Our goals for this site are to provide adaptable resources to support faculty, including those very new to these topics, who wish to add some student-centered, research-based QIS instruction to your classes.
We have materials for two different audiences, depending on the type of course you teach.
QM Course: Are you teaching quantum mechanics in a physics department to (mostly) majors, and want to add a short unit on quantum information (e.g. quantum computing or cryptography) within your course?
And: if you teach undergraduate quantum mechanics to physics students and are looking for a suite of student-centered materials to help you with the entire QM course, please visit our Adaptable curricular materials for Quantum Mechanics site.
Who is this for? An instructor teaching Quantum Mechanics who would like to add some quantum information science topics to their physics course.
(Materials are suitable for faculty new to these topics.)
Topics: We have divided our materials into three main topics.
Each is standalone - materials can be taught without any prerequisite material from the other groups. (Because of this, there are some duplicate materials in and across the groups.)
The material within each group is ordered so that the material early in that group may be used later in that same group. However, we do not intend for the materials to be used wholly as written. Rather, they will comprise a complete set that a faculty member can pick and choose from as it fits their schedule, the material already taught, and the interests of both the faculty member and student.
The “instructional materials” found on this site include: lecture notes for faculty, concept tests, homework questions, online tutorials, and assessments.
Please use and adapt whatever is helpful to you, however it will most benefit your students. Please credit our work if you share your materials beyond your own classes. Please make an effort to keep assessment materials off the open web - alter questions for your students.
Curriculum designers Dr. Steven Pollock and Dr. Gina Passante have each taught this material in their classes. Feel free to download our lecture notes - we are happy for you to use whatever material works for your class.
Please email if you try our materials and have any feedback:
steven.pollock (at) colorado.edu or gpassante (at) fullerton.edu
I teach entanglement and EPR midway through a first semester junior-level quantum mechanics course following a spins-first curriculum (using McIntyre's Quantum Mechanics textbook).
I teach qubits and teleportation late in the second semester (senior-level) quantum course, as a basic introduction to quantum information science.
Each sequence takes one week. These are all taught in "large lecture" (75+ student) settings with 3 50-minute lectures/week.
Please reach out if you have questions or feedback: steven.pollock (at) colorado.edu
CSUF is a primarily undergraduate, large, Hispanic-serving institution. I teach these materials in several of my quantum mechanics courses at the undergraduate and master's levels.
Feel free to reach out if you have questions (gpassante at fullerton dot edu).
Superposition of states in quantum mechanics is hard to think about classically - what does it mean to be both "0 and 1" (or "up" and "down", "alive" and "dead")? Many questions related to the interpretation of such a state (is this a statement about reality, or about our knowledge? Is there a difference?) were treated as philosophical for a long time. After Bell's theoretical work and then following experimental (Nobel prize-winning) efforts, we now have answers to some of these questions.
In this module, we provide an introduction to some "spooky" aspects of quantum mechanics arising from 2-particle entanglement. We work in the context set up by Einstein, Podolsky and Rosen, the so-called "EPR Paradox," and conclude with an example of a Bell inequality and an experimental test for hidden variables. In our courses, this module takes 1 week to cover.
We have written brief notes on the topics in this module to help bring non-expert faculty up to speed. View them by expanding the titles below, or download them as a single pdf file (coming soon).
Topics:
The experimental context for this module is a gedanken experiment (since realized in the lab) building on ideas from Einstein, and outlined by David Mermin in a 1985 Physics Today article: "Is the Moon There when Nobody Looks?" It is described in the language of 2-particle entangled states of spin-1/2 objects. The setup is designed so two distant observers make simultaneous measurements on their particles (each part of an entangled pair), and the correlations of outcomes they measure would lead Einstein (or anyone believing that widely separated particles cannot influence each other) to argue that the measurable physical properties (outcomes) must already be there BEFORE a measurement is made. Einstein, Podolsky, and Rosen called these "elements of reality." We now know that experiments disagree with this - and indeed, the non-classical outcomes seem to imply "spooky action at a distance."
Prerequisites for students:
The required formalism is largely described in the sections below, but to follow the notation, students need to have some prerequisite skills, including familiarity with Dirac notation for single-particle states, e.g. $\ket{+}$ for a spin-up state (in the Z-basis). (Through this section, we may casually switch notations e.g. to $\ket{\uparrow}$ or $\ket{0}$, if this makes a particular question easier to state or discuss.)
