediction algorithms”, Phys. Rev. A 55, 1703 (1997) [ arXiv:gr-qc/9610028 ].↵

5 Comments

Jasper
August 28, 2014 at 5:00 am

Very nice an clear post. I just wanted to know, if the Schmidt basis is the exact basis for a particle decohered in the position basis at any time, while the pointer basis is the asymptotic basis, then do you think that decoherence actually tells us that macroscopic systems never occur in superpositions in space? Similar asymptotic problems always show up in physics (e.g. in thermodynamics) and therefore I would tend to say that FAPP decoherence shows us that superpositions in space become localized. Would you agree? Or do you think that something more would be needed?
- Jess Riedel
  August 28, 2014 at 1:38 pm
  
  > do you think that decoherence actually tells us that macroscopic systems never occur in superpositions in space?…I would tend to say that FAPP decoherence shows us that superpositions in space become localized. Would you agree?
  Yes, one can still calculate a characteristic spatial coherence length for the macroscopic system. When a large initial superposition of the system is exposed to an environment, the coherence length begins large but is suppressed extremely quickly by decoherence. But rather being driven to zero, it approaches some finite value consistent with the fact that objects are never completely localized in space.
  - Jasper
    August 29, 2014 at 8:33 am
    
    Thanks for the elaboration and also for the great series of posts on quantum mechanics!
Daniel Ranard
September 1, 2017 at 6:04 pm

Thanks Jess for lots of great posts — I think I must still be confused though. You say that the Schmidt basis states of the reduced density matrix of a particle that is “decohered in the position basis” may not actually be localized in position. I assume you’re referring to a particle that is (more or less) already decohered, as opposed to a particle still in the process of decohering.
Are the Schimdt states spatially delocalized because (a) the Schmidt states haven’t yet approached close enough to the pointer states (in which case, why would you say the particle is already decohered?), or is it because (b) the pointer states themselves aren’t localized in position (but then, in which sense does the decoherence takes place wrt the position basis)?
If the answer is (a) and the whole point is, as you say, “all sorts of things can happen to the system that depart from its idealized dynamics,” in what sense is the particle actually decohered in the position basis during these non-ideal dynaimcs?
- Jess Riedel
  September 3, 2017 at 7:56 pm
  
  Yea, I am referring to a particle whose density matrix has become approximately, but not completely, diagonal in the position basis. In particular, the off-diagonal elements will generally have much smaller norm than they do initially. The claim is that each of the Schmidt states have the same position variance as the full state while the pointer states have much less.
  The Schmidt states can be spatially delocalized while the pointer states are not because the eigenstates of a density matrix are not continuous with the distance between density matrices. That is, two density matrices can be arbitrarily close together while having eigenstates that are always very different. For instance, the maximally mixed state of the qubit can be written
  $\rho = \frac{1}{2}\vert 0\rangle\langle 0 \vert+\frac{1}{2}\vert 1\rangle\langle 1\vert = \frac{1}{2}\vert +\rangle\langle +\vert +\frac{1}{2}\vert -\rangle \langle - \vert$
  with $\vert 0\rangle$ and $\vert 1\rangle$ orthonormal, and $\vert \pm\rangle = (\vert 0\rangle \pm \vert 1\rangle)/\sqrt{2}$ . Then we can define the perturbations
  $\rho_1 = (1-\epsilon)\rho + \epsilon \vert 1\rangle\langle 1\vert,\qquad \rho_+ = (1-\epsilon)\rho + \epsilon \vert +\rangle\langle +\vert.$
  The states $\rho_1$ and $\rho_+$ are arbitrarily close for small $\epsilon$ , but their eigenbases are always maximally different ( $\{\vert 0\rangle,\vert 1\rangle\}$ and $\{\vert +\rangle,\vert -\rangle\}$ , respectively).
  I’m not sure if the Gaussian case discussed by Page also exploits this instability of the Schmidt basis near points of degeneracy, or if the effect is different. I went back to his paper, but it looked non-trivial to fill in the procedure he sketches on page 7 and 8.