theory of quantum entanglement
TRANSCRIPT
Theory of quantum entanglement
Mark M. Wilde
Hearne Institute for Theoretical Physics,Department of Physics and Astronomy,
Center for Computation and Technology,Louisiana State University,
Baton Rouge, Louisiana, USA
On Sabbatical withStanford Institute for Theoretical Physics,
Stanford University,Stanford, California 94305
Mitteldeutsche Physik Combo, University of Leipzig, Leipzig, Germany, September 17, 2020
Mark M. Wilde (LSU) 1 / 103
Motivation
Quantum entanglement is a resource for quantum informationprocessing (basic protocols like teleportation and super-dense coding)
Has applications in quantum computation, quantum communication,quantum networks, quantum sensing, quantum key distribution, etc.
Important to understand entanglement in a quantitative sense
Adopt axiomatic and operational approaches
This tutorial focuses on entanglement in non-relativistic quantummechanics and finite-dimensional quantum systems
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Quantum states...
(Image courtesy of https://en.wikipedia.org/wiki/Quantum_state)
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Quantum states
The state of a quantum system is given by a square matrix called thedensity matrix, usually denoted by ρ, σ, τ , ω, etc. (also calleddensity operator)
It should be positive semi-definite and have trace equal to one.That is, all of its eigenvalues should be non-negative and sum up toone. We write these conditions symbolically as ρ ≥ 0 and Trρ = 1.Can abbreviate more simply as ρ ∈ D(H), to be read as “ρ is in theset of density matrices.”
The dimension of the matrix indicates the number of distinguishablestates of the quantum system.
For example, a physical qubit is a quantum system with dimensiontwo. A classical bit, which has two distinguishable states, can beembedded into a qubit.
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Interpretation of density operator
The density operator, in addition to a description of an experimentalprocedure, is all that one requires to predict the (probabilistic)outcomes of a given experiment performed on a quantum system.
It is a generalization of (and subsumes) a probability distribution,which describes the state of a classical system.
All probability distributions can be embedded into a quantum state byplacing the entries along the diagonal of the density operator.
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Mixtures of quantum states
A probabilistic mixture of two quantum states is also a quantumstate. That is, for σ0, σ1 ∈ D(H) and p ∈ [0, 1], we have
pσ0 + (1− p)σ1 ∈ D(H).
The set of density operators is thus convex.
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Mixed states and pure states
A density operator can have dimension ≥ 2 and can be written as
ρ =∑i ,j
ρi ,j |i〉〈j |,
where |i〉 ≡ ei is the standard basis and ρi ,j are the matrix elements.
Since every density operator is positive semi-definite and has traceequal to one, it has a spectral decomposition as
ρ =∑x
pX (x)|φx〉〈φx |,
where pX (x) are the non-negative eigenvalues, summing to one,and |φx〉 is a set of orthonormal eigenvectors.
A density operator ρ is pure if there exists a unit vector |ψ〉 such thatρ = |ψ〉〈ψ| (rank = 1) and otherwise it is mixed (rank > 1).
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Multiple quantum systems...
IBM 65-qubit universal quantum computer (released September 2020)
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Composite quantum systems
If the state of Alice’s system is ρ and the state of Bob’s system is σand they have never interacted in the past, then the state of the jointAlice-Bob system is
ρA ⊗ σB .
We use the system labels to say who has what.
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Composite quantum systems (ctd.)
More generally, a generic state ρAB of a bipartite system AB acts ona tensor-product Hilbert space HAB ≡ HA ⊗HB .
If |i〉Ai is an orthonormal basis for HA and |j〉Bj is anorthonormal basis for HB , then |i〉A ⊗ |j〉Bi ,j is an orthonormalbasis for HAB .
Generic state ρAB can be written as
ρAB =∑i ,j ,k,l
ρi ,k,j ,l |i〉〈k |A ⊗ |j〉〈l |B
where ρi ,k,j ,l are matrix elements
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Quantum entanglement...
Depiction of quantum entanglement taken fromhttp://thelifeofpsi.com/2013/10/28/bertlmanns-socks/
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Separable states and entangled states
If Alice and Bob prepare states ρxA and σxB based on a randomvariable X with distribution pX , then the state of their systems is∑
x
pX (x)ρxA ⊗ σxB .
Such states are called separable states [Wer89] and can be preparedusing local operations and classical communication (LOCC). No needfor a quantum interaction between A and B to prepare these states.
By spectral decomposition, every separable state can be written as∑z
pZ (z)|ψz〉〈ψz |A ⊗ |φz〉〈φz |B ,
where, for each z , |ψz〉A and |φz〉B are unit vectors.
Entangled states are states that cannot be written in the aboveform.
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Example of entangled state
A prominent example of an entangled state is the ebit (eee · bit):
|Φ〉〈Φ|AB ,
where |Φ〉AB ≡ 1√2
(|00〉AB + |11〉AB).
In matrix form, this is
|Φ〉〈Φ|AB =1
2
1 0 0 10 0 0 00 0 0 01 0 0 1
.To see that this is entangled, consider that for every |ψ〉A and |φ〉B
|〈Φ|AB |ψ〉A ⊗ |φ〉B |2 ≤1
2
⇒ impossible to write |Φ〉〈Φ|AB as a separable state.
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Tool: Schmidt decomposition
Schmidt decomposition theorem
Given a two-party unit vector |ψ〉AB ∈ HA ⊗HB , we can express it as
|ψ〉AB ≡d−1∑i=0
√pi |i〉A |i〉B , where
probabilities pi are real, strictly positive, and normalized∑
i pi = 1.
|i〉A and |i〉B are orthonormal bases for systems A and B.[√pi]i∈0,...,d−1 is the vector of Schmidt coefficients.
Schmidt rank d of |ψ〉AB is equal to the number of Schmidtcoefficients pi in its Schmidt decomposition and satisfies
d ≤ min dim(HA), dim(HB) .
Pure state |ψ〉〈ψ|AB is entangled iff d ≥ 2.
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Tool: Partial trace
The trace of a matrix X can be realized as
Tr[X ] =∑i
〈i |X |i〉,
where |i〉i is an orthonormal basis.
Partial trace of a matrix YAB acting on HA ⊗HB can be realized as
TrA[YAB ] =∑i
(〈i |A ⊗ IB)YAB(|i〉A ⊗ IB),
where |i〉Ai is an orthonormal basis for HA and IB is the identitymatrix acting on HB .
Both trace and partial trace are linear operations.
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Interpretation of partial trace
Suppose Alice and Bob possess quantum systems in the state ρAB .We calculate the density matrix for Alice’s system using partial trace:
ρA ≡ TrB [ρAB ].