In the downloadable lecture notes, we make use of a setup where measurement of a spin-1/2 object can be made along an arbitrary axis pointing in the $\hat{n}$ direction. We work with measurements constrained to the x-z plane. Students are thus expected to know the following (and if not, you will need to teach this before this unit):
Physicists continue to debate "interpretation" of quantum states, but this broader module is designed to introduce students to Bell tests and to present the experimental evidence that shows entangled states do not imply "local hidden variables" (or what Einstein called "elements of reality" in his EPR paper).
We thus begin our unit on entanglement (leading to an understanding of the EPR paradox) with a brief discussion of how one interprets a superposition state as philosophically distinct from a mixed state. We do not go into the mathematics of mixed states, as it is not needed for our discussion.
Both superposition and mixed states yield definite probabilities for measurement outcomes, but they are not the same: superposition states contain relative phase information which can lead to distinct experimental outcomes. (In a very simplistic way, one might say that a superposition is "this AND that" while a mixed state is "this OR that" with given probabilities. )
This section is also a good time to talk about the different language you are using to describe different states and what is interpretation vs experimental outcome.
To talk about entanglement, we need to introduce the notation for referring to two particles. We use the tensor product ($\otimes$) to 'join' two states (and also to join two Hilbert spaces).
The state $\ket{0} \otimes \ket{1} = \ket{01}$ (note that we often suppress the $\otimes$ symbol) is a two-particle state where the first particle is in the state $\ket{0}$ and the second particle is in the state $\ket{1}$.
Two-particle states that can be written as $\ket{\psi} \otimes \ket{\phi}$ are straightforward. For example, we can have $\frac{1}{\sqrt{2}} ( \ket{0} - \ket{1}) \otimes \frac{1}{\sqrt{3}}(\ket{0} + \sqrt{2} \ket{1})$ that can be expanded out as $\frac{1}{\sqrt{6}} ( \ket{00} + \sqrt{2}\ket{01} - \ket{10} - \sqrt{2} \ket{11})$.
Entangled states are defined as states that CANNOT be written as $\ket{\psi} \otimes \ket{\phi}$.
An example of an entangled state is $\frac{1}{\sqrt{2}} (\ket{00} + \ket{11})$. The example concept tests below help show the weirdness of entangled states.
Consider an experiment with a spin-0 source producing pairs of spin-1/2 particles in quantum state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$. This state (or equivalently, simply the fact that we start with a spin-0 system and must conserve total angular momentum) ensures that any measurement by observer #1 of the component of spin along any given direction will be perfectly anticorrelated with a measurement by observer #2 in the same direction.
How can we explain such perfect anticorrelation for well-separated observers?
A natural idea is that the particles must carry with them information about what outcome will be measured from the state. This is referred to as "local hidden variables". Einstein argued in his 1935 EPR paper that such information (or "elements of reality") must exist in order to explain perfect anticorrelations if the observers are very far apart. Otherwise, such anticorrelation would have to be an (absurd?) "spooky action at a distance".
In a highly simplified Gedanken experiment, as shown below, the observers agree to make only one of 3 choices for axis direction, axes $\hat{a}$, $\hat{b}$, or $\hat{c}$, oriented $120^{\circ}$ apart (all in the plane of the page.) The decision of what direction to measure along is made randomly, at the same time, immediately before the measurement. This is done to ensure that observers #1 and #2 cannot communicate to each other what their measurement direction choices are.
(Note: directions ($\hat{a}$, $\hat{b}$, $\hat{c}$) all lie in the plane of the page, it is easy to incorrectly misinterpret the diagram above as showing 3-D right handed coordinates)
The existence of local hidden variables would mean that each particle carries an instruction set of what would be measured along any of these three direction choices. In this experiment, with $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$, the two particles must have "opposite" instruction sets to ensure perfect anticorrelation. These hidden variables need to exist before any measurement is made and include instructions for measurements along all measurement directions.
