Quantum mechanics is the best-confirmed theory of particle dynamics today. Not only is it the basis for all digital technologies, it also serves as the theoretical foundation for our best-confirmed theories of matter (quantum field theories). On the other hand, since its inception, it has been beset with conceptual problems. In particular, there is no current consensus on just how to interpret it: What would the world be like, if it were true? In this course, students will examine the foundations of quantum mechanics theory from a conceptual and mathematical perspective and then review two proposals as to how it should be interpreted: the Ghirardi, Rimini and Weber (GRW) Interpretation and the Many Worlds Interpretation. A central part of the course will be devoted to conceptual issues surrounding quantum information theory and current applications such as quantum teleportation, quantum computing and quantum cryptography.

This course is geared toward students with college-level background in mathematics and minimal background in physics or philosophy; however the topics covered will be of interest to students in the physical and information-theoretic sciences as well as philosophy students.

#### Course Objectives

- To gain an understanding of the type of experimental results that motivated the development of quantum mechanics
- To examine the mathematical formalism of quantum mechanics and how it differs from classical mechanics
- To learn the difference between a classical bit and a quantum bit (qubit) and how quantum mechanics can be applied to cryptography
- To describe two procedures that allow quantum mechanical systems to densely encode and "teleport" information and the difference between a classical computation and a quantum computation
- To gain understanding of two popular proposals for understanding what quantum mechanics says about the world

#### Meet the Instructor

Jonathan Bain, PhD

Associate Professor of Philosophy of Science

NYU Polytechnic School of Engineering

**Module 1. Motivation: The 2-Slit and 2-Path Experiments**

*Objective: * To gain an understanding of the type of experimental results that motivated the development of quantum mechanics. To be discussed:

- Why is it hard to interpret the results of Davisson and Germer's 2-Slit experiment strictly in terms of wave phenomena? Why is it hard to interpret their results strictly in terms of particle phenomena?
- If electrons are like bar magnets with north and south poles, what should we expect when we send a beam of electrons through a Stern-Gerlach apparatus?
- Why can't we build a "Color and Hardness" box that would measure both Color and Hardness, if all we have to work with are individual Color and Hardness boxes as components?
- What does it mean to say Color and Hardness are not correlated?
- In what sense does a Hardness measurement "disrupt" the Color property?

**Required Reading**

*The Structure and Interpretation of Quantum Mechanics.* Hughes, R. I. G. Harvard, pp. 1-8; 1989.
*Quantum Mechanics and Experience.* Albert, D. Harvard, pp. 1-16; 1992.

**Links/Supplemental Reading**

**Module 2. Formalism: Vectors, Vector Spaces and Operators**

*Objective:* To examine the mathematical formalism of quantum mechanics and how it differs from classical mechanics. To be discussed:

- What motivates the use of vectors to represent states in quantum mechanics?
- Explain what the Eigenvalue-Eigenvector Rule states.
- How would you expand the state of a soft electron to measure its Color?
- According to the Born Rule, what is the probability that a measurement of the Color of a soft electron would return the value black?
- Suppose you measure the Color of a soft electron and obtain the value black. According to the Projection Postulate, what is its state after the measurement? What is the probability that a second Color measurement would return the value black?

**Required Reading**

*Quantum Mechanics and Experience. *Albert, D. Harvard, pp. 17-43; 1992.

**Links/Supplemental Reading**

**Module 3. Applications: Qubits versus ***C*-bits, Quantum Cryptography

**Objective: **To learn the difference between a classical bit and a quantum bit ('qubit'), and an understanding of how quantum mechanics can be applied to cryptography. To be discussed:

- What are the essential differences between a qubit and a classical bit?
- How might you argue that quantum information is not essentially different from classical information? How might you argue that quantum information is essentially different from classical information?
- Why does a literal interpretation of superpositions entail that a qubit encodes an arbitrarily large amount of information, of which only 1 classical bit's worth of information is accessible?
- What is meant by the claim that unknown qubits cannot be cloned?
- In the protocol for distributing a secret key using non-orthogonal states of quantum systems, what is the random element associated with Alice's encoding procedure and Bob's decoding procedure? Why is this random element important?

**Required Reading**

**Links/Supplemental Reading**

**Module 4. Applications: Dense Coding, Teleportation and Quantum Computation**

*Objective:* To describe two procedures that allow quantum mechanical systems to densely encode and "teleport" information and the difference between a classical computation and a quantum computation. To be discussed:

- What is the goal of quantum dense coding, and why is it paradoxical on a literal interpretation of qubits?
- What does the use of entangled states in dense coding and teleportation indicate about what it means to transmit information?
- In quantum teleportation, is the unknown qubit that Bob receives
*exactly* the same unknown qubit that Alice "teleported" to him? Would you submit yourself to a quantum teleporter (given a sufficiently complex machine were to be developed based on the same fundamental principles)? Why or why not?
- Why are transformations on qubits "reversible"? Why is this an initial problem in implementing a classical computer using qubits?
- In what sense can a qubit based function calculator perform an arbitrarily large number of computations in a single step?

**Required Reading**

**Links/Supplemental Reading**

**Module 5. Interpretation: The Measurement Problem, GRW and Many Worlds**

*Objective:* To gain understanding of two popular proposals for interpreting what quantum mechanics is telling us about the world. To be discussed:

- In what sense is the Measurement Problem a conflict between the Projection Postulate and the Schrödinger Dynamics?
- How does the GRW Theory attempt to solve the Measurement Problem?
- What is the problem of wavefunction tails for GRW?
- How does the Many Worlds Interpretation attempt to solve the Measurement Problem?
- In the Many Worlds Interpretation, all possible outcomes of every physical interaction always occur. Does knowledge of all these myriad worlds abrogate our moral responsibilities in this world?

**Required Reading**

*Quantum Mechanics and Experience.* Albert, D. Harvard, pp. 73-133; 1992.

**Links/Supplemental Reading**