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I got a little puzzled when thinking about two entangled fermions.

Say that we have a Hilbert space in which we have two fermionic orbitals $a$ and $b$. Then the Hilbert space $H$'s dimension is just $4$, since it is spanned by \begin{align} \{ |0\rangle, c_a^\dagger |0\rangle, c_b^\dagger |0\rangle, c_a^\dagger c_b^\dagger |0\rangle\}, \end{align} where $c_i^\dagger$ are the fermionic operators that create a fermion in orbital $i$.

Say we have a state $c_a^\dagger c_b^\dagger |0\rangle$. Then if I partition my Hilbert space into two by looking at the tensor product of the Hilbert spaces of each orbital, i.e. $H = H_a \otimes H_b$, then my state can be written as $c_a^\dagger |0\rangle_a \otimes c_b^\dagger |0\rangle_b$, from which it is obvious that this state is unentangled ($|0\rangle = |0\rangle_a \otimes |0\rangle_b$).

Now I was thinking about writing the state in first quantized i.e. a wavefunction. Let $\phi_a(r), \phi_b(r)$ be the wavefunctions of the orbitals $a$ and $b$. Then \begin{align} \psi(x_1,x_2) = \langle x_1 x_2 | c_a^\dagger c_b^\dagger |0 \rangle = \phi_a(x_1) \phi_b(x_2) - \phi_a(x_1)\phi_b(x_2). \end{align} This is where I got confused. What object is $\psi(x_1,x_2)$, i.e. what Hilbert space does it belong to? What exactly are we doing when we do $\langle x_1 x_2 | c_a^\dagger c_b^\dagger |0 \rangle$? We seem to be changing/expanding our Hilbert space by taking the position representation?

Written in this way, and assuming the same partition $H_a \otimes H_b$, the unentangled nature of the original state is no longer manifest. I'm not sure what the partition $H_a \otimes H_b$ even means in this context. Would that be saying $\psi(x_1, x_2) = \psi_a(x_1, x_2) \times \psi_b(x_1,x_2)$ where $\psi_i(x_1,x_2)$ is a linear combination of $\phi_i(x_1), \phi_i(x_2)$? This does not seem right to me.

Regardless, now I have a state written in two different but supposedly equivalent ways, with the same partition of the Hilbert space, yet it is unentangled in one way and entangled in the other.


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You cant partition a two-fermion sector of the Fock space as tensor product - underlying states are always anti-symmetriezed –  Slaviks Apr 25 '13 at 4:41
@ Slaviks : sorry, I didn't quite get what you meant. Are you saying $H = H_a \otimes H_b$ is wrong? but that's how you build up multi-fermion Fock states. each operator $c_i^\dagger$ acts in a Hilbert space which is just the vacuum and an excitation, and so for $N$ fermions $H = \bigotimes_{i=1}^N H_i$ ($2^N$ possible states). Here i'm just considering $N = 2$. –  nervxxx Apr 25 '13 at 6:34
@Slaviks - if you apply a Jordan-Wigner transformation you immediately get a tensor-product-decomposition into 2-dimensional Hilbert spaces. I think this question is an ongoing debate in the literature. There are different approaches, since it appears not to be particularly easy to deal with the (anti-)symmetrization. In my view, the answer must be found by considering the result of measurements and their correlations and therefore (probably) factorization properties of one- and two-body density matrices in the simplest cases. See e.g. arXiv:0902.1684 and PRA 67, 024301 (2003) and their refs. –  S. Gammelmark Apr 25 '13 at 8:03
@nervxxx I see what you mean. I'm saying that occupation number decomposition $H_1 \bigotimes H_2$ is not the same as (the impossible) decomposition into the tensor product of single-particle Hilbert spaces. But need to think more, especially in light of S. Gammelmark's comment. –  Slaviks Apr 25 '13 at 8:32
Indeed, there is much literature around discussing the fact that the antisymmetrization that you automatically get in multi fermion states shouldn't be considered as operationally useful "entanglement" e.g. Zanardi et al state that it's not useful to discuss entanglement without specifying the manner in which one can manipulate and probe its constituent physical degrees of freedom. In this sense entanglement is always relative to a particular set of experimental capabilities. –  twistor59 Apr 25 '13 at 9:36
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3 Answers 3

Let me remind you that the Fock space of multiple fermions is defined to be the antisymmetric (fermionic) subspace of the full Fock space

$$ \Gamma_a=\bigoplus_{n=0}^{\infty}H^{\wedge n}, $$

where $\wedge$ stands for the antisymmetric tensor product

$$ v_1\wedge\ldots\wedge v_n=\frac{1}{n!}\sum_{p\in P_n}\sigma_p v_{p(1)}\wedge\ldots\wedge v_{p(n)}. $$

Here $\sigma_p$ is the sign of the permutation $p$ in the group of permutations $P_n$.

