Quantum Entanglement and Nonlocality: Complexity challenges in quantum information processing

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Quantum Entanglement and Nonlocality Complexity
challenges in quantum information processing
Andrew Doherty collaboration with P. Parrilo, F.
Spedalieri, B. Terhal, D. Schwab
KITP Santa Barbara , 21 March 2003
Caltech MURI Center for Quantum
Networks DARPA/ARO/DDRE
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Outline
Background Quantum mechanics, Bell inequalities,
entanglement and teleportation.
Using convex relaxations to generate a hierarchy
of tests for entanglement.
Duality interpreting the primal and dual forms
physically, constructing measurements to
demonstrate entanglement, locality.
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Bell Experiment
Quantum mechanics has a tenuous hold on locality.
Alice and Bob make measurements in distant
laboratories.
In any sensible theory the outcomes of
Alices measurements are unaffected by Bobs
choice of setting. Local quantum measurements
violate this intuition.
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A New (Quantum) Engineering
In an increasing number of physical systems it
is now possible to monitor, manipulate and feed
back to individual quantum systems (atoms,
photons,...).
Imagine if the physical resources for
computation, communication, control were quantum
and not classical.
Quantum systems like this promise to allow us to
build better computers, share secret keys for
cryptography, allow a range of information
processing tasks.
Quantum analogs of control and systems theory,
information theory and computer science are being
developed and seem interestingly different.
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Teleportation
Getting states from one place to another.
Alice and Bob still in distant laboratories.
Alice is on the phone to Bob.
Quantum State.
measurements
feedback
Answer two yes or no questions and Alice can
teleport the state to Bob.
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Quantum States
States of quantum systems are described by
positive operators

The states of physical systems at different
locations are described by operators on the
tensor product space of the two systems.
Entangled state
Cant assign a single state to each subsystem
Shroedinger, EPR 1935.
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Quantum Dynamics and Measurements
Dynamics preserve positivity of states. In fact
they are completely positive maps.

where
with probability
Unitary evolution and projective measurement are
special cases. Local dynamics respect tensor
product.
If were just interested in measurements, the
possible outcomes are
Positive operator valued measure (POVM).
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Quantum Dynamics and Measurements II

Quantum states and evolutions have a nice
convexity structure.
There is a physically important duality between
states (Schroedinger picture) and observables
(Heisenberg picture).
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Quantum Dynamics and Measurements III

Local properties are encoded in the tensor
product, this leads naturally to higher order
polynomials.
Local dynamics and local measurements respect the
tensor product structure. We can average over
systems we dont care about by defining a reduced
density operator.
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Convex Optimization is the Right Tool for Many
Problems in Quantum Information and Control
Approximate state transformations Rains,
Audenaert and De Moor,
Chefles, Winter and Jozsa
Distinguishing entangled states Parrilo,
Doherty, Spedalieri
Quantum Coin Tossing Kitaev
Local Hidden Variable Constructions Terhal,
Doherty, Schwab.
All involve semidefinite programming.
http//www.arxiv.org/quant-ph
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Entangled Quantum States
Dynamics preserve positivity of states. Local
dynamics respect tensor product.

with some probability
Entanglement relates to the resources necessary
to prepare states of bipartite systems. Compare
with controllability
Werner 1989.
Entangled state cant be prepared by local
operations and classical communication. Cant be
written as convex combination of product states.
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Entanglement as a Resource
Entanglement is also useful for things, which is
why we are interested in it. Entanglement is
necessary for Bell inequality violations and
teleportation, most quantum information
processing.
In this context we would like to know whether or
not a given state is entangled, whether it is
good for teleportation, whether it violates a
Bell Inequality. For pure states problem is
simple Entangled pure states are easy to
distinguish, all are good for teleportation, all
violate some Bell inequality. For mixed states,
incomplete results.
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Distinguishing Separable from Entangled States
Doherty, Parrilo, Spedalieri Phys. Rev. Lett. 88
(2002) 187904 Terhal, Doherty Schwab,
quant-ph/0210053
For every entangled state, there is some
observable that could be measured in the lab that
shows that it is entangled. (Entanglement Witness)
As a bonus our tests construct these observables.
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Distinguishing Separable from Entangled States
Doherty, Parrilo, Spedalieri Phys. Rev. Lett. 88
(2002) 187904 Terhal, Doherty Schwab,
quant-ph/0210053
It is possible construct these extensions by a
series of efficient algorithms. (Semidefinite
programs) If an extension cannot be created at
any stage the state is entangled. Each step in
the series is able to detect the entanglement of
more states until eventually all entangled states
are detected.
Similar techniques are able to construct local
hidden variable theories for quantum states.
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First Relaxation
Restrict attention to a special form for Z
The biquadratic Hermitian form Z is restricted to
be a sum of squared magnitudes.
r is entangled if there is a Z such that
Semidefinite Program!
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Positive Partial Transpose Criterion

• First relaxation (Peres, Horodecki, 1996)
• If is not positive then is entangled.
• This is equivalent to our first relaxation.
• States with positive partial transposes (PPT
  states) may be separable.
• Several examples of entangled
• PPT states are known. The cone
• of PPT states is strictly larger than
• the cone of separable states.
• PPT criterion is necessary and
• sufficient in low dimensions
  

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NPT States are Robustly Entangled?

