Project Title: Quantum information in accelerated reference frames
Candidate number: 850020
Project number: ////
Supervisor: Vlatko Vedral
Word count: 7386
1
MPhys Project | AL
Abstract
earlier studies by confining the modes to a finite volume and by restricting the period of
acceleration to be finite.
To investigate any entanglement present in
the field after the acceleration, I will use several tools from quantum information theory.
In particular, I will use two properties of the
quantum state, the mutual information and the
twist. The first is well known, and is based on
the von Neumann entropy of a quantum state.
The second is less well known. Inspired by the
Wilson Loop in lattice gauge field theory [5],
it measures the asymmetry of correlations between quantum subsystems, and has a direct
geometric interpretation, which will be fully explained later in the report.
The report is structured as follows: in section
2, I will describe the tools from quantum information theory which I will later apply to the
field in the cavity. In section 3, I will explain
some background ideas from quantum field theory in curved spacetime necessary to describe
what happens to the cavity during its acceleration, and then apply them in detail in section 4.
In section 5, I will draw these strands together,
applying ideas from section 2 to the results of
section 4 . In section 6, I will conclude and
suggest future directions for study.
In the research field of relativistic quantum information, ideas from quantum information theory, originally applied to
quantum mechanical systems, are instead applied to systems involving relativistic quantum fields.
In this report I focus on one such recently studied system - that of an accelerating cavity containing a quantum
field. Firstly, I discuss the effect of the
acceleration on the field, showing that
particles are generated inside the cavity, and that the field becomes entangled. I then quantify the entanglement
present by computing two properties of
the quantum state - the mutual information and the twist.
1
Introduction
Quantum entanglement is a phenomenon well
known in the context of quantum mechanics, a
non relativistic theory. However, recently there
have been many investigations into the effects
that relativity, both special and general, have
on entanglement [1, 2]. Now, instead of quantum mechanics, we must use quantum field theory, and instead of finding entanglement between a finite number of separate quantum subsystems, we can find it in a single quantum field
- both between the modes of this field, and between different points in spacetime.
In this report, I investigate the effect that
a finite period of acceleration - and so by the
equivalence principle, a gravitational field - has
on a simple quantum field confined to a cavity,
as first studied in [11, 12, 13]. My focus will be
on the generation of entanglement in the field,
as a result of this acceleration.
There have been a variety of scenarios studied (examples include particle creation in the
FRW metric [9], and the consequences of an
event horizon for entanglement [3]). However,
there are several reasons to study the cavity
in particular. As well as being theoretically
tractable, it removes several artificialities of
2
Ideas From Quantum Information
In this section I introduce ideas from Quantum
Information which will be applied later, with a
focus on explaining the twist in subsections 2.52.7. All material in subsections 2.1-2.4 can be
found in many textbooks, such as [4]. An introduction to the twist can be found in [5], and
a detailed example of its application to composite quantum systems is [6]. A discussion of
the Lorentz singular value decomposition can
be found in [7].
2.1
Composite Quantum States
Suppose we have a quantum system comprised
of two subsystems, A and B, with two sets of
basis vectors, {|ai i} and {|bi i}. These basis
vectors can be combined to form a basis for
2
MPhys Project | AL
the overall system using the tensor product ⊗.
The simplest case is that of subsystem A in
state |aj i and subsystem B in state |bk i. Then
|ψi = |aj i ⊗ |bk i,
(2.1.1)
denotes the state of the composite system AB
(the explicit tensor product symbol is often
omitted). More generally, it can be shown that
{|ai i ⊗ |bj i} forms a basis for AB, i.e. a general
state of the system can be written
X
|ψi =
cij |ai i ⊗ |bj i.
