Quantum Networks Reading Group #1

Posted on by mathiashdb

These notes are the first set of summaries of the University of Washington Aeronautics and Astronautics Quantum Networks Reading Group. The first few posts will be summaries of basic quantum mechanics, roughly following “A Introduction to Quantum Control and Dynamics” by d’Alessandro. This is an excellent book that is readable to any control theorist or quantum physicist with a passing knowledge of the other discipline. This post will cover the mathematical basics of quantum mechanics, such as Hilbert spaces and BraKet notation. Next week, we will cover quantum dynamics, including the Schrodinger equation and the measurement postulate with some illustrative examples.

A Hilbert space \mathcal{H} is a complex vector space with an operation (\cdot,\cdot)\to\mathbb{C} with the following properties for x,y,z\in\mathcal{H} and \alpha\in\mathbb{C}:

  • (y,x)=(x,y)^*
  • (x+y,z)=(x,z)+(y,z)
  • (\alpha x,y) = \alpha^*(x,y)
  • (x,x)\geq 0
  • (x,x)=0~\iff~x=0

Note that the  third property says that the Hilbert space is antilinear in its first argument. This is a notational difference between physicists and mathematicians, who tend to define Hilbert spaces as antilinear in the second argument. I’m not sure if control theorists care, but analysts tend to shake their heads at physicists for this. I certainly recommend not caring as opposed to getting embroiled in such a conflict, even more so if you are one of those people that tend to prefer one of Vim or Emacs over the other a little too vigorously.

A Hilbert space is called separable if and only if there exists a dense, countable set \{x_j\} such that

(x_i, x_j) = \delta_{ij}

where \delta_{ij} is the Kronecker delta (0 if i\neq j,~1 else). This set is the basis of the space. Hilbert spaces are also complete, in that any Cauchy sequence on \mathcal{H} converges with respect to (\cdot,\cdot). For completeness (pun perhaps intended) a Cauchy sequence \{y_i\} is said to converge if for every \epsilon>0, there exists an N such that if n,m>N

\sqrt{(y_n-y_m,y_n-y_m)}<\epsilon.

For the purposes of quantum control, we usually consider finite-dimensional Hilbert spaces, where the dimension refers to the number of basis vectors. Quantum mechanics doesn’t preclude the use of infinite-dimensional Hilbert spaces, but typically when you are trying to control a physical quantum system a lot of the higher-energy states become negligible and you can reduce your analysis to a few low-energy states. If this previous sentence doesn’t make sense, worry not; we will make sense of it next week.

Dirac was an early quantum physicist who among other things, developed an interesting notation for physicists. Einstein once remarked that his own greatest contribution to physics (perhaps this is an urban legend) was his development of Einstein tensor notation, which makes tensor manipulation almost trivially easy , and so developing nice notation is nothing to snicker at. In Dirac notation, a state vector in the Hilbert space \mathcal{H} is denoted with the Ket symbol |\psi\rangle. For each Ket, there is a corresponding Bra \langle \phi |. The algebraic quantities we get out of this notation are:

  • State vector: |\psi\rangle
  • Inner product: \langle \phi |\psi \rangle
  • Outer product :| \psi \rangle \langle \phi |

The functional analyst will notice that the bra is a functional: it exists in the dual space of \mathcal{H}, and pairs with kets to map to \mathbb{C}. More explicitly, if we expand a state vector ket as

|\psi\rangle = \sum_j a_j |x_j\rangle,

the the dual element is given by

\langle\psi| = \sum_j a_j^*\langle x_j|.

The beauty of this notation will start to make sense once we actually do some computations with it starting next week.

The next thing we will discuss is the idea of spin, which here we will touch on only briefly. Spin is a really cool and deep property of particles, in particular because it has no real classical analogue. Next week we will discuss the Stern-Gerlach experiment which essentially lead to the discovery of spin. Spin is related to angular momentum, and it behaves very similarly to angular momentum on the quantum level. The spin of a particle takes on discrete values; either integer or half-integer values. A particle is said to have integer spin l\in\mathbb{Z} if the spin can take values

-l,(-l+1),\dots,(l-1),l.

Similarly, a particle is said to have half-integer spin l/2,~l\in\mathbb{Z}, if the spin takes values

-l/2,\dots,-1/2,1/2\dots,l/2.

Many common particles have half-integer spin 1/2. Since the particle can take two values of spin, -1/2 and 1/2, the corresponding Hilbert space is two dimensional. Let us choose the basis of \mathcal{H} to be such that one basis vector |0\rangle corresponds to spin up (1/2), and the other vector |1\rangle corresponds to spin down (-1/2). The state vector can be written as

|\psi \rangle = \cos\left(\dfrac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\dfrac{\theta}{2}\right)|1\rangle,

where I will fully admit that the parameters \theta,\phi are completely unmotivated in context, but I can tell you that they correspond to the angles on something called the Bloch sphere. The Bloch sphere is a 3-D sphere in which spin exists, and what I didn’t tell you before was that the state vector above was for spin along the z-axis of this Bloch sphere. We will discuss the Bloch sphere in plenty of detail later on. For now, just absorb the fact that you can arbitrarily define Hilbert space basis vectors based on whatever you want them to be (in this case they are states corresponding to individual values of spin), and that you can write the state of a particle in terms of these vectors. In the coming weeks, we will discuss exactly how to do this and when it is a good idea to do so.

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