Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871

Lecture 11 (2017)

Jon Yard QNC 3126 [email protected] http://math.uwaterloo.ca/~jyard/qic710 1 General measurements and POVMS (POVM = Positive Operator Valued Measure)

2 Prelude: projective measurements 2 2

0 1 span of 0 and 1

In both cases, there is a complete set of mutually orthogonal projectors: and

The probability of outcome j is using Tr(AB)=Tr(BA)

The collapsed state is the projected vector, but normalized 3 General measurements (1) 푚 † Let 퐴1, 퐴2, … , 퐴푚 be any matrices satisfying ෍ 퐴푗 퐴푗 = 퐼 푗=1 Corresponding measurement is a stochastic operation on 𝜌 † that, with probability Tr 퐴푗𝜌퐴푗 , produces outcome: 풋 (classical information) 퐴 𝜌퐴† 푗 푗 (the collapsed quantum state) † Tr 퐴푗𝜌퐴푗

Example 1: (퐴푗 = 휙푗 휙푗 ) (rank-1 orthogonal projectors) Consistent with our first definition of measurements

Question: what if we do the above but don’t look at 풋 ? 4 General measurements (2)

When 퐴푗 = 휙푗 휙푗 are orthogonal projectors and 𝜌 = 휓 휓 ,

† Tr 퐴푗𝜌퐴푗 = Tr 휙푗 휙푗|휓 휓|휙푗 휙푗 = 휙푗|휓 휓|휙푗 〈휙푗|휙푗〉 2 = 휙푗|휓

† 퐴푗휌퐴푗 휙푗 휙푗|휓 휓|휙푗 〈휙푗| Moreover, † = 2 = 휙푗 휙푗 Tr 퐴푗휌퐴푗 〈휙푗|휓〉

5 General measurements (3)

Example 3 (trine state measurement):

1 3 1 3 Let 휙 = 0 , 휙 = − 0 + |1〉, 휙 = − 0 − |1〉 0 1 2 2 2 2 2

2 2 1 0 Define 퐴 = 휙 〈휙 | = 0 3 0 0 3 0 0 2 1 2/3 − 2 퐴1 = 휙1 〈휙1| = 3 4 − 2 6 2 1 2/3 2 퐴2 = 휙2 〈휙2| = 3 4 2 6

† † † Then 퐴0퐴0 + 퐴1퐴1 + 퐴2퐴2 = 퐼.

If the input itself is an unknown trine state 휙푘휙푘, then the probability that classical outcome is 푘 is 2/3 = 0.6666… 6 General measurements (3)

Question: Are there states the trine measurement can’t distinguish? 1 3 1 3 휙 = 0 , 휙 = − 0 + |1〉, 휙 = − 0 − |1〉 0 1 2 2 2 2 2 2 2 1 0 퐴 = 휙 〈휙 | = 0 3 0 0 3 0 0 2 1 2/3 − 2 퐴1 = 휙1 〈휙1| = 3 4 − 2 6 2 1 2/3 2 퐴2 = 휙2 〈휙2| = 3 4 2 6 0 +푖 1 0 −푖 1 Answer: 푖 = , −푖 = (푌-eigenstates) 2 2

The trine measurement is not informationally complete! 7 General measurements (4) Often measurements arise in contexts where we only care about the classical part of the outcome (not the residual quantum state) † † The probability of outcome j is Tr 퐴푗𝜌퐴푗 = Tr 𝜌퐴푗 퐴푗

Simplified definition of such measurements 푚 ෍ 퐸 = 퐼 Let 퐸1, 퐸2, … , 퐸푚 be positive semidefinite and with 푗 푗=1

The probability of outcome j is Tr 𝜌퐸푗 . Called a POVM (Positive Operator-Valued Measure)

It is a measure valued in positive (-semidefinite) operators.8 Informationally-complete POVMs

A POVM 퐸1, 퐸2, … , 퐸푚 is informationally complete if spanℝ 퐸1, 퐸2, … , 퐸푚 = all Hermitian 푑 × 푑 matrices. Such POVMs can distinguish any states. Example: Informationally complete POVMs such that rank 퐸푗 = 1 for each 푗, i.e. such that 퐸1 = 훼1 휙1 〈휙1|, 퐸2 = 훼2 휙2 〈휙2|, … , 퐸푚 = 훼푚 휙푚 〈휙푚|, are sometimes called tight frames. (Very hard) question: Do informationally-complete POVMs exist with rank 퐸푗 = 1 for every 푗 and Tr 퐸푖퐸푗 = constant for 푖 ≠ 푗? Answer: Apparently yes (no proof yet), known as

SIC-POVMs (Symmetric Informationally Complete POVMs) 9 “Mother of all operations”

Let 퐴1,1, 퐴1,2, … , 퐴1,푘1 satisfy 푚 푘푚 † 퐴2,1, 퐴2,2, … , 퐴2,푘 2 ෍ ෍ 퐴푗,푖퐴푗,푖 = 퐼 퐴푚,1, 퐴푚,2, … , 퐴푚,푘푚 푗=1 푖=1 Then there is a quantum operation that, on input , produces 푘푚 with probability † the state: ෍ 퐴푗,푖𝜌퐴푗,푖 푖=1 j (classical information)

푘푚 † σ 퐴푗,푖𝜌퐴 푖=1 푗,푖 (the collapsed quantum state) 푘푚 † σ푖=1 Tr 퐴푗,푖𝜌퐴푗,푖

Also known as quantum instrument or heralded quantum channel Simulations among operations

11 Simulations among operations (1)

Theorem 1: any quantum operation can be simulated by applying a unitary operation on a larger quantum system: 0 0 discard 0 This specification of a quantum operation is called U the Stinespring form, or input 𝜌 𝜎 output isometric extension

푛 푚 Example: decoherence 0 훼 2 0 0 + 1 𝜌 = 0 훽 2 12 Simulations among operations (2)

Proof of Theorem 1: 푚 푛 Let 퐴1, 퐴2, … , 퐴2푘 be any 2 × 2 matrices such that

This defines a mapping from 푚 qubits to 푛 qubits:

This specification of the quantum operation is called the Kraus form

13 Simulations among operations (3) Since Set V =

the columns of V are orthonormal

Let 푈 be any unitary matrix Now, consider the circuit: with first 2푛 columns from 푉 0 0 0 푈 is a 2푚+푘 × 2푚+푘 matrix U  (and its columns partition into 푚−푛+푘 푛 2 blocks of size 2 ) 14 Simulations among operations (4) The output state of the circuit is

15 Simulations among operations (5) Tracing out the high-order 푘 qubits of this state yields exactly the output of mapping that we want to simulate

푚 + 푘– 푛 푘 0 0 discard 0 U input 𝜌 𝜎 output

푛 푚 Note: this approach is not always optimal in the number of ancilliary qubits used—there are more efficient methods 16 Simulations among operations (6) Theorem 2: any measurement can also be simulated by applying a unitary operation on a larger quantum system and then measuring:

input 𝜌 𝜎 quantum output U 0 j classical output 0 0

This is the same diagram as for Theorem 1 (drawn with the extra qubits at the bottom) but where the “discarded” qubits are measured and part of the output

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