The experiment repeats many times, and the outcome ($+$ or $-$) for each particle is recorded (but not the axis of measurement). We compute the probability after many trials that the two observers record the SAME outcome (both $+$ or both $-$). By enumerating all possibilities, we quickly establish that Prob(same) $\le$ 4/9, no matter what the distribution of "hidden instruction sets" might be. This is a Bell inequality; it arises simply from counting. It must hold for any local hidden-variable theory.
More details are found in the next section and our downloadable notes.
References: The original EPR paper is quite readable (Phys. Rev. 47, 777 (1935)), but we find the suggested experimental context of position and momentum harder for our students than the spin version. The setup we use follows Mermin's (excellent) paper "Is the moon there when nobody looks?"
The previous section on Hidden Variables set up an example of a "Bell test." An initial 2-particle state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{+-} - \ket{-+})$ is created, and well-separated observers Alice and Bob each measure one of the two particles. They choose to measure spin along one of three randomly-selected axes, $120^{\circ}$ apart. Each individual measurement is recorded only as $+$ or $-$ (they do not record the axis measured). After many measurements, they compare outcomes and determine Probability(same outcome).
A quantum calculation begins by computing Prob($++$) in the special case that observer #1 measures in the z-direction, and observer #2 is in any of the (three) $\hat{n}$ directions:
$Prob(\#1 \ is\ +\hat{z}, \#2\ is +\hat{n}) = |\ \bra{+}\ \bra{+_n}\ \ket{\psi}\ |^2$
Using $\ket{+_n} = \cos{\theta/2}\ket{+} + \sin{\theta/2} \ket{-}$, we find this probability is $\frac{1}{2}\sin^2{\theta/2}$ (see lecture notes for more details) Adding Prob(--) = Prob(++), we get (for this situation):
Prob(same) = $\sin^2{\theta/2}$ .
Averaging over the other possibilities for measurement combinations (e.g. that $\hat{n}$ can be oriented at $\theta = 0, \pm 120^{\circ}$ with equal likelihood), we conclude Prob(same) = 0.5
This outcome has been well-tested experimentally. This violation of the "Bell inequality" (Prob(Same) $\le$ 4/9) experimentally validates quantum theory and is explicitly inconsistent with local hidden variable theories.
Reference:
"Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons ", Marissa Giustina et al., Phys. Rev. Lett. 115, 250401 (2015)
Below are a collection of clicker questions and homework questions that we developed for teaching these materials. These materials can be adapted for use in your classroom, including transitioning some questions to group whiteboard activities or group exams.
(For more information on how we have implemented these types of materials in our upper-division classes, see our related AJP paper.)
Relevent Acephysics.net Tutorial (which can be done in class, alone or in groups, or at home alone) (Note: You must set up a login to Acephysics once before viewing)
EPR and entangled states - this Tutorial has questions about spin-0 Bell states and entanglement. It is framed in a context of cryptography, but does not assume any class instruction or background on quantum cryptography.
Introductory QIS courses are often taught to students from several different academic backgrounds. There will be some students from physics, computer science, engineering, math, and even chemistry backgrounds. This set of materials is intended to be a deep conceptual practice on some of the fundamental ideas in quantum information science. Some students may find these materials too easy, while others may find them too hard. That is exactly the reason for their existence! The goal is to bring all students to the same level of understanding that will set the foundation for future QIS learning.
The activities are designed to take between 30-60 minutes and can be used either in class (with paper versions) or assigned for homework (interactive online versions). If you are interested in using the online versions email us at hello[at]acephysics[dot]net to set up a page for your course where you will have access to student completion data.
Preliminary online tutorials can be found at https://acephysics.net/ (and linked directly below). These online tutorials are suitable for homework or in-class group work.
Note: By design, these materials do not cover complex (or even intermediate) QIS topics. We have chosen to focus on those materials most important at the beginning of a QIS course. These topics happen to be "physics heavy" topics, and ones for which our group's educational and research expertise is particularly well-matched.
This activity is a first step in learning about quantum computing, introducing qubit states, Dirac and matrix notation, measurement probabilities, and single-qubit X, Z, H, and I gates.
There is no prerequisite knowledge required for this activity, but previous classroom introduction to basic ideas and notation of quantum states (including Dirac and matrix notation) is useful.
There is not currently a demo page for this tutorial (coming soon!), but you can view the Tutorial using the student link by entering an email address. This will store your answers so that you can return to the activity later.
