# I Does quantum entanglement depend on the chosen basis?

#### vanhees71

Gold Member
Ok, I can live with that definition although I think it's not very clear.

First of all, "subsystem" can also refer to two (compatible) quantities for a single particle (as in the example of the SG experiment, where one spin component, usually $\sigma_z$, and position are compatible observables), which define an orthonormal basis of common eigenvectors of the corresponding self-adjoint operators.

In this general sense for me still entanglement refers to entanglement with respect to a clearly given set of observables, and also the surprising implications of entanglement (or to put it in Einstein's words, inseparability) are in terms of "stronger-than-classical" correlations of observables on a quantum system.

#### martinbn

Ok, I can live with that definition although I think it's not very clear.
Do you have a reference for a different definition?

#### PeterDonis

Mentor
"subsystem" can also refer to two (compatible) quantities for a single particle
Yes. Subsystems are best viewed as subsets of the degrees of freedom of the quantum system as a whole, not "particles".

which define an orthonormal basis of common eigenvectors of the corresponding self-adjoint operators.
You don't need to choose a basis in order to pick out subsystems this way. For example, you don't need to choose which direction you are measuring spin or which basis you want to use for configuration space (position/momentum) in order to pick out spin and configuration as different subsystems. Nor do you need to choose a basis to know whether or not those subsystems are entangled.

#### vanhees71

Gold Member
I stumble already over slide 4 on the first cited sources:

The general ket in $\mathcal{H}_A \otimes \mathcal{H}_B$ is
$$|Psi \rangle = \sum_{ij} c_{ij} |u_i \rangle \otimes |v_j \rangle,$$
where the $|u_i \rangle$ and $|v_j \rangle$ are CONSs of $\mathcal{H}_A$ and $\mathcal{H}_B$. The notation on these slides don't make sense, as becomes clear if the dimension of the two Hilbert spaces are not the same. One example is my beloved Stern-Gerlach experiment, where $\mathcal{H}_A$ is the infinite dimensional (separable) Hilbert space realizing the Heisenberg algebra of position and momentum operators and $\mathcal{H}_B$ the $(2s+1)$-dimensional Hilbert space describing spin for particles with spin $s \in \{1/2,1,\ldots \}$.

If we stick to the definition that a pure state is non-entangled (I prefer to say it's separable) iff it can be written in terms of a product state
$$|\Psi \rangle = |\psi \rangle \otimes |\phi \rangle.$$
Then the decomposition in the arbitrary product basis reads
$$|\Psi \rangle = \sum_{ij} \psi_i \phi_j |u_i \rangle \otimes v_j \rangle.$$
That means in this sense of entanglement, a state is separable iff
$$c_{ij}=\psi_i \phi_j,$$
i.e., iff the coefficients are products of complex numbers depending only on $i$ and $j$ respectively.

A basis-independent characterization then should be: A pure state is separable iff the reduced state of either subsystem is itself a pure state again. From now on I write $|u_i,v_j \rangle$ for $|u_i \rangle \otimes |v_j \rangle$. Then indeed one has
$$\hat{\rho}_A=\mathrm{Tr}_B \hat{\rho}=\sum_{i,j,k} \langle u_i,v_j|\hat{\rho}|u_k,v_j \rangle |u_i \rangle \langle u_k|.$$
For the above product state we indeed have
$$\hat{\rho}_A=\sum_{i,j,k} \langle u_i,v_j|\psi,\phi \rangle \langle \psi,\phi|u_k,v_j \rangle |u_i \rangle \langle u_k| = \sum_{i,j,k} \psi_i \psi_k^{*} |\phi_j|^2 |u_i \rangle \langle u_k|=|\phi \rangle \langle \phi|.$$
The other direction is proven in Ballentine Quantum Mechanics: If the partial trace of a pure state of a composite system is pure then necessarily this state is separable.

In other words: within this definition a pure state is entangled if its partial trace (on either of the subsystems) is mixed.