(2.1.2)
i,j
This can be further generalised to systems comprised of many subsystems. If we have a Npartite system, where the j-th subsystem has
basis vectors {|ij i}, then the state of the system can be written
X
|ψi =
ci1 ...iN |i1 i ⊗ ... ⊗ |iN i. (2.1.3)
i1 ...iN
We can also define density operators for subsystems of an overall composite state. This is
done by ‘tracing out’ the subsystems we do not
wish to represent. For the density matrix ρ of
a N-partite composite state, the partial trace
over a subsystem γ is defined as
trγ ρ =
X
hiγ |ρ|iγ i,
(2.2.4)
i
where the |iγ i are the basis states of subsystem γ. The remaining N-1 subsystem density
matrix will represent a mixed state, even if the
overall state of the system is pure.
2.3
The Von Neumann Entropy and Mutual Information
Von Neumann defined the entropy of a density
matrix as
X
S(ρ) = −tr(ρ log ρ) = −
λx log λx , (2.3.1)
x
2.2
The Density Matrix
In reality, we often don’t know the exact quantum state of a system we are dealing with.
However, if we have a complete basis of states,
we can describe the system by a statistical mixture of these states, representing our own subjective ignorance rather than inherent quantum
uncertainty. These states are known as mixed
states, and can be represented by the density
operator. If our system is in one of a set of
states |ψi i with probability pi , this operator is
defined as
X
ρ=
pi |ψi ihψi |.
(2.2.1)
where the λx are the eigenvalues of ρ. This
definition is in close analogy to a classical entropy, with a probability distribution replaced
by a density matrix, and shares many of the
properties a classical entropy does. It is commonly used to quantify the amount of information present in a density matrix [4].
Using this, a family of entropic entanglement
measures can be defined, one of which is the
mutual information. For a composite system
AB with joint density matrix ρAB , the mutual
information is defined as
S(A : B) = S(A) + S(B) − S(A, B), (2.3.2)
i
We see that, in order for (2.2.1) to represent a
valid density operator, tr(ρ) = 1, since probabilities must sum to 1, and the eigenvalues of ρ
must be non negative, since probabilities cannot be negative.
There are several useful properties of the
density operator. The average of an operator
A over a mixture of states is defined as
X
X
hĀi =
pi hAi =
pi hψi |A|ψi i, (2.2.2)
i
i
from which it can be shown that
hĀi = tr(ρA).
(2.2.3) 3
where
S(A, B) = −tr(ρAB log ρAB ),
(2.3.3)
and ρA , ρB are found by taking the partial trace
over one of the subsystems.
Intuitively, the mutual information tells us
the amount of information common to both A
and B. Zero mutual information between A and
B means that A and B are completely uncorrelated.
MPhys Project | AL
2.4
Qubits
A more general class of operations is represented by SL(2,C), the group of all 2 × 2
complex matrices with determinant 1, of which
SU (2) is a subgroup. Members of this group
represent operations which only succeed probabilistically, and are used in SLOCC.
Suppose now we consider just two qubits, A
and B. Under these operations, the two qubit
density matrix evolves as
The results of the previous subsections hold for
a wide variety of quantum systems - they could
be applied to harmonic oscillators, atomic
states, even systems with continuous degrees of
freedom, like position and momentum. In particular, we have made no assumptions about
the dimensionality of the Hilbert spaces we are
working in.
However, many measures of entanglement
and correlation are defined with reference to
a ubiquitous, and particularly simple system a two state system, known as a qubit [4].
These systems exist in a 2-dimensional
Hilbert space, with an orthogonal basis
{|0i, |1i}. The state of a qubit is, in general, a
superposition of these states
|ψi = α|0i + β|1i.
ρ0AB =
(P ⊗ Q)ρAB (P † ⊗ Q† )
,
tr((P ⊗ Q)ρAB (P † ⊗ Q† ))
(2.5.1)
where P is an operation on qubit A, Q is an
operation on qubit B, and denominator is for
normalisation.
2.6
(2.4.1)
The Hilbert Schmidt Representation
The 4×4 matrices formed from tensor products of the Pauli matrices {σ0 , σ1 , σ2 , σ3 } can
be used as a basis for representing a density
matrix of two qubits - this is known as the
Hilbert-Schmidt representation
Qubits are, conceptually, the simplest systems
it is possible to store information in - they
are the quantum analogy of a classical bit of
information. Very frequently, the composite
systems discussed in the previous sections will
turn out be comprised of qubits, as will be the
case in the following discussion.
