Entanglement assisted state preparation and channel simulation
Entanglement assisted state preparation
and channel simulation
Debbie Leung1, Peter Shor2, Andreas Winter3 (heavy lifters)
Charles H. Bennett1 (cheerleader)
1IBM, 2AT&T, 3Bristol,
ARO, NRO, ERATO, ITP,
(Northampton, MA Ford Dealership)
Robert Owen, Charles Fourier, Edward Bellamy:
Free goods & services will make everything better.
Fourier, Emma Goldman…1967 Summer of Love
Free Love will make everything better
Timothy Leary, Ken Kesey:
Free LSD will make everything better
(Aram Harrow, poster session)
Will Free Entanglement change the world?
…well then, will it at least simplify the theory of quantum
channels?
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
1
Entanglement assisted state preparation and channel simulation
• Problem of Multiple Capacities of Quantum Channels
• Classical Reverse Shannon Theorem
• Quantum Reverse Shannon Theorem (QRST)
• Oblivious Remote State Preparation
• (Simulated Quantum Time Travel)
Quantum capacities vanish.
Channel can be classically
simulated, but its classical
capacity is still enhanced by
entanglement.
Fully quantum
behavior. Channel
can cary
quantum
info directly
(Q > 0).
Noiseless
Channel
Channel can carry
quantum info, but
only with the help of
a 2-way classical side
Fully classical behavior:
channel can be classically
simulated, has no quantum
capacity, and its classical
capacity is not increased by
entanglement.
channel: (Q=0 but Q2 >0)
Channel has positive quantum capacity Q, but its
classical capacity is not increased by entanglement (Shor 0106052).
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
2
Entanglement assisted state preparation and channel simulation
Multiple Capacities of Quantum Channels
Alice
Alice
Noisy quantum channel
Bob
Q plain quantum capacity = qubits faithfully trasmitted per channel use,
via quantum error correcting codes
No simple expression
C plain classical capacity = bits faithfully trasmitted per channel use
No simple expression
Q2 classically assisted quantum capacity, i.e. qubit capacity in the
presence of unlimited 2-way classical communication, (e.g. using
entanglement distillation and teleportation)
CE
No simple expression
entanglement assisted classical capacity i.e. bit capacity in the
presence of unlimited prior entanglement between sender and
receiver. Simple expression, analogous to classical channel’s capacity
Classical Channel Capacity
x
N
y
C (N) = maxp(x)
H(X) + H(Y) − Η(X,Y)
The capacity of a classical channel noisy channel N is the
maximum, over input distributions, of the (Shannon)
entropy of the inputs X, plus that of the outputs Y, minus
their joint entropy.
Shannon’s noisy coding theorem shows that any noisy
channel N can simulate a noiseless bit channel with
efficiency approaching C(N) and fidelity approaching
1 in the limit of large block size.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
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Entanglement assisted state preparation and channel simulation
ρ
Φρ
(entangled
purification
of ρ)
R
ρ
ρQ
Q
Ν (ρ)(
Ν
CE (Ν) = maxρ
R
RQ
Ν ⊗Ι ))(Φρ)
S(ρ) + S(Ν(ρ)) − S(Ν⊗ Ι (Φρ))
Entanglement-Assisted capacity CE of a quantum channel Ν is equal
to the maximum, over channel inputs ρ, of the input (von Neumann)
entropy plus the output entropy minus their “joint” entropy (more
precisely the joint entropy of the output and a reference system entangled
with the late input) (BSST 0106052).
Entanglement-assisted capacity is thus the natural quantum generalization
of the classical capacity of a classical channel.
Ν
Α
Β
CE(Ν ))
The complicated theory of quantum channel capacity would be greatly
simplified if the Quantum Reverse Shannon Theorem (QRST) were
true: any quantum channel can be asymptotically simulated by prior
entanglement and an amount of classical communication equal to
its entanglement assisted capacity. Then, in a world full of entanglement,
all quatum channels would be qualitatively equivalent, and quantitatively
could be characterized by a single parameter.
