- By
**bruis** - Follow User

- 98 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Randomizing Quantum States' - bruis

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Randomizing Quantum States

Charles Bennett

Patrick Hayden

Debbie Leung

Peter Shor

Andreas Winter

RSP: quant-ph/0307100 RAND: quant-ph/0307104

Given >0, there exists a choice of unitaries

{Uk}, k=1,…,n such that the map

is ε-randomizing, with

Relax security criterionA CPTP map is ε-randomizing if for all states ,

1 bit of key/qubit

(1+2α) bits of key/qubit

Approximate PQCCan encrypt a quantum state using 1 secret

random bit per encrypted qubit asymptotically.

Remote state preparation:Non-oblivious teleportation

A circuit that needs no introduction:

Remote state preparation:Non-oblivious teleportation

Allow Alice to perform an arbitrary φ-dependent measurement

From randomization to RSP

Circuit for teleportation:

l qubits

k: 2l bits

Before receiving k Bob knows nothing:

(Private quantum channel!)

From randomization to RSP

Circuit for remote state preparation:

l qubits

k:l+o(l) bits

Before receiving k, Bob knows nothing:

(Approximate private quantum channel!)

Knowledge is power

All you need in this life is ignorance and confidence,

and then success is sure. -Mark Twain

Oblivious Alice (Teleportation):

1 ebit + 2 cbits per 1 qubit sent

There is no knowledge that is not power.

-Ralph Waldo Emerson

Non-Oblivious Alice (RSP):

1 ebit + 1 cbit per 1 qubit sent

Knowledge is power, if you know it about the right person.

-Ethel Watts Mumford

Randomization and nonlocality

Does an ε-randomizing R destroy all

correlations with the environment?

True for separable ρ (easy).

Not true for entangled inputs!

Fidelity with maximally mixed state small:

Rank argumentRecall ε-randomizing map:

Act on half of a maximally entangled state:

has rank no more than n~d log d

~

Y

Alice and Bob implement an LOCC measurement.

Characterizing leftover correlationsWhat does randomization map do to entangled inputs?

R

Charlie prepares maximally entangled state k then randomizes it.

k info randomized

Conclusion: LOCC cannot distinguish strong and weak randomization

Characterizing leftover correlationsWhat does randomization map do to entangled inputs?

R

X

Y

Charlie prepares maximally entangled state k then randomizes it.

Alice and Bob implement an LOCC measurement.

• Select the unitaries {Uk} at random from Haar measure.

• Show they are ε-randomizing with probability >0.

Ingredients:

• Large deviation estimate

• Discretization

Randomization proof ideasWill be concerned with distribution of:

where

In particular,

(Exponential convergence to the mean)

Large deviation estimateWill allow replacement of “all states” by “all states in M”

in definition of ε-randomization.

DiscretizationFind anε-net M:

For all pure states φ, there exists a state

φ’ in M such that

Cd

Quantum data hiding

GOAL: Charlie hides a bit from Alice and Bob, secure against LOCC

RESULT: There exist bipartite n-qubit states hiding a bit with security 2-(n-1).

[DLT, 2001]

Decodability:

Hiding qubitsGOAL: Charlie hides qubits from Alice and Bob, secure against LOCC

[dV H T, 2002]

It is possible to hide a space of dimension

in a bipartite system where each of the subsystem has

dimension d.

Asymptotically: 1 hidden qubit per 2 physical qubits.

Previous best: ~ 1 hidden qubit per ~75 physical qubits.

Data hiding summaryConclusion

- Encryption of quantum states using 1 bit of shared key per encrypted qubit
- Remote state preparation using 1 bit of communication and 1 ebit per qubit sent
- Physical operation that destroys classical correlations but not quantum
- Data hiding of 1 qubit in 2 physical qubits

Prepare superpositions of well hidden states?

Hiding a qubit: First attemptTASK: Hide an arbitrary quantum state

Hiding a qubit: First attempt

TASK: Hide an arbitrary quantum state

PROBLEM: Data hiding with pure states is impossible!

(So much for superpositions.)