ρAB
3
1 X
=
S(A, B)ij σiA ⊗ σjB .
2
(2.6.1)
i,j=0
2.5
LOCC and SLOCC Evolution
S is called the correlation matrix. It is in general real, but not necessarily symmetric. The
coefficients S(A, B)ij are given by
When studying an N-partite quantum system, we usually picture N spatially separated
parties, each with access to a single subsystem - in this case, each with a single qubit.
The parties can perform local operations on
their own qubits, and communicate with each
other via classical channels. This protocol is
known either as Local Operations with Classical Communication (LOCC), or Stochastic Local Operations with Classical Communication
(SLOCC), depending on what sort of operations the individual parties are allowed to perform.
If each party performs a deterministic evolution of their subsystem, we can represent their
single qubit operations using members of the
group SU (2) - the group of 2 × 2 unitary matrices with determinant 1. These are the reversible operations, and correspond to LOCC.
S(A, B)ij = tr(σiA ⊗ σjB ρAB ) = hσiA ⊗ σjB i.
(2.6.2)
This matrix can be used as an alternative to
the density matrix for representing operations
on quantum states.
As described in [5], we can represent SLOCC
operations in the Hilbert-Schmidt picture using members of SO+ (1, 3), the proper orthochronous Lorentz group. Elements of the
Lorentz group L satisfy the defining relation
LT ηL = η,
(2.6.3)
where η is the Minkowski metric
η = diag(1, −1, −1, −1).
4
(2.6.4)
MPhys Project | AL
SO+ (1, 3) is then a subgroup of the Lorentz
group, with the additional conditions that
L00 ≥ 1 and det(L) = 1.
Each member of SL(2,C) can be mapped to
one in SO + (1,3). Explicitly, the relation between P ∈ SL(2,C) and P ∈ SO + (1,3) is
1
Pij = tr(P † σi P σj ).
2
(2.6.5)
It can then be shown that, if the two qubit
density matrix evolves as in (2.5.1), then the
corresponding correlation matrix evolves as
S(a, b)0ij = PS(a, b)QT ,
(2.6.6)
with P, Q given by (2.6.5).
2.7
The Lorentz Singular Value Decomposition
It can be shown [7] that the correlation matrix
S(a, b) can always be decomposed as:
S(a, b) = Va Σab WbT ,
SO + (1,3),
(2.7.1)
with Va , Wb
∈
and Σab =
diag(s0 , s1 , s2 , s3 ) a diagonal matrix. This
is known as a Lorentz singular value decomposition, and the si are known as the
Lorentz singular values. Comparison with
(2.6.6) reveals a natural interpretation. Va , Wb
correspond to operations on the qubits A, B,
and Σab is a diagonal correlation matrix - in
fact, representing a mixture of bell states, as
discussed in [5].
2.8
The Twist
one, joining the last qubit to the first. The required symmetrizing operation for this link is
not necessarily the identity. We have a problem
- symmetrizing the last link will cause the first
link to de symmetrize. Though we can symettrize each link individually, we cannot simultaneously symmetrize every link in the loop.
This global degree of asymmetry is what characterises the twist. It is not a property of
any particular link, since each link can always
be symmetrized, but rather a property of the
whole loop. A graphical representation of this
procedure can be found in Ref. [5] Fig.1.
We can find the operation necessary to symmetrize a link as follows. Using (2.7.1), every
two qubit correlation matrix can be written as
S = V ΣW T ,
(2.8.1)
where V , W ∈ SO+ (1, 3). Using the defining
property of the Lorentz group (2.6.3) on W , we
can write S as
S(a, b) = (V ηW T η)(W ΣW T )
= Λ(a, b)S̃(a, b).
