The matter of separability has connections to classical mechanics. In some choice of cotangent bundle frame the coordinates and momenta q_i and p_i define the dynamics of a system according to a Hamiltonian H(p_i, q_i)
dq_i/dt = ∂H/∂p_i, dp_i/dt = -∂H/∂q_i.
We may then write the dynamics according to a general X_i = (q_i, p_i) in the phase space as
dX_i/dt = Ω_{ij}∂_jH
as a Hamiltonia flow. dX_i/dt is a tangent vector to a Hamiltonian flow. The Hamiltonian is defined according to the position and momenta variables for i = 1, ..., n. The phase space then has 2n dimensions. This differential equation is then a constraint on this system, so the Hamiltonian projects these 2n dimensions to 2n-1 dimensions on the contact manifold of solution.
The Hamiltonian is related to a function of the position variables q_i and their derivatives dq_i/dt = q'_i, called the Lagrangian, L = L(q_i, q'_i). The Lagrangian is defined on half of the phase space as a result. The relationship between the Lagrangian and the Hamiltonian is that
L = p_iq'_i - H(q_i, p_i)
This is a Legedre tranformation which maps the configuration variables of the Lagrangian to the Hamiltonian. For the configuration space a manifold M the cotangent bundle T*M is then the vector flow on the phase space. This is always a symplectic structure. A symplectic two-form is
Ω = Ω_{ij}dp_i/\dq_j
The Hamiltonian vector flow then defines Poisson brackets. In general a cotangent bundle T*M on M is symplectic, however, a symplectic structure is not always a cotangent bundle.
Quantum mechanics may now be slipped into the picture. The symplectic matrix Ω_{ij} has a pseudo-complex structure. It is a block off diagonal matrix of I_n unit matrices
Ω =
|0 I_n|
|-I_n 0|
with the feature that
Ω^2 = -I_{2n}
which has features common to i^2 = -1. The lifting of this structure to a Kahler structure gives geometric quantization. I am not going to delve into that complicated math-physics, but what it does is to replace the Poisson brackets {q_i, p_j} -- > [q_i, p_j] with the commentators of quantum mechanics. The odd thing is that this Kahler structure (a complex symplectic structure) does not permit one to access all the variables. In fact you can only access half of them --- those which pertain to the configuration variables of the Lagrangian, or a Fourier transform to the momentum variables.
This permits a particle system with some configuration variables q_i, to have a total quantum state that is an entanglement product where sub-states share these variables. The configuration variables are not unique to a single particle, but because there is this underlying phase associated with the other n variables you don't access there is then no locality to the configuration variables. For n particles in separable states, there are n configuration variables. However, if two of these variables share the same phase, the particles associated with them may have 2 less degrees of freedom. This means a quantum system with n particles can have far fewer configuration variables or degrees of freedom than just n for each particle.
A spin system has in the basis of the Pauli matrix σ_z the states |+> and |-> for spin up and down. The Pauli matrix σ_z acts on these states as
σ_z|±> = ±|±->.
Now these states are complex numbers, which means there are 2 variables for each state and thus 4 altogether. However, there are constraints, such as the probability Born rule 1 = P_+ + P_-, P_± = |a_±|^2 for a state |ψ> = a_+|+> + a_-|->, and irrelevance of a phase in real valued measurements. So this reduces the number of variables from 4 to 4 - 2 = 2. That is just what we would expect.
Now let us consider two spin systems, say two electrons. The use of electron spin state is not concrete, for these arguments hold just as well for polarization direction of photons. So we have two sets of states and operators {σ_z, |±>}^1 {σ_z, |±>}^2 denoted with an additional index I = 1,2 and we still have
σ ^i_z|±>^i = ±|±->^i.
We can form two independent states |ψ>^i = a^i_+|+>^i + a^i_-|->^i for the two spin systems. For each there are 4 variables and 2 constraints. This gives 4 degrees of freedom in total. Yet we can compose these spin states in various ways. One way of doing this is
|ψ> = (1/sqrt{2})(|+>|-> + e^{iω}|->||+>),
where I have dropped the index i, and we just implicitly see the first and second |±> as i = 1 and 2. This makes reading things clearer. The e^{iω} is a phase which for it equal + and - the state is not an eigenstate of σ ^i and is an eigenstate of σ ^i respectively. So these are singlet and triplet state configurations. I probably should not have mentioned this, but it does have some subtle implications. This is an entangled state. If you have access to |±>}^1 then you also have access to |±>}^2, and this holds no matter how far apart these states end up as. You can entangle two electrons by overlapping their wave functions. One that is done you can separate them arbitrarily far and they are still entangled.
Now let us count the degrees of freedom for this state. We have again 4 variables for each |±>}^i but now we have one constraint from Born rule and another from the "mod-out" of phases. So you have 6 independent variables. Now if you are Alice your part of the EPR pair (this entangled states between spins) you have half of these variables which is 3. This is more information than with just having access to a single spin locally with 2 variables. So something funny is going on. If you attempt to access this information as spin up or down there is then an additional variable you have no information about, thus the state of the system is undetermined and Alice can't access the quantum bits which may be teleported by Bob from his part of the entangled state. The additional bit of information needed is the manner by which Bob has selected his eigenstates, or the orientation of the Stern Gerlach apparatus (or apparatus appropriate for the observables measured. This is the key which needs to be transmitted by Bob to Alice. This must be communicated as classical information.
Cheers LC