 # University of Queensland

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University of Queensland
Quantum Noise Michael A. Nielsen University of Queensland Goals: To introduce a tool – the density matrix – that is used to describe noise in quantum systems, and to give some examples. The first two lectures today are going to introduce the basic background in quantum mechanics that you need to know in order to do quantum information science. The goal of this and the next lecture is to introduce _all_ the basic elements of quantum mechanics, using examples drawn from quantum information science. I’m going to do this assuming only elementary linear algebra, and mathematical maturity about that I would expect for a good third or fourth year undergraduate. Notice, by the way, that over the next few days I’m going to use the umbrella term “quantum information science” to encompass quantum information and computation. There are two groups of people that these lectures are for. The first is for people with little or no background in quantum mechanics, who’d like to learn the subject, or at least brush up on it. Thus, the approach today is going to be fairly elementary, and some experts may wish to go and spend some time enjoying the sights of Brisbane. However, they may also like to say, and participate in and observe an experimental approach to the teaching of quantum mechanics, an approach that I think has some very considerable advantages over the standard approach.

Fundamental point of view
Density matrices Generalization of the quantum state used to describe noisy quantum systems. Terminology: “Density matrix” = “Density operator” Quantum subsystem Ensemble Fundamental point of view

What we’re going to do in this lecture, and why we’re doing it
Most of the lecture will be spent understanding the density matrix. Unfortunately, that means we’ve got to master a rather complex formalism. It might seem a little strange, since the density matrix is never essential for calculations – it’s a mathematical tool, introduced for convenience. Why bother with it? The density matrix seems to be a very deep abstraction – once you’ve mastered the formalism, it becomes far easier to understand many other things, including quantum noise, quantum error-correction, quantum entanglement, and quantum communication.

I. Ensemble point of view

Qubit examples

Qubit example

Why work with density matrices?
Answer: Simplicity! The quantum state is: ?

Two-qubit example

Dynamics and the density matrix

Single-qubit examples
“Completely mixed state”

How the density matrix changes during a measurement

Characterizing the density matrix
What class of matrices correspond to possible density matrices?

Summary of the ensemble point of view

A simple example of quantum noise
With probability p the not gate is applied. With probability 1-p the not gate fails, and nothing happens. If we were to work with state vectors instead of density matrices, doing a series of noisy quantum gates would quickly result in an incredibly complex ensemble of states.

How good a not gate is this?
A quantum operation The fidelity measures how similar the states are, ranging from 0 (totally dissimilar), up to 1 (the same). Fidelity measures for two mixed states are a surprisingly complex topic!

How good a not gate is this?

II. Subsystem point of view
Bob Alice

II. Subsystem point of view
Bob Alice

How to calculate: a method, and an example

The example of a Bell state

III. The density matrix as fundamental object
Postulate 1: A quantum system is described by a positive matrix (the density matrix), with unit trace, acting on a complex inner product space known as state space.

Why teleportation doesn’t allow FTL communication
Alice Bob Let me give a brief explanation of an application where entanglement is a useful resource. I’ll explain one of the most striking effects, quantum teleportation.

Why teleportation doesn’t allow FTL communication
Alice Bob 01 01 To explain the protocol, it helps to assume that the system Alice wants to transmit is a spin ½ particle. All the ideas generalize easily to other systems, but for concreteness it is easiest to explain the protocol in this case. In the first part of teleportation a third party, Victor, (picture) prepares two spin one half particles (picture) in an entangled state just like that we saw before. (picture) He then transmits one of these particles to Alice, and the other to Bob. (picture) In the next stage of the protocol, Alice uses a measuring device to perform a joint measurement on her two particles (picture). I won’t describe exactly how this measurement is to be performed (picture), but suffice to say that the measurement has four possible outcomes. (picture) I’ll use a string of two bits to represent the possible outcome of the measurement, as shown here. In the next stage of the protocol Alice transmits her measurement result to Bob (picture), using an ordinary classical communication channel. At this point something rather magical happens. A simple calculation shows that Bob’s quantum state, conditional on the measurement result received from Alice (picture), is very closely related to the original state of Alice’s system. In fact, it is so closely related that by performing a simple unitary rotation of the Bloch sphere (picture), Bob is able to ensure that the final state of his system is exactly the same as the original state of Alice’s system.