The paper by Sasaki et al seems to use my notion of entanglement, while their reference 3 (the RMP by Horodeki$^4$) takes the more general definition discussed above. I still think it's more clear to state entanglement with respect to a concrete measurement on subsystems. On the other hand the more general definition is easier to state (at least for pure states).

#### martinbn

Can you write down one definition, from these papers, that is basis dependent. I only leafed through the papers, but couldn't see anything of the sort.

Gold Member

#### microsansfil

Thank for the link. Now, just like Martinbn, I don't see where it is explicitly developed that an entangled pure state can be Basis-dependent.

The concept of entanglement is much more difficult to capture for a mixed state then for a pure state. Further, the mixed-state decomposition is not unique. Moreover, in realistic systems, a pure state inevitably falls into a mixed state.

If i understand the definition of entanglement given by vanhees71 : a pure state ρ ∈ HA⊗HB is entangled if and only if the reduced state ρA is not pure. Equivalently, ρ is separable if and only if ρA is pure. Here ρA is the reduced state defined as ρA≡TrBρ. However, this definition says nothing about whether or not a entanglement state could be Basis-dependent.

/Patrick

#### atyy

Thank for the link. Now, just like Martinbn, I don't see where it is explicitly developed that an entangled pure state can be Basis-dependent.

The concept of entanglement is much more difficult to capture for a mixed state then for a pure state. Further, the mixed-state decomposition is not unique. Moreover, in realistic systems, a pure state inevitably falls into a mixed state.

If i understand the definition of entanglement given by vanhees71 : a pure state ρ ∈ HA⊗HB is entangled if and only if the reduced state ρA is not pure. Equivalently, ρ is separable if and only if ρA is pure. Here ρA is the reduced state defined as ρA≡TrBρ. However, this definition says nothing about whether or not a entanglement state could be Basis-dependent.
It's probably clearer to say "measurement setup" than "basis". Anyway, the Sasaki et al reference is pretty close to what vanhees71 has been saying (as he himself noted). They give explicit examples in section V. You can also find comments in section 1.2.4 of https://arxiv.org/abs/1302.4654.

#### zonde

Gold Member
First of all, "subsystem" can also refer to two (compatible) quantities for a single particle (as in the example of the SG experiment, where one spin component, usually $\sigma_z$, and position are compatible observables),
I'm not sure I understand your definition of "subsystem". In SG experiment you measure only position. If spin is another "subsystem" how do you even perform it's measurement?

#### martinbn

It's probably clearer to say "measurement setup" than "basis".
Well, the thread is about whether the definition of entanglement depends on the choice of basis.
You can also find comments in section 1.2.4 of https://arxiv.org/abs/1302.4654.
A dissertation at the Budapest University of Technology and Economics is hardly the definitive reference. Nevertheless, in section 1.2.4 they do not discuss dependence on basis. They discuss the factorization of the Hilbert space into a product of two spaces. My guess is that you are confusing that with the choice of basis.

#### vanhees71

Gold Member
Well, I think that the confusion is due to the fact that there seem to be the two slightly different definitions of "entanglement". Having read a bit in the nice review paper by the Horodecki family,

https://doi.org/10.1103/RevModPhys.81.865
https://arxiv.org/abs/quant-ph/0702225

I come to the conclusion that indeed the stronger assumption that a state is considered on-entangled (separable) iff the state doesn't factorize into a direct product, $\hat{\rho} = \hat{\rho}_A \otimes \hat{\rho}_B$ (for the case that one consideres the factorization in only two distinct subsystems (bipartite entanglement)). This is a basis-independent statement, but it's not a simple task to figure out for a given state whether it's entangled or not (except for a pure state for the entire system as discussed in #30). The Horodeckis seem to be the experts having investigated this question in great detail and for the most general cases. So the above cited RMP article seems to be pretty authorative.

BTW I'd say that a dissertation from a well-respected university can be considered a definitive reference since it's usually as much peer reviewed (if not with more care) as a usual peer-reviewed paper in a well-respected scientific journal. In the case of the dissertation quoted above, this is the more credible since it seems to be based on several publications in peer-reviewed well-respected scientific journals.

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