diff --git a/lec_19_quantum.md b/lec_19_quantum.md
index 2eb67f8..edf19ec 100644
--- a/lec_19_quantum.md
+++ b/lec_19_quantum.md
@@ -187,9 +187,9 @@ relatively short resource that contains essentially everything we discuss here.
 See also this [blog post of Aaronson](http://www.scottaaronson.com/blog/?p=208) for a high level explanation of Shor's algorithm which ends with links to several more detailed expositions.
 See also [this lecture](http://www.scottaaronson.com/democritus/lec14.html) of Aaronson for a great discussion of the feasibility of quantum computing (Aaronson's [course lecture notes](http://www.scottaaronson.com/democritus/default.html) and the [book](http://www.amazon.com/Quantum-Computing-since-Democritus-Aaronson/dp/0521199565) that they spawned are fantastic reads as well).
 
-__States:__ We will consider a simple quantum system that includes $n$ objects (e.g., electrons/photons/transistors/etc..) each of which can be in either an "on" or "off" state - i.e., each of them can encode a single _bit_ of information, but to emphasize the "quantumness" we will call it a _qubit_.
-A _probability distribution_ over such a system can be described as a $2^n$ dimensional vector $v$ with non-negative entries summing up to $1$, where for every $x\in\{0,1\}^n$, $v_x$ denotes the probability that the system is in state $x$.
-As we mentioned, quantum mechanics allows negative (in fact even complex) probabilities and so a _quantum state_ of the system can be described as a $2^n$ dimensional vector $v$ such that $\|v\|^2 = \sum_x |v_x|^2 = 1$.
+__States:__ We will consider a simple quantum system that includes $n$ objects (e.g., electrons/photons/transistors/etc.) each of which can be in either an "on" or "off" state - i.e., each of them can encode a single _bit_ of information, but to emphasize the "quantumness" we will call it a _qubit_.
+A _probability distribution_ over a classical system can be described as a $2^n$ dimensional vector $v$ with non-negative entries summing up to $1$, where for every $x\in\{0,1\}^n$, $v_x$ denotes the probability that the system is in state $x$.
+As we mentioned, quantum mechanics allows negative (in fact even complex) probability amplitudes and so a _quantum state_ of the system can be described as a $2^n$ dimensional unit norm vector $v$ of amplitudes such that $\|v\|^2 = \sum_x |v_x|^2 = 1$.
 
 __Measurement:__ Suppose that we were in the classical probabilistic setting, and that the $n$ bits are simply random coins.
 Thus we can describe the _state_ of the system by the $2^n$-dimensional vector $v$ such that $v_x=2^{-n}$ for all $x$.