We can then use ρA to predict the outcome of any experimentperformed on Alice’s system alone.
Partial trace generalizes marginalizing a probability distribution:
TrY
[∑x ,y
pX ,Y (x , y)|x〉〈x |X ⊗ |y〉〈y |Y]
=∑x ,y
pX ,Y (x , y)|x〉〈x |X Tr[|y〉〈y |Y ]
=∑x
[∑y
pX ,Y (x , y)
]|x〉〈x |X =
∑x
pX (x)|x〉〈x |X ,
where pX (x) ≡∑y pX ,Y (x , y).
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Purification of quantum noise...
Artistic rendering of the notion of purification(Image courtesy of seaskylab at FreeDigitalPhotos.net)
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Tool: Purification of quantum states
A purification of a state ρS on system S is a pure quantum state|ψ〉〈ψ|RS on systems R and S , such that
ρS = TrR [|ψ〉〈ψ|RS ].
Simple construction: take |ψ〉RS =∑
x
√p(x)|x〉R ⊗ |x〉S if ρS has
spectral decomposition∑
x p(x)|x〉〈x |S .
Two different states |ψ〉〈ψ|RS and |φ〉〈φ|RS purify ρS iff they arerelated by a unitary UR acting on the reference system. Necessity:
TrR [(UR ⊗ IS)|ψ〉〈ψ|RS(U†R ⊗ IS)] = TrR [(U†RUR ⊗ IS)|ψ〉〈ψ|RS ]
= TrR [|ψ〉〈ψ|RS ]
= ρS .
To prove sufficiency, use Schmidt decomposition.
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Uses and interpretations of purification
The concept of purification is one of the most often used tools inquantum information theory.
This concept does not exist in classical information theory andrepresents a radical departure (i.e., in classical information theory itis not possible to have a definite state of two systems such that thereduced systems are individually indefinite).
Physical interpretation: Noise or mixedness in a quantum state is dueto entanglement with an inaccessible reference / environment system.
Cryptographic interpretation: In the setting of quantum cryptography,we assume that an eavesdropper Eve has access to the fullpurification of a state ρAB that Alice and Bob share. This meansphysically that Eve has access to every other system in the universethat Alice and Bob do not have access to!
Advantage: only need to characterize Alice and Bob’s state in orderto understand what Eve has.
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Quantum channels...
Artistic rendering of a quantum channel(Image courtesy of Shutterstock / Serg-DAV)
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Classical channels
Classical channels model evolutions of classical systems.
What are the requirements that we make for classical channels?
1) They should be linear maps, which means they respect convexity.
2) They should take probability distributions to probabilitydistributions (i.e., they should output a legitimate state of a classicalsystem when a classical state is input).
These requirements imply evolution of a classical system is specifiedby a conditional probability matrix N with entries pY |X (y |x), sothat the input-output relationship of a classical channel is given by
pY = N pX ⇐⇒ pY (y) =∑x
pY |X (y |x)pX (x).
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Quantum channels
Quantum channels model evolutions of quantum systems.
We make similar requirements:
A quantum channel N is a linear map acting on the space of(density) matrices:
N (pρ+ (1− p)σ) = pN (ρ) + (1− p)N (σ),
where p ∈ [0, 1] and ρ, σ ∈ D(H).
We demand that a quantum channel should take quantum states toquantum states.
This means that it should be trace (probability) preserving:
Tr[N (X )] = Tr[X ]
for all X ∈ L(H) (linear operators, i.e., matrices).
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Complete positivity
Other requirement is complete positivity.
We can always expand XRS ∈ L(HR ⊗HS) as
XRS =∑i ,j
|i〉〈j |R ⊗ X i ,jS ,
and then define
(idR ⊗NS)(XRS) =∑i ,j
|i〉〈j |R ⊗NS
(X i ,jS
),
with the interpretation being that “nothing (identity channel)happens on system R while the channel N acts on system S .”
A quantum channel should also be completely positive:
(idR ⊗NS)(XRS) ≥ 0,
where idR denotes the identity channel acting on system R ofarbitrary size and XRS ∈ L(HR ⊗HS) is such that XRS ≥ 0.
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Quantum channels: completely positive, trace-preserving
A map N satisfying the requirements of linearity, trace preservation,and complete positivity takes all density matrices to density matricesand is called a quantum channel.
To check whether a given map is completely positive, it suffices tocheck whether
(idR ⊗NS)(|Φ〉〈Φ|RS) ≥ 0,
where
|Φ〉RS =1√d
∑i
|i〉R ⊗ |i〉S
and d = dim(HR) = dim(HS).
Interpretation: the state resulting from a channel acting on one shareof a maximally entangled state completely characterizes the channel.
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Choi-Kraus representation theorem
Structure theorem for quantum channels
Every quantum channel N can be written in the following form:
N (X ) =∑i
KiXK†i , (1)
where Kii is a set of Kraus operators, with the property that∑i
K †i Ki = I . (2)
The form given in (1) corresponds to complete positivity and the conditionin (2) to trace (probability) preservation. This decomposition is notunique, but one can find a minimal decomposition by taking a spectraldecomposition of (idR ⊗NS)(|Φ〉〈Φ|RS).
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Unitary channels
If a channel has one Kraus operator (call it U), then it satisfiesU†U = I and is thus a unitary matrix.1
Unitary channels are ideal, reversible channels.
Instruction sequences for quantum algorithms (to be run on quantumcomputers) are composed of ideal, unitary channels.
So if a quantum channel has more than one Kraus operator (in aminimal decomposition), then it is non-unitary.
1It could also be part of a unitary matrix, in which case it is called an “isometry.”Mark M. Wilde (LSU) 26 / 103
Measurement channels
Measurement channels take quantum systems as input and produceclassical systems as output.
A measurement channel M has the following form:
M(ρ) =∑x
Tr[Mxρ]|x〉〈x |,
where Mx ≥ 0 for all x and∑
x Mx = I .
Can also interpret a measurement channel as returning the classicalvalue x with probability Tr[Mxρ].
We depict them as
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Quantum instrument
A quantum instrument is a quantum channel with a quantum inputand two outputs: one classical and one quantum [DL70, Oza84].
It is a measurement in which we record not only the classical data,but also keep the post-measurement state in a quantum system
Evolves an input state ρ as follows:
ρ→∑x
Mx(ρ)⊗ |x〉〈x |
where Mxx is a set of completely positive maps such that the summap
∑xMx is trace preserving.