If you are assigning this activity in your course, email us at hello[at]acephyics[dot]net to get a course page where you can access student completion information.
Alternatively, you may use a paper version of the activity in your class. (download below)
Quantum Circuit Diagrams - practice with single-qubit gates represented as circuit diagrams.
Prerequisite knowledge: Basic gates and quantum states (i.e. the previous tutorial "Introduction to Quantum Gates")
There is not currently a demo page for this tutorial (coming soon!), but you can view the Tutorial using the student link by entering an email address. This will store your answers so that you can return to the activity later.
If you are assigning this activity in your course, email us at hello[at]acephyics[dot]net to get a course page where you can access student completion information.
Alternatively, you may use a paper version of the activity in your class. (download below)
Helping students describe systems with multiple qubits
Prerequisites: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions. (i.e. the sections above, on this page) There is not currently a demo page for this tutorial (coming soon!), but you can view the Tutorial using the student link by entering an email address. This will store your answers so that you can return to the activity later.
If you are assigning this activity in your course, email us at hello[at]acephyics[dot]net to get a course page where you can access student completion information.
Alternatively, you may use a paper version of the activity in your class. (download below)
An introduction to the Controlled NOT (CNOT) gate and the related concept of entanglement.
Prerequisites: Students should be familiar with quantum states, basic single-qubit gates and circuit diagram conventions, and the tensor product. (i.e. the sections above, on this page)
There is not currently a demo page for this tutorial (coming soon!), but you can view the Tutorial using the student link by entering an email address. This will store your answers so that you can return to the activity later.
If you are assigning this activity in your course, email us at hello[at]acephyics[dot]net to get a course page where you can access student completion information.
Alternatively, you may use a paper version of the activity in your class. (download below)
This activity takes students through the BB84 quantum key distribution protocol. It uses quantum circuit notation and includes the effect of an eavesdropper.
Prerequisite knowledge: Basic single qubit gates (in particular, the Hadamard, H gate). Students should know how to predict probabilistic outcomes of measurements in the computational (Z) basis on superpositions of $|0\rangle, |1\rangle, |+\rangle, |-\rangle$ states.
Access a demo version of the activity (First page only; responses not saved)
Here is a full version of the Tutorial. (Initial login to acephysics.net required)
If you are assigning this activity in your course, email us at hello[at]acephyics[dot]net to get a course page where you can access student completion information.
We are PER research faculty teaching at very different institutions - different in class sizes and setups, student demographics, institutional research-focus - but all interested in helping introduce undergraduates to basic elements of Quantum Information Science. We have all taught a variety of quantum courses for many years
The materials you will find here are not meant to be taken as givens, this is not a "fixed curriculum" that you are supposed to fully adopt (or reject). We hope that you will be inspired by some of the activities, notes, concept-tests, homeworks and more, and will borrow and adapt them for your own situation and students. We do not all use exactly the same materials ourselves.
We have borrowed where we can from PER literature on Quantum Mechanics (and tried - but apologize up front where we occasionally have failed - to appropriately credit the the hard development work of others!) We do not claim that these materials are "Research-validated" (they are still under development!), but cheerfully present sometimes half-baked or partially-tested materials that we might argue are "research-based", a vaguer but perhaps more realistic description.
We welcome feedback and suggestions. If you make significant changes or additions, and particularly if you have classroom evidence that suggests it works well - let us know. We hope someday this site will be flexible enough to allow for community-sharing of new resources, and will work towards making that happen. (See contact information at the bottom of the page)
CONTACT INFORMATION:
This page was built by Steven Pollock (steven.pollock (at) colorado.edu), Gina Passante (gpassante (at) Fullerton.edu), and Bethany Wilcox (bethany.wilcox (at) colorado.edu). Contact us with questions or feedback.
We are funded in part by NSF DUE- 2012147 and 2011958: Collaborative Research: Connecting Spins-First Quantum Mechanics Instruction to Quantum Information Science
PLEASE USE AND ADAPT whatever is helpful to you, however it will most benefit your students. Please credit our work if you share your materials beyond your own classes. Please make an effort to keep assessment materials off the open web - alter questions for your students.
©2022 University of Colorado Boulder
and California State University, Fullerton
Funded by the National Science Foundation
grants 2011958 and 2012147