• No PPT states are useful for teleportation
  (Horodecki3 1999)
• No known violations of Bell inequalities by PPT
  states.
• While it is frequently possible to extract pure
  entangled states from mixed ones, this is
  impossible for PPT states.
• Rare apparently, zero measure
• in infinite dimension

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Example Form
Equally not all positive biquadratic Hermitian
forms can be written as a sum of squares Here is
an example in three variables
Choi 1975.
Proof of positivity was originally by means of
arithmetic-geometric inequality.
Semidefinite programming methods allow us to
generate short proofs both of positivity and
indecomposibility.
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Second Relaxation
We get a less trivial program if we consider a
larger set of entanglement witnesses.
minimize
subject to
If the minimum is less than zero then the state
is entangled and we have constructed an
indecomposible entanglement witness by explicit
search.
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Results
Tested on many examples of bound entangled states
in , , , and

Computation scales at worst like
Bound entangled states taken from
Horodecki 1997, HHH 1999, Horodecki, Lewenstein
2000, Bruss, Peres 1999, Bennett et al. 1999
Dont know of an entangled state that doesnt
have an entanglement witness of this kind.
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Sum of Squares Decomposition
Numerical work can also provide short proof of
positivity. (Choi form for example)
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Higher Order Relaxations
In general it is possible to create a hierarchy
of tests, each of them a semidefinite program,
corresponding to searching over the set of
entanglement witnesses such that
We can show that for k sufficiently large any
entangled state will be detected by this
hierarchy. Complete sequence of easy tests,
but in fact NP-Hard problem if we increase the
system size (Gurvits).
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Some Physical Insight
We can understand the primal semidefinite program
physically
Imagine r is separable
Consider the state
State of any copy of A and B is r
State has positive partial transposes.
Can relate these objects to Bell inequalities.
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Constructing Symmetric Extensions
This problem can be cast as a semidefinite
program.
Partial trace condition
Expand in basis for matrices
Just consider basis elements
minimize subject to
Extension exists if optimum less than one.
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Semidefinite Programs
minimize
subject to
If conditions can be satisfied problem is feasible
Minimize linear function over intersection of
affine subspace with cone of positive matrices.
Excellent numerical methods available.
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Semidefinite Programming Duality
Consider a feasible point
So for feasible points
Constraints guarantee that feasible primal values
bound dual optimum and vice versa.
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Dual Problem
The optimization dual to the one that constructs
extensions searches for an entanglement witness.
maximize subject to
If optimum is greater than one (no extension
exists) Z is an entanglement witness.
As a test for entanglement this is neither weaker
nor stronger than partial transpose criterion.
Fewer variables means easier to deal with
analytically.
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Bell Experiment
Quantum mechanics has a tenuous hold on locality.
Alice and Bob make measurements in distant
laboratories.
In any sensible theory the outcomes of
Alices measurements are unaffected by Bobs
choice of setting. Local quantum measurements
violate this intuition.
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Bell Experiment Deterministic
Shared randomness.
measurements
measurements
outcomes
outcomes
Each of Alices measurements has some outcome
with probability one, independent of the
measurement made by Bob.
Alices outcomes.
Bobs outcomes.
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Bell Experiment II
Shared randomness.
Each of the deterministic outcomes occurs with
some probability.
Bell polytope.
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Bell Experiment Quantum
Quantum State.
measurements
measurements
outcomes
outcomes
Each measurement is described by a POVM.
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Local Hidden Variable Theory
Reproduces the measurement probabilities of a
quantum mechanical Bell experiment as a convex
combination of the deterministic classical
outcomes.
If this is not possible the state violates some
Bell inequality for this number of settings.
Entanglement is necessary for such a violation.
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Strategy
No Bell inequality violations if different POVMs
commute or if Bob only has one setting.
Try to replace r by another state for which LHV
is obvious and
Extensions of r fit the bill if they are suitably
symmetric.
Extension for r
Whichever spaces are traced over.
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Extensions and LHVs
If r has a (s,1)-symmetric extension then it does
not violate a Bell inequality for s settings for
Alice and any number of settings for Bob.
Imaginary Bell experiment Alice and Bob share
and Alice performs each of her POVMs on a
different one of her copies of system A.
(Essentially only a single POVM)
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Comments
Result works for all POVMs (not just projective
measurements) and any number of measurement
