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

Lecture 2 (2017)

Jon Yard QNC 3126 [email protected]

1 Superdense coding

2 How much classical information in n qubits?

2푛1 complex numbers apparently needed to describe an arbitrary 푛-qubit pure quantum state:

훼000 000 + 훼001 001 + 훼010 010 + ⋯ + 훼111 111

Does this mean that an exponential amount of classical information is somehow “stored” in 푛 qubits?

Not in an operational sense ... For example, Holevo’s Theorem (from 1973) implies: one cannot convey more than 푛 classical bits of information in 푛 qubits

3 Holevo’s Theorem

Easy case: Hard case (the general case):

b1 b1 b b ψ 2 ψ 2 U b3 b3 n qubits n qubits bn bn 0 U bn+1 0 b 푏1, 푏2, … , 푏푛 certainly n+2 m qubits 0 cannot convey more bn+3 0 than 푛 bits! bn+4 0 bn+m Proof is beyond the scope of this course (but see CS 766 / QIC 820) 4 Superdense coding (prelude) Suppose that Alice wants to convey two classical bits to Bob sending just one qubit

푎푏

Alice Bob 푎푏

By Holevo’s Theorem, this is impossible

5 Superdense coding In superdense coding, Bob is allowed to send a qubit to Alice first

푎푏

Alice Bob 푎푏

How can this help?

6 How superdense coding works 1. Bob creates the state |00〉 + |11〉 and sends the first qubit to Alice 2. Alice: If 푎 = 1, apply 푋 to her qubit. Recall that 0 1 푋 = Then if 푏 = 1, apply 푍 to her qubit. 1 0 1 0 Then send the qubit back to Bob. 푍 = 0 −1 푎푏 (푍푏푋푎 ⊗ 퐼)(|00〉 + |11〉)

00 |00〉 + |11〉 = |Ψ+〉 01 00 − 11 = |Ψ 〉 − Bell basis 10 01 + 10 = |Φ+〉

11 01 − 10 = |Φ−〉 3. Bob measures the two qubits in the Bell basis 7 Measurement in the Bell basis

Specifically, Bob applies input output |00〉 + |11〉 |00〉 |00〉 − |11〉 |01〉 H |01〉 + |10〉 |10〉 |01〉 − |10〉 −|11〉 to his two qubits ... and then measures them, yielding 푎푏

This concludes superdense coding

8 Teleportation

9 Recap

• n-qubit quantum state: 2푛-dimensional unit vector

• Unitary op: 2푛2푛 linear operation 푈 such that 푈†푈 = 퐼 (where 푈† denotes the conjugate transpose of 푈) 푈|0000〉 = the 1st column of 푈 푈 0001 = the 2nd column of 푈 the columns of 푈 : : : : : : are orthonormal 푈 1111 = the (2푛)th column of 푈

10 Incomplete measurements (I)

Measurements up until now The orthogonal subspaces are with respect to orthogonal can have other dimensions: one-dimensional subspaces: 2 2

0 1 span of 0 and 1 (qutrit)

11 Incomplete measurements (II)

Such a measurement on 훼0 0 + 훼1 1 + 훼2 2

(renormalized)

훼 0 + 훼 1 with prob 훼 2 + 훼 2 results in 0 1 0 1 2 |2〉 with prob 훼2

12 Measuring the first qubit of a two-qubit system

훼00 00 + 훼01 01 + 훼10 01 + 훼11 11

Defined as the incomplete measurement with respect to the two dimensional subspaces: • span of 00 & 01 (all states with first qubit 0), and • span of 10 & 11 (all states with first qubit 1)

0, 훼 00 + 훼 01 with prob 훼 2 + 훼 2 Result is 00 01 00 01 2 2 1, 훼10 10 + 훼11 11 with prob 훼10 + 훼11 13 Easy exercise: show that measuring the first qubit and then measuring the second qubit gives the same result as measuring both qubits at once

14 Teleportation (prelude) Suppose Alice wishes to convey a qubit to Bob by sending just classical bits 훼 0 + 훽|1〉

훼 0 + 훽|1〉 If Alice knows 훼 and 훽, she can send approximations of them ―but this still requires infinitely many bits for perfect precision Moreover, if Alice does not know 훼 or 훽, she can at best acquire one bit about them by a measurement 15 Teleportation scenario In teleportation, Alice and Bob also start with a Bell state 1 1 00 + 11 훼0 + 훽1 2 2

and Alice can send two classical bits to Bob

Note that the initial state of the three qubit system is 1 1 훼 훼 훽 훽 훼0 + 훽1 00 + 11 = 000 + 011 + 100 + 111 2 2 2 2 2 2 16 How teleportation works

Initial state 1 1 훼 0 + 훽 1 ) 00 + |11〉 2 2 훼 훼 훽 훽 = 000 + 011 + 100 + 111 2 2 2 2 1 1 1 = 00 + |11〉 훼 0 + 훽 1 ) 2 2 2 1 1 1 + 01 + |10〉 훼 1 + 훽 0 ) 2 2 2 1 1 1 + 00 − |11〉 훼 0 − 훽 1 ) 2 2 2 1 1 1 + 01 − |10〉 훼 1 − 훽 0 ) 2 2 2 Protocol: Alice measures her two qubits in the Bell basis and sends the result to Bob (who then “corrects” his state) 17 Measuring the Bell basis H Alice applies to her two qubits, yielding:

1 00 훼 0 + 훽 1 ) 2 (00, 훼|0〉 + 훽 1 ) with prob. ¼ 1 + |01〉 훼 1 + 훽 0 ) (01, 훼 1 + 훽|0〉) with prob. ¼ 2 1 + |10〉 훼 0 − 훽 1 ) (10, 훼 0 − 훽|1〉) with prob. ¼ 2 1 + |11〉 훼 1 − 훽 0 ) (11, 훼 1 − 훽|0〉) with prob. ¼ 2

Then Alice sends her two classical bits to Bob, who then adjusts his qubit to be 훼|0〉 + 훽 1 depending on the values of the bits. 18 Bob’s adjustment procedure

Bob receives two classical bits 푎, 푏 from Alice, and: if 푏 = 1 he applies 푋 to qubit 0 1 1 0 푋 = , 푍 = if 푎 = 1 he applies 푍 to qubit 1 0 0 −1 yielding: 00, 훼|0〉 + 훽 1 01, 푋(훼 1 + 훽|0〉) = 훼|0〉 + 훽 1 10, 푍(훼 0 − 훽|1〉) = 훼|0〉 + 훽 1 11, 푍푋(훼 1 − 훽|0〉) = 훼|0〉 + 훽 1

Note that Bob acquires the correct state in each case

19 Summary of teleportation

0 + 1 H Alice a 00 + 11 b Bob X Z 0 + 1

Quantum circuit exercise: try to work through the details of the analysis of this teleportation protocol

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