Β
ξ1⊗ξ2...⊗ξm
Α
≈Ν ⊗m(ξ1⊗ξ2...⊗ξm)
≈mCE(Ν )
classical bits)
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
4
Entanglement assisted state preparation and channel simulation
The QRST is now known to hold for many
special classes of channels (e.g. CQ channels,
Bell-diagonal channels…) and, through recent
work of Shor and Winter, for all (quantum
discrete memoryless) channels when their
inputs are drawn from a fixed distribution ρ.
But first let’s look at the classical reverse
Shannon theorem.
Classical Shannon Theorem:
A noisy channel can simulate a noiseless channel
Alice
=
Bob
Homer Simpson's Reverse Shanon's Theorem:
A noiseless channel can simulate a noisy channel.
Alice
=
Bob
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
5
Entanglement assisted state preparation and channel simulation
Alice
=
Bob
A Better Reverse Shannon Theorem (quant-ph/0106052)
In the presence of shared random information between sender and receiver,
a noiseless channel can asymptotically simulate a noisy one of equal capacity.
Alice
Common
Random
Source
=
Bob
Therefore, in the presence of shared random information,
all classical noisy channels are asymptotically equivalent.
Simulation Method: Alice and Bob first preagree on a sparse
set SR of 2n(C+δ) n-bit strings, using their shared random info R.
Alice receives
input string x
Next she simulates the
channel locally to get a
provisional output y
Then she picks y' in SR
at same distance from x
as y was, and tells Bob
its index using n(C+δ) bits.
Range of d values for which SR
typically includes at least one
member y' at distance d from x.
n
0
Distribution of Hamming distances d=|x-y|
induced by noisy channel.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
6
Entanglement assisted state preparation and channel simulation
In the large m limit, sending m bits through the noisy channel
can be simulated by sending about mC noiseless “intrinsic”
bits, which Alice chooses with the help of the input,
Alice
Common
Random
Source
intrinsic
Bob
extrinsic
and about m(1-C) “extrinsic” random bits, which
have nothing to do with the channel input, and so can
be preagreed before Alice receives the input.
A similar situation exists with POVMs, which can be thought of as
QC channels, ie channels with a quantum input and classical output.
For simplicity we assume the inputs to be qubit states uniformly
distributed on the surface of the Bloch sphere.
Consider the “BB84” POVM , whose elements are projectors onto
four states equally spaced around the equator of the Bloch sphere.
The Homer Simpson way of doing this POVM is to have Alice
perform the POVM, then tell Bob the 2 bit result that she found. A
more economical way of doing it is to have Alice and Bob preagree
on one bit of extrinsic information,---which basis Alice will use--then have Alice perform an X or Y measurement and tell Bob the
1-bit result. The 4-outcome POVM has thus been faithfully
expressed as a mixture of two 2-outcome measurements, where the
choice of which of these to perform can be made extrinsically, ie
without regard to the input state on which the measurement is to be
performed.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
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Entanglement assisted state preparation and channel simulation
A less trivial case is the “trine” POVM, comprising three projectors
equally spaced around the equator. Thinking of this as a QC
channel, it takes a qubit state as input and produces an output 0, 1,
or 2. This POVM cannot be exactly expressed as a mixture of 2outcome measurements. But asymptotically, in the limit of large
block size l, the performance of l trine measurements can be
approximated as an extrinsic mixture of about (3/2) l blockwise
measurements, each with about 2 l intrinsic outcomes.
Thus, on large blocks, the input/output statistics of trine
measurements can be simulated by about 1.58 bits of preagreed
random information and about 1 bit of classical communication per
trine measurement.
This may be thought of as the dual of the fact that, when used as a
CQ channel, the trine states asymptotically can communicate 1 bit
per channel use (Holevo, Schumacher, Westmoreland).