outcomes for each setting.
Since all separable states have symmetric
extensions there is an LHV of this kind for every
Bell experiment on every separable state.
Many entangled states (PPT and NPT) have
(1,s)-symmetric quasi-extensions for small s
(only). (e.g. real UPB states, Werner states.)
There must be other kinds of LHV theories that
are relevant for entangled states. (Result of
Werner.)
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Summary and Outlook
Its possible to extend previously known tests
for entanglement to a whole hierarchy that is in
a sense complete.
These tests amount to looking for local hidden
variable theories for states, or for observables
that prove that the state is entangled.
Theory is it possible to understand the set of
states that violate local realism or that are
useful for teleportation? Might this be a better
way to define entanglement?.
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Outstanding questions
Is our second relaxation complete in low
dimensions? Can we find entangled states not
detected by the second relaxation? (Are there
doubly bound entangled states?) Does our
sequence of tests eventually detect all
entangled states? (Or can it be made to?) Is the
separability problem exponentially hard? Can we
use similar techniques to find product state
decompositions (or Bell inequality violations)?
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Some Progress
Doherty, Parrilo, Spedalieri Phys. Rev. Lett. 88
(2002) 187904 Terhal, Doherty Schwab,
quant-ph/0210053
If two systems are highly entangled they are less
able to be entangled with other systems.
On the other hand separable states can be shared
out as much as you like.
r is separable
Consider the state (extension)
State of any copy of A and B is r
Pure entangled states and most mixed ones dont
have extensions like this.
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Quasi-Extensions and LHVs
Checking the LHV requires verifying the equality
This holds as a result of the partial trace and
symmetry properties of and the normalization
of the POVMs.
Positivity of only serves to guarantee
positivity of the probabilities. So in fact we
only need
only needs to be positive on product
states quasi-extension
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Constructing Symmetric Extensions
This problem can be cast as a semidefinite
program.
Partial trace condition
Expand in basis for matrices
Just consider basis elements
minimize subject to
Extension exists if optimum less than one.
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Semidefinite Programs
Primal
Dual
maximize
minimize
subject to
subject to
If conditions can be satisfied problem is feasible
Minimize linear function over intersection of
affine subspace with cone of positive matrices.
(separability, distillation Rains, CPM
optimization Audenaert De Moor)
Excellent numerical methods available.
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Semidefinite Programming Duality
Consider a feasible point
So for feasible points
Constraints guarantee that feasible primal values
bound dual optimum and vice versa.
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Semidefinite Programming Duality II
A nice sub-class of problems are termed strictly
feasible.
If there is a feasible point such that
then the primal and dual optimal values are the
same and there is a point attaining
these values.
Numerical methods tend to attempt to minimize the
difference between the two optimal values
(duality gap).
For the strictly feasible case we are guaranteed
certificates of optimality as well as points
attaining the optimum.
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Dual Problem
The optimization dual to the one that constructs
extensions searches for an entanglement witness.
maximize subject to
If optimum is greater than one (no extension
exists) Z is an entanglement witness.
As a test for entanglement this is neither weaker
nor stronger than partial transpose criterion.
Fewer variables means easier to deal with
analytically.
46
Results
Tested on many examples of bound entangled states
in
Computation scales at worst like
Bound entangled states taken from
Horodecki 1999, Bennett et al. 1999 Horodecki,
Lewenstein 2000, Bruss, Peres 1999
Analytically, extensions exist for real UPB
states and s2, and Werner states for sltd.
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Summary and Outlook
Described a method of constructing local hidden
variable theories for quantum states that works
for all separable states and some entangled ones.
Construction works for a fixed number of settings
for Alice (say) but for any POVMs with any number
of outcomes.
Numerically and analytically tractable since a
semidefinite program.
Other kinds of local hidden variable exist
(Werner). Perhaps it would be fruitful to find a
more complete characterization of LHV theories.
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Conclusions
Have described a hierarchy of new separability
criteria.
These are computationally and theoretically
attractive since they require solving a
semidefinite program.
This suggests that there are interesting
subclasses of the PPT entangled states, members
of which are as yet unidentified.
Solution of the program involves the construction
of explicit entanglement witnesses. Equally
given indecomposible positive maps from the
literature we can construct bound entangled
states and simplified proofs of positivity.
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A New Separability Criterion
Suppose r is separable
Consider the state

• supported on symmetric subspace
• an extension of r
• has positive partial transposes