Thus the confusion here comes from the fact that $c_a^{\dagger}c_b^{\dagger}|0\rangle\neq|ab\rangle$ as you seem to state.

Recall that the creation and annihilation operators are defined within the occupation number representation, i.e. $c_a^{\dagger}c_b^{\dagger}|0\rangle=|11\rangle$, where the first number denotes the number of fermions in state $a$ while the second denotes the number of fermions in state $b$. On the other hand, states written in the occupation number representation are defined to be properly antisymmetrized (for fermions) many-body basis states, as forced upon us by the particle indistinguishability. Therefore they are defined within the fermionic Fock space. Any textbook will show so, take a look at the first chapter of Many-Body Quantum Theory in Condensed Matter Physics: An Introduction by Bruus and Flensberg for example. For two fermions described via a single particle basis $\{|a\rangle,|b\rangle\}$ one possible choice is:

$$ |11\rangle=\frac{1}{\sqrt{2}}\left(|ab\rangle-|ba\rangle\right). $$ Therefore

$$ c_a^{\dagger}c_b^{\dagger}|0\rangle=\frac{1}{\sqrt{2}}\left(c_a^{\dagger}|0\rangle_1\otimes c_b^{\dagger}|0\rangle_2-c_b^{\dagger}|0\rangle_1\otimes c_a^{\dagger}|0\rangle_2\right) $$

The familiar anticommutativity of these operators is now obvious from this from

$$ c_b^{\dagger}c_a^{\dagger}|0\rangle=\frac{1}{\sqrt{2}}\left(c_b^{\dagger}|0\rangle_1\otimes c_a^{\dagger}|0\rangle_2-c_a^{\dagger}|0\rangle_1\otimes c_b^{\dagger}|0\rangle_2\right)=-c_a^{\dagger}c_b^{\dagger}|0\rangle $$

In fact one of the great advantages of creation and annihilation operators is that they include the antisymmetry (for fermions) of the wave function implicitly.

Dotting this with $\langle x_1x_2|$ we obtain:

$$ \langle x_1x_2|c_a^{\dagger}c_b^{\dagger}|0\rangle=\frac{1}{\sqrt{2}}\left(\phi_a(x_1)\phi_b(x_2)-\phi_b(x_1)\phi_a(x_2)\right). $$

Thus there is no inconsistency, both representations show that the particles are entangled.

On the other hand dotting $\langle x_1x_2|$ with $c_a^{\dagger}|0\rangle_1\otimes c_b^{\dagger}|0\rangle_2$ would simply produce

$$ \psi(x_1,x_2)=\phi_a(x_1)\phi_b(x_2) $$

Therefore there is no inconsistency here also, however, as I said, the important thing to remember is that

$$ c_a^{\dagger}c_b^{\dagger}|0\rangle\neq c_a^{\dagger}|0\rangle_1\otimes c_b^{\dagger}|0\rangle_2 $$

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No, that is not correct. With the choice of partitioning of the Hilbert space I have specified in the problem, the two subsystems are unequivocally unentangled. I cite, for example, the use of the orbital cut in the entanglement spectrum of QH states (go look it up) - in there, IQH states are unentangled because they have the form $|\psi\rangle = \prod_i c_i^\dagger |0\rangle$. –  nervxxx Jun 3 '13 at 17:39
Next, $c_a^{\dagger}c_b^{\dagger}|0\rangle=\frac{1}{\sqrt{2}}\left(c_a^{\dagger}|0 \rangle_1 \otimes c_b^{\dagger}|0\rangle_2-c_b^{\dagger}|0\rangle_1\otimes c_a^{\dagger}|0\rangle_2\right)$ is wrong. The multiparticle Fock state vacuum is built up from the tensor product of individual vacuums $\prod_i |0\rangle_i$. Then a Fock state representation arises from acting the creation operators on this: $c_a^\dagger c_b^\dagger |0\rangle = |0\rangle_i \cdots c_a^\dagger |0\rangle_a \otimes \cdots c_b^\dagger |0\rangle_b \cdots|0\rangle_j$ –  nervxxx Jun 3 '13 at 17:43
The anticommutativty arises because of the canonical anticommutation relations $\{ c_i, c_j^\dagger \} = \delta_{ij}$. This then results in $c_a^\dagger c_b^\dagger |0\rangle = -c_b^\dagger c_a^\dagger |0\rangle$. the whole point of the Fock state number representation is that it treats the particles immediately as indistinguishable and does not have an unphysical label (the positions, $x_1, x_2, \cdots$) for each particle like a first quantized notation (wavefunction) would. The first quantized notation is really cumbersome and awkward, because what it does –  nervxxx Jun 3 '13 at 17:47
is to say that each particle is distinguishable with position labels $x_1, x_2, \cdots$, but then it enforces the rule that they're actually indistinguishable and should be odd under a swap of labels, leading to the Slater determinant / antisymmetrization scheme. So, -1. –  nervxxx Jun 3 '13 at 17:49
@nervxxx, I have edited my answer to further explain my point. However I would also like you to provide me with concrete references, since "go look it up" is not one. –  mgphys Jun 4 '13 at 8:57
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I'm posting a modified version of my comment as an answer, as more people will see it this way.

