Ph CS 219A Lecture 2 Density Operator

Density Operators and Open Quantum Systems

1. Why Study Open Quantum Systems?

• A closed quantum system is perfectly isolated and follows the usual rules of quantum mechanics.
• An open system interacts with an environment that we cannot fully observe or control. Real laboratory systems—including quantum computers—are always open to some degree.
• These interactions can cause decoherence, which contributes to errors in quantum computation. Understanding decoherence lays the groundwork for studying quantum error correction.
• A useful model is to treat the system and its environment together as one closed system, then describe only the part of interest.

2. Review of the Rules for Closed Systems

The lecture reviews the basic mathematical framework:

• States are rays in a complex Hilbert space. A state vector is usually normalized to length one; multiplying it by an overall phase does not change the physical state. Relative phases in a superposition do matter.
• Observables are Hermitian operators. They can be expressed as sums of real eigenvalues times orthogonal projectors.
• Measurements follow the Born rule: the probability of an outcome is the squared length of the state’s projection into the corresponding eigenspace. After measurement, the state is projected into that eigenspace and normalized.
• Time evolution follows the Schrödinger equation and is unitary for a closed system.
• Composite systems are described by tensor-product Hilbert spaces.

The contrast between deterministic unitary evolution and probabilistic measurement is part of the measurement problem in quantum mechanics. The lecture notes the issue but does not explore it in depth.

3. Qubits, Measurement Bases, and Interference

• A qubit is a two-dimensional quantum system, with basis states conventionally written (|0\rangle) and (|1\rangle).
• The Pauli operators (\sigma_x, \sigma_y, \sigma_z) provide common observables. Measuring one corresponds to asking about the qubit along a particular axis.
• A general pure qubit state can be represented on the Bloch sphere using two angles:

[ |\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle. ]

The basis states (|0\rangle) and (|1\rangle) lie at opposite poles, while other points represent other pure states.

• Unlike a classical coin, a qubit can be measured in many different bases. The relative phase between components of a superposition affects measurement results.
• For example, equal superpositions of (|0\rangle) and (|1\rangle) with a plus or minus sign produce opposite outcomes when measured along the (x)-axis. This illustrates interference of amplitudes: alternatives can add constructively or destructively.

With many identically prepared qubits, measurements of (\sigma_x, \sigma_y, \sigma_z) can be used to estimate the state’s Bloch-sphere coordinates. A single measurement generally cannot determine an unknown state.

4. Density Operators: Describing What a Subsystem Can Observe

When a system is entangled with an inaccessible environment, its state generally cannot be described by a single state vector. Instead, it is described by a density operator (or density matrix).

For a joint state (|\psi\rangle_{AB}), the state available to observer A is

[ \rho_A = \operatorname{Tr}_B\bigl(|\psi\rangle\langle\psi|\bigr), ]

where (\operatorname{Tr}_B) is the partial trace over system B.

The density operator gives A’s predictions for every measurement on A:

[ \langle M_A\rangle = \operatorname{Tr}(\rho_A M_A). ]

It contains all the information needed to predict measurements on A alone, though it does not retain all information about correlations between A and B.

A density operator has three defining properties:

• It is Hermitian.
• It is positive semidefinite: its expectation value in any state is nonnegative.
• It has unit trace.

Its eigenvalues are nonnegative and sum to one. They can be interpreted as probabilities in an ensemble of states:

• A pure state has one nonzero eigenvalue.
• A mixed state has more than one nonzero eigenvalue.

For measurements on A alone, a density operator is indistinguishable from a preparation in which one of several states was chosen probabilistically. The mixture need not arise from anyone actually performing a measurement; it is sufficient that the available measurements cannot distinguish the situations.

5. Schmidt Decomposition and Entanglement

Any pure state of two systems can be written, with suitable choices of basis, in Schmidt form:

[ |\psi\rangle_{AB} = \sum_i \sqrt{p_i}\,|i\rangle_A|i’\rangle_B. ]

The (p_i) are probabilities and are the nonzero eigenvalues of both systems’ reduced density operators. This decomposition is closely related to the singular-value decomposition of a matrix.

• If there is only one nonzero term, the joint state is a product state and is not entangled.
• If there is more than one nonzero term, the state is entangled.
• The number of nonzero terms is the Schmidt rank.

The lecture emphasizes that local operations and classical communication—actions performed separately in the two laboratories and coordinated by ordinary messages—cannot create shared entanglement from an initially unentangled state. Creating it requires a joint quantum operation or quantum communication.

6. Entanglement, Correlations, and No Instantaneous Messaging

For a maximally entangled pair, Alice and Bob can find strongly correlated outcomes when they measure in matching bases. The correlations are not limited to one basis, unlike correlations between ordinary classical bits.

However, Alice cannot use shared entanglement to send Bob a message faster than light:

• Bob’s reduced state remains the maximally mixed state, regardless of whether or how Alice measures.
• Bob’s own measurement outcomes remain random.
• Alice and Bob can later compare results through classical communication to verify their correlations, but Alice’s choice does not let Bob determine a message on his own.

7. Coherent Superpositions, Mixtures, and the Quantum Eraser

• In a coherent superposition, relative phases matter and amplitudes can interfere.
• In an incoherent mixture, the system is described by probabilities for alternatives; there is no phase relationship between them that can produce interference.
• If information exists—even in principle—about which alternative occurred, the alternatives cannot interfere in the same way.

The lecture illustrates this with a Stern–Gerlach-style example: an electron’s spin can become correlated with its position, leaving a record of whether its spin was up or down along an axis. If that record is erased so the alternatives can no longer be distinguished, they can again be combined coherently.

In an entangled pair, Bob can measure in a different basis—for example, the (x)-basis—and thereby conditionally prepare a corresponding pure state for Alice. The lecture calls this a quantum eraser in its example: measuring in the alternative basis removes the relevant which-outcome information. Alice can use the resulting information only after Bob communicates his measurement result; this does not enable instantaneous messaging.

Speakers and Sources Featured

• Speaker: John Preskill, lecturer.
• Source mentioned: The course lecture notes, Chapter 2.
• Named concepts and references discussed: Max Born (Born rule), Paul Dirac (Dirac notation), Wolfgang Pauli (Pauli operators), and the Stern–Gerlach experiment. These are references within the lecture, not additional speakers.