Probability of obtaining outcome x :
Tr[Mx(ρ)]
and post-measurement state in this case is
Mx(ρ)
Tr[Mx(ρ)]
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Purifications of quantum channels
Recall that we can purify quantum states and understand noise asarising due to entanglement with an inaccessible reference system.
We can also purify quantum channels and understand a noisy processas arising from a unitary interaction with an inaccessible environment.
Stinespring’s theorem [Sti55]
For every quantum channel NA→B , there exists a pure state |0〉〈0|E and aunitary matrix UAE→BE ′ , acting on input systems A and E and producingoutput systems B and E ′, such that
NA→B(ρA) = TrE ′ [UAE→BE ′(ρA ⊗ |0〉〈0|E )(UAE→BE ′)†].
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Summary of quantum states and channels
Every quantum state is a positive, semi-definite matrix with traceequal to one.
Quantum states of multiple systems can be separable or entangled.
Quantum states can be purified (this notion does not exist in classicalinformation theory).
Quantum channels are completely positive, trace-preserving maps.
Preparation channels take classical systems to quantum systems, andmeasurement channels take quantum systems to classical systems.
Quantum channels can also be purified (i.e., every quantum channelcan be realized by a unitary interaction with an environment, followedby partial trace). This notion also does not exist in classicalinformation theory.
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Distance measures...
(Image courtesy of https://imgflip.com/memegenerator/One-Does-Not-Simply
and quote from page 1 of Fuchs’ thesis: https://arxiv.org/pdf/quant-ph/9601020)
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Function of a diagonalizable matrix
If an n × n matrix D is diagonal with entries d1, . . . , dn, then for afunction f , we define
f (D) =
g(d1) 0 · · · 0
0 g(d2)...
.... . . 0
0 · · · 0 g(dn)
where g(x) = f (x) if x ∈ dom(f ) and g(x) = 0 otherwise.
If a matrix A is diagonalizable as A = KDK−1, then for a function f ,we define
f (A) = Kf (D)K−1.
Evaluating the function only on the support of the matrix allows forfunctions such as f (x) = x−1 and f (x) = log x .
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Trace distance
Define the trace norm of a matrix X by ‖X‖1 := Tr[√X †X ].
Trace norm induces trace distance between two matrices X and Y :
‖X − Y ‖1 .
For two density matrices ρ and σ, the following bounds hold
0 ≤ 12 ‖ρ− σ‖1 ≤ 1.
LHS saturated iff ρ = σ and RHS iff ρ is orthogonal to σ.
For commuting ρ and σ, normalized trace distance reduces tovariational distance between probability distributions along diagonals.
Has an operational meaning as the bias of the optimal successprobability in a hypothesis test to distinguish ρ from σ [Hel67, Hol72].
Does not increase under the action of a quantum channel:
‖ρ− σ‖1 ≥ ‖N (ρ)−N (σ)‖1 .
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Fidelity
Fidelity F (ρ, σ) between density matrices ρ and σ is [Uhl76]
F (ρ, σ) := ‖√ρ√σ‖21.
For pure states |ψ〉〈ψ| and |φ〉〈φ|, reduces to squared overlap:
F (|ψ〉〈ψ|, |φ〉〈φ|) = |〈ψ|φ〉|2.
For density matrices ρ and σ, the following bounds hold:
0 ≤ F (ρ, σ) ≤ 1.
LHS saturated iff ρ and σ are orthogonal and RHS iff ρ = σ.
Fidelity does not decrease under the action of a quantum channel N :
F (ρ, σ) ≤ F (N (ρ),N (σ)).
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Uhlmann’s theorem
Uhlmann’s theorem [Uhl76] states that
F (ρS , σS) = maxUR
|〈ψ|RSUR ⊗ IS |φ〉RS |2,
where |ψ〉RS and |φ〉RS purify ρS and σS , respectively.
A core theorem used in quantum Shannon theory, and in other areassuch as quantum complexity theory and quantum error correction.
Since it involves purifications, this theorem has no analog in classicalinformation theory.
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Relations between fidelity and trace distance
Trace distance is useful because it obeys the triangle inequality, andfidelity is useful because we have Uhlmann’s theorem.
The following inequalities relate the two measures [FvdG98], whichallow for going back and forth between them:
1−√F (ρ, σ) ≤ 1
2‖ρ− σ‖1 ≤
√1− F (ρ, σ).
Sine distance√
1− F (ρ, σ) [Ras06] has both properties (triangleinequality and Uhlmann’s theorem).
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Entanglement theory...
(Image courtesy of Jurik Peter / Shutterstock)
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Reminder of separable and entangled states
If Alice and Bob prepare states ρxA and σxB based on a randomvariable X with distribution pX , then the state of their systems is∑
x
pX (x)ρxA ⊗ σxB .
Such states are called separable states [Wer89] and can be preparedusing local operations and classical communication (no need for aquantum interaction between A and B to prepare these states).
Pure state entangled iff Schmidt rank ≥ 2 (thus, easy to decide if apure state is entangled)
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Motivation from Bell experiment
Bell experiment [Bel64] consists of spatially separated parties A andB performing local measurements on a state ρAB .
A flips a coin and gets outcome x .
Then performs a measurement Γ(x)a a with outcome a.
B flips a coin and gets outcome y .
Then performs a measurement Ω(y)b b with outcome b.
Conditional probability of observing a and b given x and y :
p(a, b|x , y) = Tr[(Γ(x)a ⊗ Ω
(y)b )ρAB ]
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Motivation from Bell experiment (ctd.)
Separable states have local hidden variable theory in a Bellexperiment (AKA shared randomness strategy)
Suppose that ρAB is separable, so that
ρAB =∑λ
p(λ)σλA ⊗ ωλB
Then conditional probability given by
p(a, b|x , y) =∑λ
p(λ)p(a|x , λ)p(b|y , λ)
where p(a|x , λ) = Tr[Γ(x)a σλA], p(b|y , λ) = Tr[Ω
(y)b ωλA]
Using a separable state, correlations achievable are simulable by aclassical strategy
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Loophole-free Bell test...
Picture of loophole-free Bell test at TU Delft(Image taken from http://hansonlab.tudelft.nl/loophole-free-bell-test/)
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NP-hardness of deciding entanglement
Computationally hard to decide if a state is separable or entangled.
More precisely, consider the following computational decision problem:
Given a mathematical description of the density operator ρAB as amatrix of rational entries and ε > 0. Decide whether
ρAB ∈ SEP(A :B)
or infσAB∈SEP(A:B)
1
2‖ρAB − σAB‖1 ≥ ε.
Decision problem NP-hard to solve for ε ≤ 1poly(dA,dB) [Gur04, Gha10].