I think the confusion hinges crucially on what kind of partitioning you're doing. The $\nu=1$ QH state is pure under orbital partitioning, but not under "particle partitioning". Maybe arXiv:0905.4204 will help. IIRC, they work out a simple example about this detail, in the 2nd section.

@nervxxx, Your 2-particle state might be pure under orbital partitioning, but it is entangled under particle partitioning. Due to the antisymmetrization, it looks like a singlet Bell state.

So the bottomline is that entanglement is completely dependent on how you choose to partition your system. The subtlety is not widely appreciated. For a nuanced discussion, see this article http://rspa.royalsocietypublishing.org/content/463/2085/2277.full

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Yes, thanks. That's what I concluded eventually. But what irks me (and the reason for the whole long exchange above) is that almost everyone else doesn't understand the Fock space (with single particle Hilbert space of dim $M$) is isomorphic to the tensor product of $M$ orbital spaces. One can see this from looking at the dimension of the Hilbert spaces - both are $2^M$, and one has a 1-1 between the two spaces, so they are isomorphic! That implies that one can actually decompose the Fock space into a tensor product - except that because this isomorphism only works for the full spaces, –  nervxxx Sep 21 '13 at 5:32
one cannot decompose the singlet wavefunction into a tensor product as $c_a^\dagger | 0 \rangle \otimes c_b^\dagger | 0 \rangle$ and represent it as some $f(x_1) \otimes g(x_2)$. –  nervxxx Sep 21 '13 at 5:35
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We have to be careful with the bra-ket formalism and its meaning. Unlike $|x_1\rangle$, I am not sure that the notation $|x_1 x_2\rangle$ where $x_1$ and $x_2$ are positional coordinates makes any sense. In literature [1] the notation $|ab\rangle$ designates the Slater determinant or Hartree-Fock state, i.e.:

$$|ab\rangle=c_a^{\dagger}c_b^{\dagger}|0\rangle=\phi_a(x_1) \phi_b(x_2)-\phi_a(x_2) \phi_b(x_1)$$

My feeling is that your confusion is related to mixing of the occupation numbers formalism and real space representation.

[1] Szabo, Ostlund, "Modern quantum chemistry: introduction to advanced electronic structure theory"

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If somebody will explain meaning standing behind |r1 r2> notation, I will be very grateful. –  freude Apr 25 '13 at 12:11
$\newcommand\ket[1]{\left|#1\right>} \ket{x_1x_2}$ is a common shorthand for $\ket{x_1}\otimes\ket{x_2}$. In general, what is written inside a ket is just a label: its meaning is heavily context dependent. Mixing the two representations is not uncommon, as long as there is no confusion. –  Frédéric Grosshans Apr 26 '13 at 6:04
ok, I totally agree with that. But let us concern a particular case from the question above, i.e. the case when $x_1$ and $x_2$ are quantum numbers labeling the eigenstates of the positional operator $\hat{x}$. What is the meaning of decomposing $\ket{x_1}\otimes\ket{x_2}$ ? Also, I am curious if there is any references containing such notation for coordinate space. It would be interesting to read more about. –  freude Apr 26 '13 at 7:16
I untastood $x_1$ and $x_2$ as labelling the position of the two fermions. But then, of course the ket $\ket{x_1x_2}$ is not antisymmetrized and does not correspond to an allowed state. Only $\ket{x_1x_2} -\ket{x_2x_1}$ is a valid state. –  Frédéric Grosshans Apr 26 '13 at 7:29
With Fermions, the situation is complicated by the fact that you impose a global symmetry. $\newcommand\ket[1]{\left|#1\right>}$ $\ket{x1x2}\ket{\uparrow\uparrow}$ is not a valid state, but $\ket{x1x2}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}$ is –  Frédéric Grosshans Apr 26 '13 at 10:13
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