In what sense is the Wigner function a quasiprobability distribution?

For the upteenth time I have read a paper introducing the Wigner function essentially like this:

The Wigner-representation of a quantum state \rho is a real-valued function on phase space definedActually, they usually use a more confusing definition. See my post on the intuitive definition of the Wigner function.a   (with \hbar=1) as

(1)   \begin{align*} W_\rho(x,p) \equiv \int \! \mathrm{d}\Delta x \, e^{i p \Delta x} \langle x+\Delta x /2 \vert \rho \vert x-\Delta x /2 \rangle. \end{align*}

It’s sort of like a probability distribution because the marginals reproduce the probabilities for position and momentum measurements:

(2)   \begin{align*} P(x) \equiv \langle x \vert \rho \vert x \rangle = \int \! \mathrm{d}p \, W_\rho(x,p) \end{align*}

and

(3)   \begin{align*} P(p) \equiv  \langle p\vert \rho \vert p \rangle = \int \! \mathrm{d}x \, W_\rho(x,p). \end{align*}

But the reason it’s not a real probability distribution is that it can be negative.

The fact that W_\rho(x,p) can be negative is obviously a reason you can’t think about it as a true PDF, but the marginals property is a terribly weak justification for thinking about W_\rho as a “quasi-PDF”. There are all sorts of functions one could write down that would have this same property but wouldn’t encode much information about actual phase space structure, e.g., the Jigner“Jess” + “Wigner” = “Jigner”. Ha!b   function

    \[J_\rho(x,p) \equiv P(x)P(p) = \langle x \vert \rho \vert x \rangle \langle p \vert \rho \vert p \rangle,\]

which tells as nothing whatsoever about how position relates to momentum.

Here is the real reason you should think the Wigner function W_\rho is almost, but not quite, a phase-space PDF for a state \rho:

  1. Consider an arbitrary length scale \sigma_x, which determines a corresponding momentum scale \sigma_p = 1/2\sigma_x and a corresponding setNot just a set of states, actually, but a Parseval tight frame. They have a characteristic spatial and momentum width \sigma_x and \sigma_p, and are indexed by \alpha = (x,p) as it ranges over phase space.c   of coherent states \{ \vert \alpha \rangle \}.
  2. If a measurement is performed on \rho with the POVM of coherent states \{ \vert \alpha \rangle \langle \alpha \vert \}, then the probability of obtaining outcome \alpha is given by the Husimi Q function representation of \rho:

    (4)   \begin{align*} Q_\rho(\alpha) = \langle \alpha \vert \rho \vert \alpha \rangle. \end{align*}

  3. If \rho can be constructed as a mixture of the coherent states \{ \vert \alpha \rangle \}, thenOf course, the P function cannot always be defined, and sometimes it can be defined but only if it takes negative values.
[continue reading]

State-independent consistent sets

In May, Losada and Laura wrote a paperM. Losada and R. Laura, Annals of Physics 344, 263 (2014).a   pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained historiesGell-Mann and Hartle usually use the term “fine-grained set of histories” to refer to a set generated by the finest possible partitioning of histories in path integral (i.e. a point in space for every point in time), but this is overly specific. As far as the consistent histories framework is concerned, the key mathematical property that defines a fine-grained set is that it’s an exhaustive and exclusive set where each history is constructed by choosing exactly one projector from a fixed orthogonal resolution of the identity at each time.b  ). Let the elementary histories \alpha = (a_1, \dots, a_N) be defined by projective decompositions of the identity P^{(i)}_{a_i}(t_i) at time steps t_i (i=1,\ldots,N), so that

(1)   \begin{align*} P^{(i)}_a &= (P^{(i)}_a)^\dagger \quad \forall i,a \\ P^{(i)}_a P^{(i)}_b &= \delta_{a,b} P^{(i)}_a \quad \forall i,a,b\\ \sum_{a} P^{(i)}_a (t_i) &= I \quad  \forall i,k \\ C_\alpha &= P^{(N)}_{a_N} (t_N) \cdots P^{(1)}_{a_1} (t_1) \\ I &= \sum_\alpha C_\alpha = \sum_{a_1}\cdots \sum_{a_N} C_\alpha \\ \end{align*}

where C_\alpha are the class operators. Then Losada and Laura showed that the following two conditions are equivalent

  1. The set is consistent“Medium decoherent” in Gell-Mann and Hartle’s terminology. Also note that Losada and Laura actually work with the obsolete condition of “weak decoherence”, but this turns out to be an unimportance difference. For a summary of these sorts of consistency conditions, see my round-up.c   for any state: D(\alpha,\beta) = \mathrm{Tr}[C_\alpha \rho C_\beta^\dagger] = 0 \quad \forall \alpha \neq \beta, \forall \rho.
  2. The Heisenberg-picture projectors at all times commute: [P^{(i)}_{a} (t_i),P^{(j)}_{b} (t_j)]=0 \quad \forall i,j,a,b.

However, this is not as general as one would like because assuming the set of histories is elementary is very restrictive. (It excludes branch-dependent sets, sets with inhomogeneous histories, and many more types of sets that we would like to work with.) Luckily, their proof can be extended a bit.