This means that if widely believed conjectures in theoretical computerscience are true, the best classical or quantum algorithms will haverunning time exponential in dA × dB .
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Positive partial transpose criterion
Even if it is NP-hard to decide whether a state is separable orentangled, we can look for one-way criteria.
Positive partial transpose criterion [Per96, HHH96]
The following transpose map is a positive map:
T (X ) :=∑i ,j
|i〉〈j |X |i〉〈j |
That is, T (X ) ≥ 0 if X ≥ 0
The transpose map is called “partial transpose” if it acts on one shareof a bipartite operator XAB :
TB(XAB) := (idA⊗TB)(XAB) =∑i ,j
(IA ⊗ |i〉〈j |B)XAB(IA ⊗ |i〉〈j |B)
A state that has a positive partial transpose is said to be a PPT state
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Positive partial transpose criterion (ctd.)
A separable state σAB has a positive partial transpose because
TB(σAB) = TB
(∑λ
p(λ)ρλA ⊗ ωλB
)=∑λ
p(λ)ρλA ⊗ TB(ωλB) ≥ 0
Thus, SEP ⊂ PPT
Containment is strict because ∃ PPT entangled states [Hor97]
Contrapositive: if ρAB has a negative partial transpose, then it isentangled
Example: Applying TB to maximally entangled state gives anoperator proportional to unitary SWAP operator, which has negativeeigenvalues
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Computational complexity and PPT states
Much easier computationally to work with PPT than with SEP
For a Hermitian operator MAB , consider the optimizations
maxσAB∈SEP
Tr[MABσAB ] vs. maxσAB∈PPT
Tr[MABσAB ]
The first is NP-hard, while the second is efficiently computable as asemi-definite program:
maxσAB∈PPT
Tr[MABσAB ] =
maxσABTr[MABσAB ] : σAB ≥ 0,TB(σAB) ≥ 0,Tr[σAB ] = 1
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Resource theory of entanglement...
(Image courtesy of https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.025001)
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Resource theory of entanglement
Idea of general resource theory is that some states are free andothers are resourceful [CG19].
Resource theory of entanglement was the first resource theoryconsidered in QIT [BDSW96]
Entanglement is useful for tasks like teleportation [BBC+93],super-dense coding [BW92], and quantum key distribution [Eke91], sothese are the resourceful states
Separable states are the free states
What are the free operations?
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LOCC channel
In the theory of entanglement and quantum communication, one oftenassumes that Alice and Bob can communicate classical data for free.
Paradigm is local op.’s and classical comm. (LOCC) [BDSW96].
A one-way LOCC channel from Alice to Bob consists of Aliceperforming a quantum instrument, sending classical outcome to Bob,who performs a quantum channel conditioned on the classical data.
An LOCC channel consists of finite, but arbitrarily large number of1-way LOCC channels from Alice to Bob and then from Bob to Alice.
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LOCC channel (ctd.)
An LOCC channel can be written as a separable channel LAB→A′B′ :
LAB→A′B′(ρAB) =∑z
(EzA→A′ ⊗FzB→B′)(ρAB),
where EzA→A′z and FzB→B′z are sets of completely positive, trace
non-increasing maps, such that LAB→A′B′ is a completely positive,trace-preserving map (quantum channel).
However, the converse is not true. There exist separable channelsthat are not LOCC channels [BDF+99]
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Depiction of LOCC
ρA0B0 ρAkBkAlice
Bob
Mx1x∈X1
Nx1x∈X1
x1
Round 1L1,→
A0B0→A1B1
Mx2x∈X2
Nx2x∈X2
x2
Round 2L2,←
A1B1→A2B2
Mxkx∈Xk
Nxkx∈Xk
xk
Round kLk,→
Ak−1Bk−1→AkBk
· · ·
· · ·
· · ·
(Figure designed by Sumeet Khatri)
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LOCC channels preserve the set of separable states
If σAB is separable and an LOCC channel LAB→A′B′ acts on it, theresulting state is separable because
LAB→A′B′(σAB) =∑z
(EzA→A′ ⊗FzB→B′)
(∑λ
p(λ)ρλA ⊗ ωλB
)=∑z,λ
p(λ) EzA→A′(ρλA)⊗Fz
B→B′(ωλB)
Thus, one cannot create entanglement by the action of LOCC onseparable states.
So it is reasonable for LOCC to be the set of free operations in theresource theory of entanglement, due to this property in addition tothe physical motivation
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Axiomatic approach to quantifying entanglement
Basic axiom for entanglement measure [HHHH09]
An entanglement measure E is a function that does not increase underthe action of LOCC. That is, E is an entanglement measure if the followinginequality holds for every state ρAB and LOCC channel LAB→A′B′ :
E (A;B)ρ ≥ E (A′;B ′)ω
where ωA′B′ := LAB→A′B′(ρAB).
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Consequence of basic axiom [HHHH09]
Implies that E is minimal and constant on separable states becauseone can get from one separable state to another by LOCC:
E (A;B)σ = c ∀σAB ∈ SEP(A :B)
Conventional to set c = 0
Thus, E (A;B)ρ ≥ 0 for every state ρAB
and E (A;B)σ = 0 if σAB ∈ SEP(A :B)
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Other desirable properties for an entanglement measure
Faithfulness
E (A;B)σ = 0 if and only if σAB ∈ SEP(A :B)
Invariance under classical communication
E (AX ;B)ρ = E (A;BX )ρ =∑x
p(x)E (A;B)ρx
for a classical–quantum state:
ρXAB :=∑x
p(x)|x〉〈x |X ⊗ ρxAB
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Other desirable properties (ctd.)
Convexity
For a convex combination ρ :=∑
x p(x)ρxAB of states,∑x
p(x)E (A;B)ρx ≥ E (A;B)ρ
Additivity
For a tensor-product state ρA1A2B1B2 = τA1B1 ⊗ ωA2B2 ,
E (A1A2;B1B2)ρ = E (A1;B1)τ + E (A2;B2)ω
Mark M. Wilde (LSU) 55 / 103
Selective LOCC monotonicity
Let LxAB→A′B′x be a collection of maps, such that L↔AB→A′B′ is anLOCC channel of the form:
L↔AB→A′B′ =∑x
LxAB→A′B′ ,
where each map LxAB→A′B′ is completely positive such that the summap L↔AB→A′B′ is trace preserving.
Furthermore, each map LxAB→A′B′ can be written as follows:
LxAB→A′B′ =∑y
Ex ,yA→A′ ⊗Fx ,yB→B′ ,
where Ex ,yA→A′x and Fx ,yB→B′x are sets of completely positive maps.