Winter's POVM Compression theorem quant-ph/0109050
(building on Massar & Popescu '00, Winter & Massar '01)
Given a density matrix ρ and a POVM a = {aj}, define the
one-shot output probabilities λj=Tr ρaj., and the square root
ensemble ρj = (√
√ρ) aj (√
√ρ) / λj realizing ρ. Then for any
tolerance ε>0, there exists a block size l and a POVM B,
which is a good approximation to A=a⊗l, and where B can be
expressed as a convex combination B=Σ
Σν xνBν of constituent
POVMs Bν each having at most M outcomes, where log M ≈
l (S(ρ) - Σj λj S(ρj) ) is the Holevo information of the square
root ensemble. The approximation is good in the sense that for
any ensemble {qk , σk } realizing ρ⊗l, if one sums over possible
inputs k and outputs J=j1 , j2 … jl , of the two POVMs, the
probability difference Σk,J |Tr qkσk (AJ - BJ)| will be < ε.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
8
Entanglement assisted state preparation and channel simulation
Sketch of Shor’s proof of QRST for fixed sources, using POVM
compression theorem. Alice’s wants to simulate a general noisy
channel Ν, using shared entanglement and as little classical
communication to Bob as possible. Let Ν
b e defined by the
Kraus operators {Aj j=1…δ} so on input state ξ the channel output is
Σj Aj ξ Aj † . Let Φin and Φout denote projectors onto maximally
entangled states of sized to the input and output dimensions of Ν.
Let Uj be dout dimensional generalized Pauli matrices.
Generalized Teleportation: Given input ξ, Alice performs a POVM
with elements (I⊗ U*j A*k) Φin (I⊗ ATk UTj) on the input ξ and her
half of a specimen of Φout, after which she tells Bob only j, the index
of which Pauli she performed. He undoes the Pauli, and is left with
Ν (ξ). Τhis uses 2 log dout bits of classical communication.
• Using Winter’s theorem Alice and Bob approximate this POVM by
another with an intrinsic cost of S(ρ) + S(Ν(ρ) )−S(( Ι ⊗Ν)(Ψρ)
where Ψρ is a purification of the input density matrix ρ.
ξ
Ν
Compressed
Generalized
Bell Measurement
CE(Ν ))
UkT
Entanglement
for Teleportation
Ν (ξ)
Shared Randomness
for POVM Compression
(can be created using more
entanglement)
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
9
Entanglement assisted state preparation and channel simulation
Teleportation versus Remote State Preparation
ξ
Α
Β
ξ
In teleportation Alice needs only a single specimen of the
unknown state, and can learn nothing about it. Bob receives
a single specimen of the state, but learns nothing more about
it. Teleportation is thus oblivious for both Alice and Bob.
Conventional Remote State Preparation 0006044
ξ
Α
Asymptotic
classical cost 1 bit
instead of 2 bits.
Β
ξ
In Remote State Preparation, Alice starts with complete
classical information about the state to be prepared. Bob gets
a specimen of it, but existing protocols leave Bob with
various byproducts that tell him more about it. Thus these
protocols are thus oblivious neither for Alice nor for Bob.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
10
Entanglement assisted state preparation and channel simulation
New Result (Leung, Shor):
If a remote state preparation protocol, using forward
communication and shared entanglement only, is exact and
oblivious for Bob, then it must be oblivious for Alice.
In other words, it can be performed without Alice classically
knowing, or learning, what state she is preparing. Formally,
Alice’s actions can be simulated by a POVM on an unknown
state (provided as a quantum input) and the entanglement she
shares with Bob. Therefore, if a remote state preparation
protocol leaks no extra information to Bob, it must, like
teleportation, cost 2 bits of classical communication per qubit
remotely prepared.
This is like fact that politicians typically find it necessary to
use more words to avoid conveying unwanted information.
