Advanced Computer Architecture Lab University of Michigan Quantum Noise and Distance Patrick Cassleman More Quantum Noise and Distance Measures for Quantum

Advanced Computer Architecture LabUniversity of Michigan

Quantum Noise and DistancePatrick Cassleman

More Quantum Noise and Distance More Quantum Noise and Distance Measures for Quantum Information Measures for Quantum Information

(Some of Ch8 and Ch 9)(Some of Ch8 and Ch 9)Patrick Cassleman

EECS 59811/29/01



OutlineOutline• Types of Quantum Noise

– Bit Flip– Phase Flip– Bit-phase Flip– Depolarizing Channel– Amplitude Damping– Phase Damping

• Distance measures for Probability Distributions• Distance measures for Quantum States



Background – The Bloch SphereBackground – The Bloch Sphere• Remember :

• The numbers and define a point on the unitthree-dimensional sphere

12

sin02

cos ie

0

1



Examples of Quantum Noise – Bit FlipExamples of Quantum Noise – Bit Flip• A bit flip channel flips the state of a qubit from |0> to |1> with

probability 1-p• Operation Elements:

01

1011

10

01

1

0

pXpE

pIpE



Examples of Quantum Noise – Bit FlipExamples of Quantum Noise – Bit Flip• Bloch sphere representation:

– Before -After

x

z

y



Examples of Quantum Noise - Phase FlipExamples of Quantum Noise - Phase Flip• Corresponds to a measurement in the |0>, |1> basis, with the

result of the measurement unknown• Operation Elements:

10

0111

10

01

1

0

pZpE

pIpE



Examples of Quantun Noise – Phase FlipExamples of Quantun Noise – Phase Flip• Bloch vector is projected along the z axis, and the x and y

components of the Bloch vector are lost



Examples of Quantum Noise – Bit-phase FlipExamples of Quantum Noise – Bit-phase Flip• A combination of bit flip and phase flip• Operation Elements:

0

011

10

01

1

0

i

ipYpE

pIpE



Examples of Quantum Noise – Bit-phase FlipExamples of Quantum Noise – Bit-phase Flip• Bloch vector is projected along y-axis, x and z components of the

Bloch vector are lost



Examples of Quantum Noise – Depolarizing Examples of Quantum Noise – Depolarizing ChannelChannel

• Qubit is replaced with a completely mixed state I/2 with probability p, it is left untouched with probability 1-p

• The state of the quantum system after the noise is:

)1(2

)( ppIE



Examples of Quantum Noise –Examples of Quantum Noise – Depolarizing Channel Depolarizing Channel

• The Bloch sphere contracts uniformly



Examples of Quantum Noise –Examples of Quantum Noise –Depolarizing ChannelDepolarizing Channel

• Quantum Circuit Representation

(1-p)||+p|



Examples of Quantum Noise – Examples of Quantum Noise – Amplitude DampingAmplitude Damping

• Noise introduced by energy dissipation from the quantum system– Emitting a photon

• The quantum operation:

1100)( EEEEEAD




• Operation Elements:

• can be thought of as the probability of losing a photon

• E1 changes |1> into |0> - i.e. losing energy• E0 leaves |0> alone, but changes amplitude of |

1>

00

0

10

01

1

0

E

E

2sin




• Quantum Circuit Representation:

in out

0 )(yR



Examples of Quantum Noise –Examples of Quantum Noise – Amplitude Damping Amplitude Damping

• Bloch sphere Representation:• The entire sphere shrinks toward the north pole, |0>



Examples of Quantum Noise – Phase DampingExamples of Quantum Noise – Phase Damping• Describes the loss of quantum information without the loss of

energy• Electronic states perturbed by interacting with different charges• Relative phase between energy eigenstates is lost• Random “phase kick”, which causes non diagonal elements to

exponentially decay to 0• Operation elements:

• = probability that photon scattered without losing energy

10

010E

0

001E



Examples of Quantum Noise – Phase DampingExamples of Quantum Noise – Phase Damping• Quantum Circuit Representation:

• Just like Amplitude Damping without the CNOT gate

in out

0 )(yR



Box 8.4 – Why Shrodinger’s Cat Doesn’t WorkBox 8.4 – Why Shrodinger’s Cat Doesn’t Work• How come we don’t see superpositions in the world we observe?• The book blames: the extreme sensitivity of macroscopic

superposition to decoherence• i.e it is impossible in practice to isolate the cat and the atom in

their box– Unintentional measurements are made

• Heat leaks from the box• The cat bumps into the wall• The cat meows• Phase damping rapidly decoheres the state into either alive or dead



Distance measures for Probability DistributionsDistance measures for Probability Distributions• We need to compare the similarity of two probability distributions• Two measures are widely used: trace distance and fidelity• Trace distance also called L1 distance or Kalmogorov distance• Trace Distance of two probability distributions px and qx:

• The probability of an error in a channel is equal to the trace distance of the probability distribution before it enters the channel and the probability distribution after it leaves the channel

x

xxxx qpqpD ||2

1),(



Distance measures for Probability DistributionsDistance measures for Probability Distributions• Fidelity of two probability distributions:

• Fidelity is not a metric, when the distributions are equal, the fidelity is 1

• Fidelity does not have a clear interpretation in the real world

x

xxxx qpqpF2

1),(



Distance measures for Quantum StatesDistance measures for Quantum States• How close are two quantum states?• The trace distance of two quantum states and :

• If and commute, then the quantum trace distance between and is equal to the classical trace distance between their eigenvalues

• The trace distance between two single qubit states is half the ordinary Euclidian distance between them on the Bloch sphere

||2

1),( trD



Trace Preserving Quantum OperationsTrace Preserving Quantum Operations are Contractive are Contractive

• Suppose E is a trace preserving quantum operation. Let and be density operators. Then

• No physical process ever increases the distance between two quantum states

),())(),(( DEED



Fidelity of Two Quantum StatesFidelity of Two Quantum States

• When and commute (diagonal in the same basis), degenerates into the classical fidelity, F(ri, si) of their eigenvalue distributions

• The fidelity of a pure state and an arbitrary state :

• That is, the square root of the overlap

2/12/1),( trF

),(F



Uhlmann’s TheoremUhlmann’s Theorem• Given and are states of a quantum system Q, introduce a

second quantum system R which is a copy of Q Then:

• Where the maximizaion is over all purifications |> of and |> of into RQ

• Proof in the book

||max),(,

F



Turning Fidelity into a MetricTurning Fidelity into a Metric• The angle between states and is:

• The triangle inequality:

• Fidelity is like an upside down version of trace distance– Decreases as states become more distinguishable– Increases as states become less distinguishable– Instead of contractivity, we have monotonicity

),(arccos),( FA

),(),(),( AAA



Monotonicity of FidelityMonotonicity of Fidelity• Suppose E is a trace preserving quantum operation, let and be density operators, then:

• Trace distance and Fidelity are qualitatively equivalent measures of closeness for quantum states– Results about one may be used to deduce equivalent results about the

other– Example:

),())(),(( FEEF

2),(1),(),(1 FDF



ConclusionsConclusions• Quantum Noise is modeled as an operator on a state and the

environment• Quantum Noise can be seen as a manipulation of the Bloch

sphere• Fidelity and Trace distance measure the relative distance

between two quantum states• Quantum noise and distance will be important in the

understanding of quantum error correction next week

Documents

Advanced Computer Architecture Lab University of Michigan Quantum Noise and Distance Patrick Cassleman More Quantum Noise and Distance Measures for Quantum