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What is the meaning of the entries of a density matrix?
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Density matrices are a convenient way of describing states of quantum
systems in contact with an environment. (State vectors = wave functions
are appropriate only for isolated systems at zero absolute temperature,
though they can be used in an approximate way in thermally isolated
contexts. But contact with an environment means positive temperature.)
If the quantum system has only a finite number n of levels,
the density matrix is an n x n matrix; otherwise it is
a linear operator on Hilbert space (but nevertheless called a matrix).
The real use for density matrices is to compute expectations
= trace (rho f)
for quantities f of interest. Indeed, rho is just a collection of
numbers enabling one to calculate these expectations.
The fact that the constant 1 must have expectation 1 leads to the
restriction that
sum_k rho_kk = trace rho = 1.
Apart from that, rho must be a Hermitian, positive semidefinite matrix,
to satisfy the requirements of statistics. (See quant-ph/0303047 for
details.) For small systems, all such density matrices can indeed be
approximately realized in practice.
Since diagonal entries of a semidefinite matrix are always nonnegative,
the p_k:=rho_kk are nonnegative numbers summing to 1 and thus look like
probabilities. What the components mean depends on the basis used.
In particluar, if the basis consists of eigenstates of a Hamiltonian,
and the eigenvalues E_k are all nondegenerate, a diagonal element
rho_kk can be interpreted as the probability that upon measuring the
energy of the system one will find the value E_k.
If f is a function of the Hamiltonian H, and the basis used consists of
eigenstates |k> of H, with H|k>=E_k|k> then the density matrix rho
has entries rho_jk = . If one now calculates the expectation
of a function f(H), the equation f(H)|k>=f(E_k)|k> implies that
= trace (rho f(H)) = sum_k
= sum_k = sum_k f(E_k)
= sum_k rho_kk f(E_k).
If we average the results f(E) of a number of measurements of the
energy, where the energy E_k is measured with probability p_k,
we get
= sum_k p_k f(E_k).
Thus, to match the expectations no matter which function we are
averaging, we need to take p_k=rho_kk. This gives the claimed
probability interpretation of the diagonal entries.
Off-diagonal elements have no simple interpretation.
Usually one does not look at off-diagonal elements at all, but they
are important in intermediate steps of calculations.
Close to absolute zero temperature, and assuming the absence of
degeneracy, (but also in certain other, well prepared nearly
isolated systems), quantum state have the property that all columns
of the density matrix are nearly parallel to a wave function psi
that is conventionally normalized to have norm 1,
psi^*psi=1.
(In Dirac language, this says =1; see the FAQ entry for bras
and kets.). This vector psi, which is clearly determined only up to a
complex number of absolute value 1, is called the wave vector
(or, in infinite dimensions, the wave function) of the state.
Idealizing this situation, one describes such quantum systems by states
in which all columns of the density matrix are exactly parallel to some
nonzero wave vector psi. (Such matrices are called rank 1 matrices;
the wave vector, also referred to as a wave function, is defined
only up to a phase factor.)
Then the k-th column is a multiple c_k psi of psi. The fact that rho
is Hermitian forces each row to be a multiple of psi^*. But this implies
that c_k is a multiple of phi^*_k, so that rho is a multiple of
psi psi^*. Since psi is normalized, the multiplication factor is just
the trace, and since the trace is 1 we find
rho = psi psi^* for any rank 1 density matrix.
If we now calculate the probability of measuring the energy E_k, we find
p_k = rho_kk = = = ,
and since is just the complex conjugate of ,
we end up with
p_k = ||^2.
This is Born's squared amplitude formula for calculating probabilities.
Thus one sees that the traditional wave vector calculus is just a
special case of the density matrix calculus, appropriate (only) for
the study of tiny, well-prepared nearly isolated systems and for
systems close to zero absolute temperature. For the study of ordinary
matter under ordinary conditions, one needs to represent states
by density matrices.
Everything that is done with wave vectors can also be done with
density matrices, or equivalently with the associated expectation
mapping. Indeed, everything becomes simpler that way, much closer
to classical mechanics, and much less weird-looking.
See quant-ph/0303047 for an exposition of the foundations of quantum
mechanics (including the probability interpretation, uncertainty
relations, nonlocality, and Bell's theorem) in terms of expectations.