Quantum Noise Michael A. Nielsen University of Queensland Goals: 1. To introduce a tool the density matrix that is used to describe noise in quantum systems, and to give some examples.
Density matrices Generalization of the quantum state used to describe noisy quantum systems. Terminology: Density matrix = Density operator Ensemble p, Quantum subsystem 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 Imagine that a quantum system is in the state with probability. p We do a measurement described by proectors P k. ( ) Probability of outcome k = Pr k state p Probability of outcome k = k = P k p k = p tr( ) Pk where ρ p is the density matrix. ρ completely determines k tr ( ρp ) k all measurement statistics.
Qubit examples Suppose = 0 with probability 1. 1 1 0 Then ρ = 0 0 = 1 0. 0 = 0 0 Suppose = 1 with probability 1. 0 0 0 Then ρ = 1 1 = 0 1 =. 1 0 1 0 + i 1 Suppose = with probability 1. 0 + i 1 0 i 1 1 1 1 1 i Then ρ = = 1 i =. i i 1
Qubit example Suppose = 0 with probability p, and = 1 with probability 1 p. ( p ) Then ρ = p 0 0 + 1 1 1 1 0 0 0 p 0 = p ( 1 p). 0 0 + = 0 1 0 1 p Measurement in the 0, 1 basis yields Pr( 0) = tr( ρ 0 0 ) = 1 p 0 1 0 0 1 p 0 = p. ( ) Similarly, Pr 1 1. = p
Why work with density matrices? Answer: Simplicity! The quantum state is: 0 with probability 0.1 1 with probability 0.1 0 + 1 with probability 0.15 0 1 with probability 0.15 0 + i 1 with probability 0.5 0 i 1 with probability 0.5 1 0 ρ = 1 0
Two-qubit example Suppose = 00 with probability p, and = with probability 1 p. Then: ρ = p 00 00 + 1 1 01 + 10 01 + 10 1 0 0 1 1 p = p 1 0 0 0 + 0 0 1 0 0 1 1 0 p 0 0 0 1 p 1 p 0 0 = 1 p 1 p 0 0 0 0 0 0 ( p) ( )( ) 01 + 10
Dynamics and the density matrix Suppose we have a quantum system in the state with probability. p The quantum system undergoes a dynamics described by the unitary matrix. U The quantum system is now in the state U with probability p. The initial density matrix is ρ =. The final density matr is ρ p ix ρ ' U. ' = UρU. = p U ( ). = U pu U
Single-qubit examples Suppose = 0 with probability p, and = 1 with probability 1 p. p 0 Then ρ =. 0 1 p 1 p 0 Suppose an X gate is applied. Then ρ' = X ρx =. 0 p Suppose = 0 and = 1 with equal probabilities. I Then ρ =. Completely mixed state Suppose any unitary gate U is applied. I I Then ρ ' = U U =. 1
How the density matrix changes during a measurement Worked Exercise : Suppose a measurement described by proectors P k is performed on an ensemble giving rise to the density matrix ρ. If the measurement gives result k show that the corresponding post-measurement density matrix is ' Pkρ Pk ρk =. tr P ρp ( ) k k
Characterizing the density matrix What class of matrices correspond to possible density matrices? Suppose ρ = p is a density matrix. Then tr( ρ ) = tr( ) = = 1 p p For any vector a, a ρ a = p a a = a 0 p Summary : tr ( ρ ) =1 and ρ is a positive matrix. ( ρ) Exercise : Given that tr =1 and ρ is a positive matrix, show that there is some set of states and probabilities p such that ρ= p.