Let’s forget that we have any projectors P^{(i)}_{a} and just consider a consistent set \{ C_\alpha \}.… [continue reading]

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.… [continue reading]

How to think about Quantum Mechanics—Part 2: Vacuum fluctuations

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

Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding“Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper”, although I’m sure there’s another term for this used by philosophers of science.)a  .

Here is everything you need to know:

  1. A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
  2. Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160]b  
  3. “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
  4. A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.
[continue reading]

Lindblad Equation is differential form of CP map

The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.Here’s an example of a memory effect: An atom immersed in an electromagnetic field can be in one of two states, excited or ground. If it is in an excited state then, during a time interval, it has a certain probability of decaying to the ground state by emitting a photon. If it is in the ground state then it also has a chance of becoming excited by the ambient field. The situation where the atom is in a space of essentially infinite size would be Markovian, because the emitted photon (which embodies a record of the atom’s previous state of excitement) would travel away from the atom never to interact with it again. It might still become excited because of the ambient field, but its chance of doing so isn’t influenced by its previous state. But if the atom is in a container with reflecting walls, then the photon might be reflected back towards the atom, changing the probability that it becomes excited during a later period.a   Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.

Given an arbitrary orthonormal basis L_n of the space of operators on the N-dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product \langle A,B \rangle = \mathrm{Tr}[A^\dagger B]), the Lindblad equation takes the following form:

(1)   \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\left(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\right) , \end{align*}

with \hbar=1.… [continue reading]

Potentials and the Aharonov–Bohm effect

[This post was originally “Part 1” of my HTTAQM series. However, it’s old, haphazardly written, and not a good starting point. Therefore, I’ve removed it from that series, which now begins with “Measurements are about bases”. Other parts are here: 1,2,3,4,5,6,7,8. I hope to re-write this post in the future.]

It’s often remarked that the Aharonov–Bohm (AB) effect says something profound about the “reality” of potentials in quantum mechanics. In one version of the relevant experiment, charged particles are made to travel coherently along two alternate paths, such as in a Mach-Zehnder interferometer. At the experimenter’s discretion, an external electromagnetic potential (either vector or scalar) can be applied so that the two paths are at different potentials yet still experience zero magnetic and electric field. The paths are recombined, and the size of the potential difference determines the phase of the interference pattern. The effect is often interpreted as a demonstration that the electromagnetic potential is physically “real”, rather than just a useful mathematical concept.


The magnetic Aharanov-Bohm effect. The wavepacket of an electron approaches from the left and is split coherently over two paths, L and R. The red solenoid in between contains magnetic flux \Phi. The region outside the solenoid has zero field, but there is a non-zero curl to the vector potential as measured along the two paths. The relative phase between the L and R wavepackets is given by \Theta = e \Phi/c \hbar.

However, Vaidman recently pointed out that this is a mistaken interpretation which is an artifact of the semi-classical approximation used to describe the AB effect.… [continue reading]

A dark matter model for decoherence detection

[Added 2015-1-30: The paper is now in print and has appeared in the popular press.]

One criticism I’ve had to address when proselytizing the indisputable charms of using decoherence detection methods to look at low-mass dark matter (DM) is this: I’ve never produced a concrete model that would be tested. My analysis (arXiv:1212.3061) addressed the possibility of using matter interferometry to rule out a large class of dark matter models characterized by a certain range for the DM mass and the nucleon-scattering cross section. However, I never constructed an explicit model as a representative of this class to demonstrate in detail that it was compatible with all existing observational evidence. This is a large and complicated task, and not something I could accomplish on my own.

I tried hard to find an existing model in the literature that met my requirements, but without luck. So I had to argue (with referees and with others) that this was properly beyond the scope of my work, and that the idea was interesting enough to warrant publication without a model. This ultimately was successful, but it was an uphill battle. Among other things, I pointed out that new experimental concepts can inspire theoretical work, so it is important that they be disseminated.

I’m thrilled to say this paid off in spades. Bateman, McHardy, Merle, Morris, and Ulbricht have posted their new pre-print “On the Existence of Low-Mass Dark Matter and its Direct Detection” (arXiv:1405.5536). Here is the abstract:

Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB).

[continue reading]

Diagonal operators in the coherent state basis

I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.I posted the answer on Physics.SE too since they encourage the answering of one’s own question. How lonely is that?!?a  

Q: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?
A: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.

The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which X, P, and X+P are all approximately diagonal in the basis of wavepackets, but RXR^\dagger is not, where R is the unitary operator which maps

(1)   \begin{align*} \vert x \rangle \to (\vert x \rangle + \mathrm{sign}(x) \vert - x \rangle) / \sqrt{2}. \end{align*}

(This operator creates a Schrodinger’s cat state by reflecting about x=0.)

For two different coherent states \vert \alpha \rangle and \vert \beta \rangle, we want to require an approximately diagonal operator A to satisfy \langle \alpha \vert A \vert \beta \rangle \approx 0, but we only want to do this if \langle \alpha \vert \beta \rangle \approx 0. For \langle \alpha \vert \beta \rangle \approx 1, we sensibly expect \langle \alpha \vert A \vert \beta \rangle to be within the eigenspectrum of A.

One might consider the negativity of the Wigner-Weyl transformCase has a pleasingly gentle introduction.b   of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]