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Randomizing Quantum States. Charles Bennett Patrick Hayden Debbie Leung Peter Shor Andreas Winter RSP: quant-ph/0307100 RAND: quant-ph/0307104. 1 bit of key per bit of message necessary and sufficient [Shannon]. One-time pad. Message. Shared key. Eavesdropper learns nothing:.

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randomizing quantum states

Randomizing Quantum States

Charles Bennett

Patrick Hayden

Debbie Leung

Peter Shor

Andreas Winter

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

private quantum channels

Eavesdropper learns nothing:

Lower bound on key length:

[BR,AMTW 2000]

Private quantum channels
relax security criterion

MAIN RESULT:

Given >0, there exists a choice of unitaries

{Uk}, k=1,…,n such that the map

is ε-randomizing, with

Relax security criterion

A CPTP map is ε-randomizing if for all states ,

approximate pqc

Security:

1 bit of key/qubit

(1+2α) bits of key/qubit

Approximate PQC

Can encrypt a quantum state using 1 secret

random bit per encrypted qubit asymptotically.

remote state preparation non oblivious teleportation
Remote state preparation:Non-oblivious teleportation

A circuit that needs no introduction:

remote state preparation non oblivious teleportation1
Remote state preparation:Non-oblivious teleportation

Allow Alice to perform an arbitrary φ-dependent measurement

from randomization to rsp
From randomization to RSP

Circuit for teleportation:

l qubits

k: 2l bits

Before receiving k Bob knows nothing:

(Private quantum channel!)

from randomization to rsp1
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
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
Randomization and nonlocality

Does an ε-randomizing R destroy all

correlations with the environment?

True for separable ρ (easy).

Not true for entangled inputs!

rank argument

Fidelity with maximally mixed state small:

Rank argument

Recall ε-randomizing map:

Act on half of a maximally entangled state:

has rank no more than n~d log d

~

characterizing leftover correlations

X

Y

Alice and Bob implement an LOCC measurement.

Characterizing leftover correlations

What does randomization map do to entangled inputs?

R

Charlie prepares maximally entangled state k then randomizes it.

characterizing leftover correlations1

Separable state!

k info randomized

Conclusion: LOCC cannot distinguish strong and weak randomization

Characterizing leftover correlations

What 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.

detecting entanglement

(conjectured)

Separable

states

Distillable

states

Product

states

Non-distillable

states

Detecting entanglement
randomization proof ideas

Strategy:

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

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

Ingredients:

• Large deviation estimate

• Discretization

Randomization proof ideas
large deviation estimate

Will be concerned with distribution of:

where

In particular,

(Exponential convergence to the mean)

Large deviation estimate
discretization

Will allow replacement of “all states” by “all states in M”

in definition of ε-randomization.

Discretization

Find anε-net M:

For all pure states φ, there exists a state

φ’ in M such that

Cd

combine not as bad at it looks

defn

ε-net

union bound

large deviations

Combine (Not as bad at it looks)
quantum data hiding

? ?

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]

hiding qubits

Security:

Decodability:

Hiding qubits

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

[dV H T, 2002]

security

Similar calculation:

Choose

Security

Hinges on showing

(Rank one projectors)

data hiding summary

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 summary
conclusion
Conclusion
  • 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
overview
Act I

LOCC data hiding for quantum states

(quant-ph/0207147)

Act II

From RSP to PQC to data hiding

Overview
slide30

A few years ago in

a lab moderately

far away...

nonlocality without entanglement

? ?

0 1 2

0

1

2

Not always possible:

Nonlocality without entanglement

[BDFMRSSW, 1999]

hiding a qubit first attempt1
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
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
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

minor algebra

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 algebra
defeating the cbit hiding

The 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 hiding

Conclusion from previous slide: there is a k such that L0Lk

imperfect hiding
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.

multipartite cbit hiding

 Authorized sets can recover the secret using quantum communication

Multipartite cbit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn i

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

multipartite qubit hiding

This authorization structure

is impossible due to no-cloning!

Multipartite qubit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn

 Authorized sets can recover the secret using quantum communication

multipartite qubit hiding1

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

Multipartite qubit hiding

A

E

B

D

C

 With LOCC alone the five parties cannot learn

 Authorized sets can recover the secret using quantum communication

quantum secret sharing
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
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
Act II

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

on the meaning of
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
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
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
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!

competing visions
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 visions
summary
Summary
  • 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
Building the POVM

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

from rsp to 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.)

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