Summary of the ensemble point of view Definition: with probability p The density matrix for a system in state is ρ p. Dynamics : ρ ρ' = UρU. Measurement: A measurement described by proectors Pk gives result k with probability tr ( Pk ρ ), and the postmeasurement density matrix is ρk =. ' Pkρ Pk tr ( ρ ) ( P ρp ) Characterization: tr =1, and ρ is a positive matrix. Conversely, given any matrix satisfying these properties, there exists a set of states and probabilities p such that ρ= p. k k
A simple example of quantum noise With probability p the not gate is applied. X With probability 1-p the not gate fails, and nothing happens. ρ p ( ) = ( 1 ) ppx X+ p p ( 1 ) = px ρx + p 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. ρ
How good a not gate is this? X How "good" a not gate is this, for a particular input? ( ) ( 1 ) ρ E ρ px ρx + p ρ A quantum operation We compare the ideal output, X, to the actual output. The usual way two states a and b are compared is to compute the fidelity, or overlap: F( a, b) a b. The fidelity measures how similar the states are, ranging from 0 (totally dissimilar), up to 1 (the same). To compare a with σ = p φ φ we compute the fidelity, F( a, σ) a σ a. Fidelity measures for two mixed states are a surprisingly complex topic!
How good a not gate is this? X ( ) ( 1 ) ρ E ρ px ρx + p ρ How "good" a not gate is this, for a particular input? We compare the ideal output, X, to the actual output. The fidelity of the gate is thus F( X, E ) XE X ( ) = ( ) = + ( 1 ) ( ) p p X X ( 1 ) = p + p X The fidelity ranges between p, for = 0, and 1, for = 0 + 1 /.
II. Subsystem point of view Bob Alice P = α k l kl kl = (( ) ) α α ( ) kl mn Pr( ) tr P I = * tr( P I k l m n ) klmn * ( ) * = α α l n P klmn kl mn I k m = α α l P klm kl ml k ( ) = α α * tr P klm kl ml k l = tr ( P ) ρa k l klm known as the reduced density matrix of system A. * where ρa α α is kl ml
II. Subsystem point of view Bob Alice P = α k l kl kl ρ A A B ( P ) ρa Pr( ) = tr, where ρ tr α α ( ) klm kl * ml k l is the reduced density matrix for sys tem A. All the statistics for measurements on system A can be recovered from ρ. A
How to calculate: a method, and an example An alternative, more convenient definition for the partial trace is to define: tr ( ) tr( ) B a1 a b1 b a1 a b1 b = b b a a 1 1 Then extend the definition linearly to arbitrary matrices. Exercise: Show that this new definition agrees with ( ) * the old, that is, tr B = α α k l when klm kl ml = α k l. A B kl kl Example: If the system is in the state a b then ρ = tr a a b b = b b a a = a a ( )
A B The example of a Bell state 00 + 11 Example: Suppose =. Then the reduced density matrix for the first system is given by: ρ = tr = = = ( ) ( ) + ( ) + ( ) + ( ) tr 00 00 tr 00 11 tr 11 00 tr 11 11 I B B B B 0 0 + 1 1 From Alice's point of view, it's ust like having the state 0 1 with probability, and the state 1 with probability. Under dynamics and measurement, the density matrix behaves ust as it does in the ensemble point of view. 1
III. The density matrix as fundamental obect 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. A system in state ρ with probability p has density matrix pρ. Postulate : The dynamics of a closed quantum system are described by ρ ρ' = UρU. Postulate 3: A measurement described by proectors Pk gives result k with probability tr ( Pk ρ ), and the postmeasurement density matrix is ρk =. ' Pkρ Pk tr ( P ρp ) Postulate 4: We take the tensor product to find the state space of a composite system. The state of one component is found by taking the partial trace over the remainder of the system. k k
Why teleportation doesn t allow FTL communication Alice Bob
Why teleportation doesn t allow FTL communication Alice Bob 01 01
Why teleportation doesn t allow FTL communication Alice Bob The initial state for the protocol is 00 + 11 Bob's initial reduced density matrix is ust the reduced density matrix for a Bell state, ρ =. I B
Why teleportation doesn t allow FTL communication Alice Bob B + B Z + B X + B XZ 1 3 4 with probability ; 01 1 B B B B 1 4 Z with probability ; 1 4 X with probability 1 3 4 XZ with probability. 1 4 4 ; and
Why teleportation doesn t allow FTL communication Alice Bob Bob's final reduced density matrix is thus ρ ' B ( B B + ) tr A 1 1... = 4 + Z Z + X X + XZ ZX = 4 * * * * α αβ α αβ β α β β α β + + + * * * * α β β α β β αβ α αβ α = 4 I =
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!
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!