Mark M. Wilde (LSU) 56 / 103
Selective LOCC monotonicity (ctd.)
Set p(x) := Tr[LxAB→A′B′(ρAB)], and for x such that p(x) 6= 0, set
ωxAB :=
1
p(x)LxAB→A′B′(ρAB).
If the classical value of x is not discarded, then the given state ρAB istransformed to the ensemble (p(x), ωx
AB)x via LOCC.
E satisfies selective LOCC monotonicity if
E (ρAB) ≥∑
x∈X :p(x)6=0
p(x)E (ωxAB),
for every ensemble (p(x), ωxAB)x∈X that arises from ρAB via LOCC
as specified above.
Interpretation: Entanglement does not increase on average under LOCC
Mark M. Wilde (LSU) 57 / 103
Proving convexity and selective LOCC monotonicity
Let E be a function that, for every bipartite state ρAB , is
1 invariant under classical communication and
2 obeys data processing under local channels, in the sense that
E (A;B)ρ ≥ E (A′;B ′)ω,
for all channels NA→A′ and MB→B′ , where
ωA′B′ := (NA→A′ ⊗MB→B′)(ρAB).
Then E is convex and is a selective LOCC monotone.
Mark M. Wilde (LSU) 58 / 103
Operational approach to quantifying entanglement
Axiomatic approach to quantifying entanglement starts with thebasic axiom and lists out properties that are desirable for anentanglement measure to obey
A conceptually different approach is the operational approach:Certain information-theoretic tasks quantify the amount ofentanglement present in a quantum state
These approaches intersect when trying to establish bounds on theoptimal rates of the operational tasks
Most prominent tasks are entanglement distillation andentanglement dilution
Mark M. Wilde (LSU) 59 / 103
Entanglement distillation
One-shot distillable entanglement of a bipartite state ρAB :
E εD(A;B)ρ := supd∈N,L∈LOCC
log2 d :
1
2
∥∥∥LAB→AB(ρAB)− ΦdAB
∥∥∥1≤ ε,
where ΦdAB
:= 1d
∑i ,j |i〉〈j |A ⊗ |i〉〈j |B
Distillable entanglement of ρAB :
ED(A;B)ρ := infε∈(0,1)
lim infn→∞
1
nE εD(An;Bn)ρ⊗n
where E εD(An;Bn)ρ⊗n evaluated on ρ⊗nAB
Strong converse distillable entanglement of ρAB :
ED(A;B)ρ := supε∈(0,1)
lim supn→∞
1
nE εD(An;Bn)ρ⊗n
Mark M. Wilde (LSU) 60 / 103
Depiction of entanglement distillation
L↔ ΦABρ⊗nAB
An A
Bn B
Alice
Bob
(Figure designed by Sumeet Khatri)
Mark M. Wilde (LSU) 61 / 103
Entanglement dilution
One-shot entanglement cost of a bipartite state ρAB :
E εC (A;B)ρ := infd∈N,L∈LOCC
log2 d :
1
2
∥∥∥LAB→AB(ΦdAB
)− ρAB∥∥∥
1≤ ε,
Entanglement cost of ρAB :
EC (A;B)ρ := supε∈(0,1)
lim supn→∞
1
nE εC (An;Bn)ρ⊗n
Strong converse entanglement cost of ρAB :
EC (A;B)ρ := infε∈(0,1)
lim infn→∞
1
nE εC (An;Bn)ρ⊗n
Mark M. Wilde (LSU) 62 / 103
Relating distillable entanglement and entanglement cost
For all ε1, ε2 ≥ 0 such that ε1, ε2 ≤ 1, [Wil20]
E ε1D (A;B)ρ ≤ E ε2
C (A;B)ρ + log2
(1
1− ε′),
where ε′ := (√ε1 +
√ε2)2
Second law like statement: cannot get out much moreentanglement than we invest
Applying definitions, conclude that
ED(A;B)ρ ≤ EC (A;B)ρ.
Asymptotically, cannot get out more than we invest
Mark M. Wilde (LSU) 63 / 103
Bounding distillable entanglement and entanglement cost
Difficult to compute distillable entanglement and entanglement cost
Next best approach: Establish lower and upper bounds
Upper bound on EC : Entanglement of formation [BDSW96]
Upper bound on ED : Rains relative entropy [Rai01, ADMVW02]
Upper bound on ED & lower bound on EC : Squashed entanglement
Each of these is an entanglement measure
Mark M. Wilde (LSU) 64 / 103
Entanglement entropy
Recall Schmidt-rank entanglement criterion for pure bipartitestates ψAB : ψAB is entangled if and only its Schmidt rank ≥ 2
Use entropy of reduced state to decide whether ψAB is entangled:
H(A)ψ := −Tr[ψA log2 ψA]
where ψA = TrB [ψAB ]. Called entropy of entanglement
H(A)ψ = 0 if ψAB is separable and H(A)ψ > 0 if ψAB is entangled
For pure bipartite states, entanglement theory simplifies immensely:
ED(A;B)ψ = EC (A;B)ψ = H(A)ψ
for every pure bipartite state ψAB [BBPS96].
Mark M. Wilde (LSU) 65 / 103
Entanglement of formation
To get an entanglement measure for a mixed state ρAB , take so-calledconvex roof of entanglement entropy [BDSW96]:
EF (A;B)ρ := inf(p(x),ψx
AB)x
∑x
p(x)H(A)ψx : ρAB =∑x
p(x)ψxAB
Decompose ρAB into a convex combination of pure states andevaluate expected entanglement entropy.
Called entanglement of formation
Mark M. Wilde (LSU) 66 / 103
Properties of entanglement of formation
EF monotone under selective LOCC, convex, subadditive, and faithful.
Reduces to entropy of entanglement for pure bipartite states
EF is non-additive [Sho04, Has09].
Also NP-hard to compute in general [Hua14], but can calculate it forcertain special classes of states.
Mark M. Wilde (LSU) 67 / 103
Entanglement of formation and entanglement cost
Entanglement of formation is an upper bound on entanglement cost:
EF (A;B)ρ ≥ EC (A;B)ρ,
for every bipartite state ρAB [BDSW96].
