## Quantum Entanglement from Feedback Control

Now André Carvalho from the physics department at Australian National University in Canberra is talking about “Quantum feedback control for entanglement production”. He’s in a theory group with strong connections to the atom laser experimental group at ANU. This theory group works on measurement and control theory for Bose-Einstein condensates and atom lasers.

The good news: recent advances in real-time monitoring allows the control of quantum systems using feedback.

The big question: can we use feedback to design the system dynamics to produce and stabilize entangled states?

Start by considering two atoms in a cavity, interacting with a laser. Think of each atom as a 2-state system — so the Hilbert space of the pair of atoms is $\mathbb{C}^2 \otimes \mathbb{C}^2$

We’ll say what the atoms are doing using not a pure state (a unit vector) but a mixed state (a density matrix). The atoms’ time evolution will be described by Lindbladian mechanics. This is a generalization of Hamiltonian mechanics that allows for dissipative processes — processes that increase entropy! A bit more precisely, we’re talking here about the quantum analogue of a Markov process. Even more precisely, we’re talking about the Lindblad equation: the most general equation describing a time evolution for density matrices that is time-translation-invariant, Markovian, trace preserving and completely positive.

As time passes, an initially entangled 2-atom state will gradually ‘decohere’, losing its entanglement.

But next, introduce feedback. Can we do this in a way that makes the entanglement become large as time passes?

With ‘homodyne monitoring’, you can do pretty well. But with ‘photodetection monitoring’, you can do great! As time passes, every state will evolve to approach the maximally entangled state: the ‘singlet state’. This is the density matrix $| \psi \rangle \langle \psi |$

corresponding to the pure state $|\psi \rangle = \frac{1}{\sqrt{2}} (\uparrow \otimes \downarrow - \downarrow \otimes \uparrow)$

So: the system dynamics can be engineered using feedback to product and stabilize highly entangled state. In fact this is true not just for 2-atom systems, but multi-atom systems! And at least for 2-atom systems, this scheme is robust against imperfections and detection inefficiencies. The question of robustness is still under study for multi-atom systems.

For more details, try:

• A. R. R. Carvalho, A. J. S. Reid, and J. J. Hope, Controlling entanglement by direct quantum feedback.

Abstract:
We discuss the generation of entanglement between electronic states of two atoms in a cavity using direct quantum feedback schemes. We compare the effects of different control Hamiltonians and detection processes in the performance of entanglement production and show that the quantum-jump-based feedback proposed by us in Phys. Rev. A 76 010301(R) (2007) can protect highly entangled states against decoherence. We provide analytical results that explain the robustness of jump feedback, and also analyse the perspectives of experimental implementation by scrutinising the effects of imperfections and approximations in our model.

How do homodyne and photodetection feedback work? I’m not exactly sure, but this quote helps:

In the homodyne-based scheme, the detector registers
a continuous photocurrent, and the feedback Hamiltonian
is constantly applied to the system. Conversely, in
the photocounting-based strategy, the absence of signal
predominates and the control is only triggered after a
detection click, i.e. a quantum jump, occurs.

### 8 Responses to Quantum Entanglement from Feedback Control

1. John Baez says:

By the way, I think the way the Lindblad equation is usually written is quite annoying. The first term is fine, but the second term, that big fat sum, is disgusting. I’m sure that with the correct algebraic concepts one could write down this second term in a more conceptual way. But I’m not sure what these concepts are!

The elements called $L_n$ are a basis of the traceless matrices on our Hilbert space $\mathbb{C}^N$ — that is, a basis of $\mathfrak{sl}(N,\mathbb{C})$. So, presumably the relevant algebra involves $\mathfrak{sl}(N,\mathbb{C})$. However, what’s going on with the expression $\rho L_m^\dagger L_n - L_m L_n^\dagger \rho + 2 L_n \rho L_m ?$

where $\rho$ is a positive matrix of trace 1? Are we dealing with some sort of representation of $\mathfrak{sl}(N,\mathbb{C})$ on the vector space of self-adjoint operators, or something?

Or maybe a representation of $\mathfrak{sl}(N^2,\mathbb{C})$?

• Eric says:

I wish you didn’t ask this question. This is the type of question that gets stuck in your head (like a song you can’t kick) and you can’t rest until you solve it :)

My gut tells me it’s coming from some form of inner product.

• John Baez says:

Eric wrote:

I wish you didn’t ask this question.

Heh. Someone knows the answer already, I’m sure. Let’s hope they tell us.

• Jacob Biamonte says:

Ladies and gents: on a related note, there is a great short review article by Nielsen which shows techniques that can be used to linearise the Lindblad equation: check out Roth’s lemma.

I think category theorists will like to see Roth’s lemma, as it’s the linear algebra behind sliding boxes around wires, operator vector duality (called map-state duality in quantum computing) and all that other great stuff. They define operations such as vec(|j>|k>) = |k>|j>, etc.

A friend of mine James Whitfield visited and learned some string diagram theory by presenting the diagrammatics behind this review paper.

• John Baez says:

It seems strange to speak of ‘linearizing’ the Lindblad equation, because it’s already obviously a linear differential equation (see here).

It also seems odd, from a mathematician’s viewpoint, to define an operation ‘vec’ which takes a matrix and thinks of it as a vector, since the space of matrices is quite obviously a vector space.

It similarly seems odd to prove that the map $B \mapsto A B C$

is linear, where $A, B, C$ are matrices, because again this is quite well-known.

But I’m sure there’s also lots of good stuff in this paper.

• Jacob Biamonte says:

The main reason is to solve the Lindblad equation as a matrix eigenvalue equation using a computer program.

Yeah, there is some funny stuff there, but I still think worth the read.

• Tim van Beek says:

This is an opportunity to learn how me talking about climate science looks like to experts, I think that’s comparably funny :-)

• John Baez says:

You’re asking great questions, Tim. And I see only one climate science expert in the discussion of “week304”: Nathan. Maybe there are more lurking around… but I haven’t seen them posting yet.

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