[WSHV,2000]

2nd simplest idea

THE PLAN: Use classical hidden bits as key to randomize a qubit

PROPERTIES: 1) can be recovered using quantum communication

2) Naïve attacks fail (AB1 to find key then rotate B2)

PROBLEM: Alice and Bob can attack AB1B2

Actually, not a problem

Any method to learn about by LOCC will provide a

method to defeat the original cbit hiding scheme.

Will argue the contrapositive:

Assume there is an LOCC operation L (with output on

Bob’s system alone) and two input states to the hiding

map E such that

CLAIM: Not all Li can be the same TPCP map. If they were, then

by linearity:

This says that L would never reveal any information about

the input state, violating the hypothesis that L defeats the

qubit hiding scheme.

Minor algebraThe attack: 1) Bob prepares a local maximally entangled state on B2B3

2) Alice and Bob apply L to AB1B2

3) Bob performs a measurement on B2B3

By Choi, there is a measurement that can partially distinguish L0 andLk

Defeating the cbit hidingConclusion from previous slide: there is a k such that L0Lk

Imperfect hiding

Wish to limit distinguishability through LOCC:

For all input states and attacks.

If the original 2n bit hiding scheme has security ,

then <2n+1 .

Not so bad: security of classical hiding schemes appears

to improve exponentially with number of qubits used.

Authorized sets can recover the secret using quantum communication

Multipartite cbit hidingA

E

B

D

C

With LOCC alone the five parties cannot learn i

All monotonic access structures are possible: [Eggeling, Werner 2002]

is impossible due to no-cloning!

Multipartite qubit hidingA

E

B

D

C

With LOCC alone the five parties cannot learn

Authorized sets can recover the secret using quantum communication

Problem for generalizing construction: B must be in all authorized sets

Multipartite qubit hidingA

E

B

D

C

With LOCC alone the five parties cannot learn

Authorized sets can recover the secret using quantum communication

Quantum secret sharing

A

e.g. ((2,3)) threshold scheme

E

B

D

C

Secure against quantum communication in unauthorized sets but

secret can be recovered by quantum communication in authorized sets.

All monotonic threshold schemes not violating no-cloning [CGL,1999]

All monotonic schemes not violating no-cloning [G,2000]

Hiding distributed quantum data

A

Logical Pauli operators

Multipartite cbit

hiding states

E

B

D

C

Resulting state provides strengthening of quantum secret sharing:

Secure against classical communication between all parties

Secure against quantum communication in unauthorized sets

Secret can be recovered only by quantum communication in authorized sets.

All monotonic schemes not violating no-cloning

Act II

Fig. 1: Glimpse of a master magician’s workshop

On the meaning of “ “

For any probability density P() on states in Cd and >0 there exists

a choice of unitaries {Us}, s=1,…,S such that

and

Compare to the perfect private quantum channel:

To achieve =0 requires log M = 2 log d.

Another version

There exists a choice of unitaries {Ups}, p=1,…,P, s=1,…,S such

that for all states in Cd

and

Can randomize every n-qubit state using 1 secret

random bit and 1 public random bit per qubit.

Consequences

- Universal remote state preparation with only 1 ebit + 1 cbit per qubit
- (No shared random bits necessary)
- Weakly randomized maximally entangled states indistinguishable from maximally mixed states using LOCC
- (Not just 1-way LOCC as sketched earlier)

Application to data hiding

R

Consider ensemble of randomized states with V chosen using Haar measure

Rank bound on entropy

So we can do coding to get about n hidden bits using nxn bipartite states!

Faction 1

Destroying classical correlations requires only 1 rbit per qubit

Destroying quantum correlations requires 2 rbits per qubit

Faction 2

Randomizing an arbitrary pure quantum state requires 1 public rbit and 1 secret rbit per qubit

Competing visionsSummary

- Described a method for hiding qubits given one for hiding bits (construction and proof not restricted to data hiding)
- Outlined a connection between LOCC data hiding, private quantum channels and remote state preparation

Building the POVM

Alice wishes to send φ. By the definition of ε-randomization:

From RSP to randomization

Circuit for teleportation:

n qubits

i: 2n bits

Before receiving i, Bob knows nothing:

(“Private quantum channel”, “Quantum one-time pad”, etc.)

Download Presentation

Connecting to Server..