Regularized entanglement of formation is equal to entanglement cost[HHT01]:
EC (A;B)ρ := E regF (A;B)ρ,
where
E regF (A;B)ρ := lim
n→∞
1
nEF (An;Bn)ρ⊗n
Mark M. Wilde (LSU) 68 / 103
Logarithmic negativity
Another simple entanglement measure for a bipartite state ρAB islogarithmic negativity [VW02, Ple05]:
EN(A;B)ρ := log2 ‖TB(ρAB)‖1
Mark M. Wilde (LSU) 69 / 103
Properties of logarithmic negativity
Non-negative: EN(A;B)ρ ≥ 0 for every state ρAB
Faithful on PPT states: EN(A;B)ρ = 0 if and only if ρAB is PPT
Selective LOCC monotone [Ple05]
Upper bound on distillable entanglement [VW02]:
ED(A;B)ρ ≤ EN(A;B)ρ
This implies that PPT states have no distillable entanglement!
However, there are better entanglement-measure upper bounds forthe distillable entanglement
Mark M. Wilde (LSU) 70 / 103
Quantum relative entropy
Quantum relative entropy of a state ρ and a positive semi-definiteoperator σ is defined as [Ume62]
D(ρ‖σ) := Tr[ρ(log2 ρ− log2 σ)]
Standard definition with operational meaning [HP91, ON00]
Mark M. Wilde (LSU) 71 / 103
Properties of quantum relative entropy
Data-processing inequality for quantum relative entropy: Let ρ be astate, σ a positive semi-definite operator, and N a quantum channel.Then [Lin75]
D(ρ‖σ) ≥ D(N (ρ)‖N (σ))
For every state ρ and positive semi-definite operator σ satisfyingTr[σ] ≤ 1,
D(ρ‖σ) ≥ 0
andD(ρ‖σ) = 0 if and only if ρ = σ
Mark M. Wilde (LSU) 72 / 103
Relative entropy of entanglement
Relative entropy of entanglement of a bipartite state ρAB[VPRK97, VP98]:
ER(A;B)ρ := infσAB∈SEP(A:B)
D(ρAB‖σAB)
Mark M. Wilde (LSU) 73 / 103
Properties of relative entropy of entanglement
LOCC monotone: consequence of data-processing of relative entropyand set of separable states preserved by LOCC
Selective LOCC monotone: from data-processing inequality, jointconvexity of relative entropy
Convex
Faithful on separable states
Reduces to entropy of entanglement for pure bipartite states
Upper bound on distillable entanglement
Mark M. Wilde (LSU) 74 / 103
Entang. of formation and relative entropy of entang.
For every bipartite state ρAB ,
EF (A;B)ρ ≥ ER(A;B)ρ
Simple proof: Let (p(x), ψxAB)x be an arbitrary pure-state
decomposition of ρAB . Then∑x
p(x)H(A)ψx =∑x
p(x)ER(A;B)ψx ≥ ER(A;B)ρ
We used the fact that REE equals entropy of entanglement for purestates and REE is convex
Mark M. Wilde (LSU) 75 / 103
Rains relative entropy
Rains relative entropy [Rai01, ADMVW02]:
R(A;B)ρ := infσAB∈PPT′(A:B)
D(ρAB‖σAB)
wherePPT′(A : B) := σAB : σAB ≥ 0, EN(σAB) ≤ 0
Mark M. Wilde (LSU) 76 / 103
Properties of Rains relative entropy
LOCC monotone: follows from data processing for relative entropyand the fact that PPT′(A : B) is preserved under LOCC
Reduces to entropy of entanglement for pure bipartite states
Rains relative entropy is a tighter upper bound on distillableentanglement than other entanglement measures [Rai01]:
ED(A;B)ρ ≤ R(A;B)ρ ≤ minER(A;B)ρ,EN(A;B)ρ
with R(A;B)ρ ≤ ER(A;B)ρ following because
SEP(A : B) ⊂ PPT(A : B) ⊂ PPT′(A : B),
Mark M. Wilde (LSU) 77 / 103
Calculating Rains relative entropy
Can be calculated efficiently using Matlab, with CVX, CVXQuad, andQuantInf packages [Wil18]:
na = 2 ; nb = 2 ;rho = randRho ( na∗nb ) ; % G e n e r a t e random s t a t e
c v x b e g i n sdpv a r i a b l e tau ( na∗nb , na∗nb ) h e r m i t i a n ;m i n i m i z e ( q u a n t u m r e l e n t r ( rho , tau )/ l o g ( 2 ) ) ;tau >= 0 ;norm nuc ( Tx ( tau , 2 , [ na nb ] ) ) <= 1 ;
c v x e n d
r a i n s r e l e n t = c v x o p t v a l ;
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Squashed entanglement...
(Image courtesy of https://levelup.gitconnected.com/quantum-key-distribution-for-everyone-f08dd5646f33)
Mark M. Wilde (LSU) 79 / 103
Mutual information
Mutual information of a bipartite state ρAB defined as [Str65]
I (A;B)ρ := D(ρAB‖ρA ⊗ ρB)
As it turns out, this has an equivalent expression
I (A;B)ρ = infσA,τB
D(ρAB‖σA ⊗ τB)
Measures how distinguishable the state ρAB is from a product state
This is not useful as a measure of entanglement because it measuresall correlations, including classical correlations
Mark M. Wilde (LSU) 80 / 103
Mutual information and separable states
Suppose that ρAB is a separable state, so that we can write it as
ρAB =∑x
p(x)σxA ⊗ τ xB .
There exists an extension of this state to a classical system X :
ωABX :=∑x
p(x)σxA ⊗ τ xB ⊗ |x〉〈x |X
Conditioned on value in classical system, state is product and thus∑x
p(x)I (A;B)σxA⊗τ
xB
= 0
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Conditional mutual information
Conditional mutual information of a tripartite state κABE is definedas
I (A;B|E )κ := H(AE )κ + H(BE )κ − H(ABE )κ − H(E )κ
Strong subadditivity entropy inequality [LR73a, LR73b]:
I (A;B|E )κ ≥ 0
for every tripartite state κABE .