Future:
•Prove QRST for an arbitrary sequence of input states,
not necessarily drawn from a fixed distribution
•Apply to other problems in channel capacity and
remote preparation.
• Average vs worst case cost of remote preparation
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
11
Entanglement assisted state preparation and channel simulation
(Unpublished “work” by B. Schumacher and CHB)
Interaction with one’s past self using an exotic
physical time machine, such as a Wormhole
time-reversed portion of trajectory
ψ
U
ψ′
Grandfather type paradoxes give rise to constraints on
allowable combinations of interaction and initial state.
Time travel can be simulated using entanglement and post selection
Bell
measurement
1
ψ
Φ0
2
3
Φ0?
U
2’
yes
Post
selection
ψ′
If the post selection was successful, qubit 1 may be viewed as a
time-reversed version, and qubit 3 may as a time-traveled version of
qubit 2’. The overall input-output state mapping is what it would
have been with a physical time machine, for 0,1 basis states, and for
arbitrary superpositions. Pure inputs are mapped to pure outputs.
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
12
Entanglement assisted state preparation and channel simulation
U = CNOT, ψ = |0
|0〉〉
Φ0?
|0〉〉
|0
Success
probability
= ½
yes
Φ0
ψ′ =
Post selection
(|0
|0〉〉 + |1
|1〉〉) /√2
(|000
|000〉〉 +|101
|101〉〉) /√2
(|000
|000〉〉 + |111
|111〉〉) /√2
Classical Post Selection can also be used to simulate time travel,
but then the output will generally be a mixed state, whereas
quantumly it is a pure state.
b
0
2
2’
2’ = b
?
Success
probability
= 1/ 2
yes
Guess bit b
ε{0,1} and
make 2
copies of
it.
b
3
Post selection
Output =
0 or 1 equiprobably.
000 or 10
1011
equiprobably
000 or 11
1111
equiprobably
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
13
Entanglement assisted state preparation and channel simulation
Φ0
Φ0?
Φ0
U
Ψ
may not be
maximally
entangled
yes
Postselection
Any output state Ψ can be synthesized from an initial maximally
entangled state by such a simulated time travel circuit.
Simple Case: U = Swap
Φ0?
ψ
Φ0
Success
probability
= 1
yes
Post selection
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
ψ′ =ψ
14
Entanglement assisted state preparation and channel simulation
U = CNOT, ψ = |1
|1〉〉
gives rise to Grandfather paradox.
Φ0?
|1〉〉
|1
Success
probability
= 0
yes
Φ0
Undefined
Post selection
Output
(|010
|010〉〉 +|111
|111〉〉) /√2
(|010
|010〉〉 + |101
|101〉〉) /√2
U = CNOT,
|0〉〉 +β|1
|1〉〉
ψ = α|0
Φ0?
ψ
Φ0
Success
probability
= α2 / 2
yes
Post selection
ψ′ =
(|0
|0〉〉 + |1
|1〉〉) /√2
α (|000
|000〉〉 +|101
|101〉〉) /√2
+β (|010
|010〉〉 +|11
|1111〉) /√2
α (|000
|000〉〉 +|111
|111〉〉) /√2
+β (|010
|010〉〉 +|10
|1011〉) /√2
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
15
Entanglement assisted state preparation and channel simulation
Q. Is it time travel?
A.
It depends on what your definition of “is” is.
Q. Doesn’t it imply cloning?
A. No. The older and younger versions of same qubit
are not independent and cannot be separately
measured to get a better state estimate. By
measuring the younger, the older is affected.
Q. What is the output state when the post selection fails?
A. Shut up! (More seriously, the output state of the
procedure is defined as what it would be if the post
selection succeeded, so it is only undefined when the
success probability is zero.)
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
16
Entanglement assisted state preparation and channel simulation
Charles Bennett, IBM, Watson (ITP Quantum Info Conference 12/7/01)
17