Why teleportation doesn’t allow FTL communication
Alice Bob To explain the protocol, it helps to assume that the system Alice wants to transmit is a spin ½ particle. All the ideas generalize easily to other systems, but for concreteness it is easiest to explain the protocol in this case. In the first part of teleportation a third party, Victor, (picture) prepares two spin one half particles (picture) in an entangled state just like that we saw before. (picture) He then transmits one of these particles to Alice, and the other to Bob. (picture) In the next stage of the protocol, Alice uses a measuring device to perform a joint measurement on her two particles (picture). I won’t describe exactly how this measurement is to be performed (picture), but suffice to say that the measurement has four possible outcomes. (picture) I’ll use a string of two bits to represent the possible outcome of the measurement, as shown here. In the next stage of the protocol Alice transmits her measurement result to Bob (picture), using an ordinary classical communication channel. At this point something rather magical happens. A simple calculation shows that Bob’s quantum state, conditional on the measurement result received from Alice (picture), is very closely related to the original state of Alice’s system. In fact, it is so closely related that by performing a simple unitary rotation of the Bloch sphere (picture), Bob is able to ensure that the final state of his system is exactly the same as the original state of Alice’s system.

Why teleportation doesn’t allow FTL communication
Alice Bob 01 To explain the protocol, it helps to assume that the system Alice wants to transmit is a spin ½ particle. All the ideas generalize easily to other systems, but for concreteness it is easiest to explain the protocol in this case. In the first part of teleportation a third party, Victor, (picture) prepares two spin one half particles (picture) in an entangled state just like that we saw before. (picture) He then transmits one of these particles to Alice, and the other to Bob. (picture) In the next stage of the protocol, Alice uses a measuring device to perform a joint measurement on her two particles (picture). I won’t describe exactly how this measurement is to be performed (picture), but suffice to say that the measurement has four possible outcomes. (picture) I’ll use a string of two bits to represent the possible outcome of the measurement, as shown here. In the next stage of the protocol Alice transmits her measurement result to Bob (picture), using an ordinary classical communication channel. At this point something rather magical happens. A simple calculation shows that Bob’s quantum state, conditional on the measurement result received from Alice (picture), is very closely related to the original state of Alice’s system. In fact, it is so closely related that by performing a simple unitary rotation of the Bloch sphere (picture), Bob is able to ensure that the final state of his system is exactly the same as the original state of Alice’s system.

Why teleportation doesn’t allow FTL communication
Alice Bob To explain the protocol, it helps to assume that the system Alice wants to transmit is a spin ½ particle. All the ideas generalize easily to other systems, but for concreteness it is easiest to explain the protocol in this case. In the first part of teleportation a third party, Victor, (picture) prepares two spin one half particles (picture) in an entangled state just like that we saw before. (picture) He then transmits one of these particles to Alice, and the other to Bob. (picture) In the next stage of the protocol, Alice uses a measuring device to perform a joint measurement on her two particles (picture). I won’t describe exactly how this measurement is to be performed (picture), but suffice to say that the measurement has four possible outcomes. (picture) I’ll use a string of two bits to represent the possible outcome of the measurement, as shown here. In the next stage of the protocol Alice transmits her measurement result to Bob (picture), using an ordinary classical communication channel. At this point something rather magical happens. A simple calculation shows that Bob’s quantum state, conditional on the measurement result received from Alice (picture), is very closely related to the original state of Alice’s system. In fact, it is so closely related that by performing a simple unitary rotation of the Bloch sphere (picture), Bob is able to ensure that the final state of his system is exactly the same as the original state of Alice’s system.

Why teleportation doesn’t allow FTL communication
Alice Bob Bob’s reduced density matrix after Alice’s measurement is the same as it was before, so the statistics of any measurement Bob can do on his system will be the same after Alice’s measurement as before! To explain the protocol, it helps to assume that the system Alice wants to transmit is a spin ½ particle. All the ideas generalize easily to other systems, but for concreteness it is easiest to explain the protocol in this case. In the first part of teleportation a third party, Victor, (picture) prepares two spin one half particles (picture) in an entangled state just like that we saw before. (picture) He then transmits one of these particles to Alice, and the other to Bob. (picture) In the next stage of the protocol, Alice uses a measuring device to perform a joint measurement on her two particles (picture). I won’t describe exactly how this measurement is to be performed (picture), but suffice to say that the measurement has four possible outcomes. (picture) I’ll use a string of two bits to represent the possible outcome of the measurement, as shown here. In the next stage of the protocol Alice transmits her measurement result to Bob (picture), using an ordinary classical communication channel. At this point something rather magical happens. A simple calculation shows that Bob’s quantum state, conditional on the measurement result received from Alice (picture), is very closely related to the original state of Alice’s system. In fact, it is so closely related that by performing a simple unitary rotation of the Bloch sphere (picture), Bob is able to ensure that the final state of his system is exactly the same as the original state of Alice’s system.

Fidelity measures for quantum gates
Research problem: Find a measure quantifying how well a noisy quantum gate works that has the following properties: It should have a simple, clear, unambiguous operational interpretation. It should have a clear meaning in an experimental context, and be relatively easy to measure in a stable fashion. It should have “nice” mathematical properties that facilitate understanding processes like quantum error-correction. Candidates abound, but nobody has clearly obtained a synthesis of all these properties. It’d be good to do so!