Mark M. Wilde (LSU) 82 / 103
Conditional mutual information with classical conditioning
For a classical–quantum state of the form κABX , where
κABX :=∑x
p(x)κxAB ⊗ |x〉〈x |X
the conditional mutual information evaluates to
I (A;B|X )κ =∑x
p(x)I (A;B)κx
Mark M. Wilde (LSU) 83 / 103
Squashed entanglement with classical extension
Motivated by this, we could define an entanglement measure for abipartite state ρAB as
Esq,c(A;B)ρ :=1
2infωABX
I (A;B|X )ω : TrX [ωABX ] = ρAB
where the extension system X is classical
It is equal to zero for every separable state, by picking the extensionsystem as we did previously
It is also a selective LOCC monotone
Mark M. Wilde (LSU) 84 / 103
Squashed entanglement
Squashed entanglement measure for a bipartite state ρAB defined as
Esq(A;B)ρ :=1
2infωABE
I (A;B|E )ω : TrE [ωABE ] = ρAB
where the extension system E is quantum [CW04] (see also[Tuc99, Tuc02])
Infimum seems to be necessary because no known upper bound onsize of quantum system E in the optimization
Mark M. Wilde (LSU) 85 / 103
Properties of squashed entanglement
Reduces to entanglement entropy for pure states [CW04]
Selective LOCC monotone [CW04]
Convex [CW04]
Additive [CW04]
Faithful [BCY11, LW18]
Upper bound on distillable entanglement and lower bound onentanglement cost [CW04]:
ED(A;B)ρ ≤ Esq(A;B)ρ ≤ EC (A;B)ρ
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Example: Generalized amplitude damping channel (GADC)
Generalized amplitude damping channel (GADC) is a two-parameterfamily of channels described as follows:
Aγ,N(ρ) = A1ρA†1 + A2ρA
†2 + A3ρA
†3 + A4ρA
†4,
where γ,N ∈ [0, 1] and
A1 =√
1− N
(1 00√
1− γ
), A2 =
√γ(1− N)
(0 10 0
),
A3 =√N
(√1− γ 0
0 1
), A4 =
√γN
(0 01 0
).
The parameter γ characterizes loss, and N describes environmental noise.
Mark M. Wilde (LSU) 87 / 103
Physical realization of GADC by beamsplitter interaction
γ1–γ
A B
E
θ(N)
for qubit thermal state θ(N) = (1− N)|0〉〈0|+ N|1〉〈1|
Mark M. Wilde (LSU) 88 / 103
Bounds on distillable entanglement of GADC [KSW20]
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.1
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.2
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.3
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.4
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.45
0.0 0.2 0.4 0.6 0.8 1.0γ
0.0
0.2
0.4
0.6
0.8
1.0N = 0.5
Q↔,LBRCI Q↔,UB
1 Q↔,UB2 Q↔,UB
3 Q↔,UB4 Q↔,UB
5
Figure: Distillable entanglement lies in the shaded region of each plot. Squashedentanglement bounds are in blue and magenta. Rains-like bound in gold (fromapproximately teleportation-simulable channel argument).
Mark M. Wilde (LSU) 89 / 103
Example: Dephased Bell state
Ent. Cost
Dist. Ent.
0.2 0.4 0.6 0.8 1.0q
0.2
0.4
0.6
0.8
1.0
Rate
Figure: Plot of distillable entanglement and entanglement cost of the dephasedBell state (1− q)ΦAB + qZBΦABZB , where ΦAB := 1
2
∑i,j∈0,1 |i〉〈j |A ⊗ |i〉〈j |B .
Mark M. Wilde (LSU) 90 / 103
Summary of entanglement measures
Distillable entanglement
Entanglement of formation
Entanglement cost
Squashed entanglement
Relative entropy of ent.
Rains relative entropy
Figure: Arrow → indicates ≥ and light blue ones are operational measures
Mark M. Wilde (LSU) 91 / 103
Conclusion
Entanglement is a resource for operational tasks like teleportation,super-dense coding, and quantum key distribution
Goal of entanglement theory is to quantify entanglement
Two approaches: axiomatic and operational approaches
Entanglement measures like entanglement of formation, squashedentanglement, and Rains relative entropy are useful and serve asbounds on operational entanglement measures like distillableentanglement and entanglement cost
Mark M. Wilde (LSU) 92 / 103
Open questions
Are there other interesting and useful entanglement measures?Efficiently computable and operationally meaningful (See [WW20] forrecent progress)
Find examples of states for which we can calculate squashedentanglement
Squashed entanglement not even known to be computable. Is ituncomputable?
What is the relevance of these entanglement measures in other areasof physics?
Mark M. Wilde (LSU) 93 / 103
References I
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[BBC+93] Charles H. Bennett, Gilles Brassard, Claude Crepeau, Richard Jozsa,Asher Peres, and William K. Wootters. Teleporting an unknown quantumstate via dual classical and Einstein-Podolsky-Rosen channels. PhysicalReview Letters, 70(13):1895–1899, March 1993.
[BBPS96] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and BenjaminSchumacher. Concentrating partial entanglement by local operations.Physical Review A, 53(4):2046–2052, April 1996.arXiv:quant-ph/9511030.
[BCY11] Fernando G.S.L. Brandao, Matthias Christandl, and Jon Yard. Faithfulsquashed entanglement. Communications in Mathematical Physics,306(3):805–830, September 2011. arXiv:1010.1750.
Mark M. Wilde (LSU) 94 / 103
References II
[BDF+99] Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor,Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters.Quantum nonlocality without entanglement. Physical Review A,59(2):1070–1091, February 1999. arXiv:quant-ph/9804053.
[BDSW96] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, andWilliam K. Wootters. Mixed-state entanglement and quantum errorcorrection. Physical Review A, 54(5):3824–3851, November 1996.arXiv:quant-ph/9604024.
[Bel64] John Stewart Bell. On the Einstein-Podolsky-Rosen paradox. Physics,1:195–200, 1964.
[BW92] Charles H. Bennett and Stephen J. Wiesner. Communication via one- andtwo-particle operators on Einstein-Podolsky-Rosen states. Physical ReviewLetters, 69(20):2881–2884, November 1992.
[CG19] Eric Chitambar and Gilad Gour. Quantum resource theories. Reviews ofModern Physics, 91(2):025001, April 2019. arXiv:1806.06107.
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References III
[CW04] Matthias Christandl and Andreas Winter. “Squashed entanglement”: Anadditive entanglement measure. Journal of Mathematical Physics,45(3):829–840, March 2004. arXiv:quant-ph/0308088.
[DL70] Edward B. Davies and J. T. Lewis. An operational approach to quantumprobability. Communications in Mathematical Physics, 17:239–260, 1970.
[Eke91] Artur K. Ekert. Quantum cryptography based on Bell’s theorem. PhysicalReview Letters, 67(6):661–663, August 1991.
[FvdG98] Christopher A. Fuchs and Jeroen van de Graaf. Cryptographicdistinguishability measures for quantum mechanical states. IEEETransactions on Information Theory, 45(4):1216–1227, May 1998.arXiv:quant-ph/9712042.
[Gha10] Sevag Gharibian. Strong NP-hardness of the quantum separabilityproblem. Quantum Information and Computation, 10(3):343–360, March2010. arXiv:0810.4507.
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References IV
[Gur04] Leonid Gurvits. Classical complexity and quantum entanglement. Journalof Computer and System Sciences, 69(3):448–484, 2004.arXiv:quant-ph/0303055.
[Has09] Matthew B. Hastings. Superadditivity of communication capacity usingentangled inputs. Nature Physics, 5:255–257, April 2009. arXiv:0809.3972.
[Hel67] Carl W. Helstrom. Detection theory and quantum mechanics. Informationand Control, 10(3):254–291, 1967.
[HHH96] Micha l Horodecki, Pawe l Horodecki, and Ryszard Horodecki. Separabilityof mixed states: necessary and sufficient conditions. Physics Letters A,223(1-2):1–8, November 1996. arXiv:quant-ph/9605038.
[HHHH09] Ryszard Horodecki, Pawe l Horodecki, Micha l Horodecki, and KarolHorodecki. Quantum entanglement. Reviews of Modern Physics,81(2):865–942, June 2009. arXiv:quant-ph/0702225.
Mark M. Wilde (LSU) 97 / 103
References V
[HHT01] Patrick M. Hayden, Michal Horodecki, and Barbara M. Terhal. Theasymptotic entanglement cost of preparing a quantum state. Journal ofPhysics A: Mathematical and General, 34(35):6891, September 2001.arXiv:quant-ph/0008134.
[Hol72] Alexander S. Holevo. An analogue of statistical decision theory andnoncommutative probability theory. Trudy MoskovskogoMatematicheskogo Obshchestva, 26:133–149, 1972.
[Hor97] Pawel Horodecki. Separability criterion and inseparable mixed states withpositive partial transposition. Physics Letters A, 232(5):333–339, August1997. arXiv:quant-ph/9703004.
[HP91] Fumio Hiai and Denes Petz. The proper formula for relative entropy andits asymptotics in quantum probability. Communications in MathematicalPhysics, 143(1):99–114, December 1991.
[Hua14] Yichen Huang. Computing quantum discord is NP-complete. New Journalof Physics, 16(3):033027, March 2014. arXiv:1305.5941.
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References VI
[KSW20] Sumeet Khatri, Kunal Sharma, and Mark M. Wilde. Information-theoreticaspects of the generalized amplitude damping channel. Physical ReviewA, 102(1):012401, July 2020. arXiv:1903.07747.
[Lin75] Goran Lindblad. Completely positive maps and entropy inequalities.Communications in Mathematical Physics, 40(2):147–151, June 1975.
[LR73a] Elliott H. Lieb and Mary Beth Ruskai. A fundamental property ofquantum-mechanical entropy. Physical Review Letters, 30(10):434–436,March 1973.
[LR73b] Elliott H. Lieb and Mary Beth Ruskai. Proof of the strong subadditivity ofquantum-mechanical entropy. Journal of Mathematical Physics,14:1938–1941, 1973.
[LW18] Ke Li and Andreas Winter. Squashed entanglement, k-extendibility,quantum Markov chains, and recovery maps. Foundations of Physics,48:910–924, February 2018. arXiv:1410.4184.
Mark M. Wilde (LSU) 99 / 103
References VII
[ON00] Tomohiro Ogawa and Hiroshi Nagaoka. Strong converse and Stein’slemma in quantum hypothesis testing. IEEE Transactions on InformationTheory, 46(7):2428–2433, November 2000. arXiv:quant-ph/9906090.
[Oza84] Masanao Ozawa. Quantum measuring processes of continuousobservables. Journal of Mathematical Physics, 25(1):79–87, 1984.
[Per96] Asher Peres. Separability criterion for density matrices. Physical ReviewLetters, 77(8):1413–1415, August 1996. arXiv:quant-ph/9604005.
[Ple05] Martin B. Plenio. Logarithmic negativity: A full entanglement monotonethat is not convex. Physical Review Letters, 95(9):090503, August 2005.arXiv:quant-ph/0505071.
[Rai01] Eric M. Rains. A semidefinite program for distillable entanglement. IEEETransactions on Information Theory, 47(7):2921–2933, November 2001.arXiv:quant-ph/0008047.
[Ras06] Alexey E. Rastegin. Sine distance for quantum states. February 2006.arXiv:quant-ph/0602112.
Mark M. Wilde (LSU) 100 / 103
References VIII
[Sho04] Peter W. Shor. Equivalence of additivity questions in quantuminformation theory. Communications in Mathematical Physics,246(3):453–472, 2004. arXiv:quant-ph/0305035.
[Sti55] William F. Stinespring. Positive functions on C*-algebras. Proceedings ofthe American Mathematical Society, 6:211–216, 1955.
[Str65] Ruslan L. Stratonovich. Information capacity of a quantumcommunications channel. i. Soviet Radiophysics, 8(1):82–91, January1965.
[Tuc99] Robert R. Tucci. Quantum entanglement and conditional informationtransmission. September 1999. arXiv: quant-ph/9909041.
[Tuc02] Robert R. Tucci. Entanglement of distillation and conditional mutualinformation. Februrary 2002. arXiv:quant-ph/0202144.
[Uhl76] Armin Uhlmann. The “transition probability” in the state space of a*-algebra. Reports on Mathematical Physics, 9(2):273–279, April 1976.
Mark M. Wilde (LSU) 101 / 103
References IX
[Ume62] Hisaharu Umegaki. Conditional expectations in an operator algebra IV(entropy and information). Kodai Mathematical Seminar Reports,14(2):59–85, 1962.
[VP98] Vlatko Vedral and Martin B. Plenio. Entanglement measures andpurification procedures. Physical Review A, 57(3):1619–1633, March1998. arXiv:quant-ph/9707035.
[VPRK97] Vlatko Vedral, Martin B. Plenio, M. A. Rippin, and Peter L. Knight.Quantifying entanglement. Physical Review Letters, 78(12):2275–2279,March 1997. arXiv:quant-ph/9702027.
[VW02] Guifre Vidal and Reinhard F. Werner. Computable measure ofentanglement. Physical Review A, 65:032314, February 2002.arXiv:quant-ph/0102117.
[Wer89] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosencorrelations admitting a hidden-variable model. Physical Review A,40(8):4277–4281, October 1989.
Mark M. Wilde (LSU) 102 / 103
References X
[Wil18] Mark M. Wilde. Entanglement cost and quantum channel simulation.Physical Review A, 98(4):042338, October 2018. arXiv:1807.11939.
[Wil20] Mark M. Wilde. Relating one-shot distillable entanglement to one-shotentanglement cost. Unpublished note, September 2020.
[WW20] Xin Wang and Mark M. Wilde. Cost of quantum entanglement simplified.Physical Review Letters, 125(4):040502, July 2020. arXiv:2007.14270.
Mark M. Wilde (LSU) 103 / 103