(2.8.2)
Thus Λ−1 (a, b), applied to qubit A will symmetrize the link. We now define the twist of
a quantum state as the trace of the product of
each of these symmetrizing operations around
the loop
1
ξ = tr(Λ(a, z)...Λ(c, b)Λ(b, a)),
4
(2.8.3)
where the 41 normalises the twist to 1 if the total transformation is the identity. As discussed
in [5], these results bear strong resemblance to
those found in lattice gauge theory (for which
a short introduction can be found in [8]). For
example, consider the transformation of Λ(a, b)
under SLOCC operations Ua and Ub on quibts
A and B
We now have the tools necessary to define the
twist. It is most easily understood as a procedure to be carried out on a quantum state, as
described below.
For an N-partite system, imagine each qubit
as point on an abstract lattice. Now choose a
closed loop on this lattice running through all
Λ(a, b) = V ηW T η
these points. For every two qubit link in this
→ Ua V ηW T UbT η
loop, a density matrix and correlation matrix
can be defined by tracing out all other qubits.
= Ua (V ηW T η)Ub−1
Now, for each link in turn, act on the first
= Ua Λ(a, b)Ub−1 .
(2.8.4)
qubit with an operation such that the correlation matrix S(a, b) for that link becomes symΛ(a, b) transforms exactly like the parallel
metric. Go around the loop, applying this pro5 transporter.
cedure to every link, until you reach the last
MPhys Project | AL
3
Ideas From Quantum Field Theory
We will not use (3.1.3) much in practice, but
the ability to define orthogonal modes is key to
almost all of the following results, so it is worth
noting.
Using (3.1.3) we can define a complete set of
orthonormal mode solutions {ui (x)}, and expand the field in terms of them
X
φ(x) =
ui ai + u∗i a†i .
(3.1.4)
In this section, I will briefly summarise some
general results from quantum field theory in
curved spacetime - a more comprehensive account can be found in textbooks such as [9]. It
should be emphasised that this material does
not describe a theory of quantum gravity, but
rather lies at a middle ground similar to that
of semiclassical light-matter interactions. In
this theory, a background metric is assumed,
on which we may define quantum fields, but
the metric itself is not affected by these fields
- it is only the ‘stage’ upon which the fields
evolve.
Many of the ideas in section 3.1 are also only
slight extensions of those found in flat spacetime, for which one reference is [10].
3.1
i
Quantisation is achieved by adopting the usual
commutators
[an , a†m ] = δnm
[an , am ] = [a†n , a†m , ] = 0.
3.2
In curved spacetime, instead of the Minkowski
metric η µν there is a general metric
g µν (x),where x denotes a general co-ordinate
on the spacetime manifold. Associated with
the metric is a line element
j
(3.1.1)
Since both sets of modes are complete, we can
express one set in terms of the other
X
uj =
αji ui + βji u∗i .
(3.2.2)
In section 4 we will consider a scalar, massless bosonic field φ, for which the relevant field
equation is the massless Klein-Gordon equation. In curved spacetime, this is generalised
to
1
i
This transformation is known as a Bogoliuobv
transformation, and the matrices αji , βji are
called Bogoliubov matrices. Schematically, we
can represent this transformation by a matrix
multiplication
u
α β
u
=
,
(3.2.3)
u∗
β ∗ α∗
u∗
1
g µν ∇µ ∇ν φ = (−g)− 2 ∂µ [(−g)− 2 g µν ∂ν φ] = 0,
(3.1.2)
where ∇ν is a covariant derivative, and g ≡
det(g µν ) is the determinant of the metric. Just
as in flat spacetime, we also define a scalar
product
Z
1
(φ1 , φ2 ) = −i
φ1 (∂µ φ∗2 )[−g(x)] 2
where (u, u∗ )T represents a vector of all the
modes and their conjugates ({ui }, {u∗i })T , and
α, β are the Bogoliubov matrices, which we
group into an overall transformation
α β
M=
.
(3.2.4)
β ∗ α∗
Σ
−
1
(∂µ φ1 )φ∗2 [−g(x)] 2
The Bogoliubov Tranformation
The principal of covariance tells us that we
should be able to express physical phenomena
in any co-ordinate system we choose. In particular, we can consider the existence of a second
set of orthogonal modes {uj (x)}, and expand
the field φ in terms of them as well
X
φ(x) =
uj aj + u∗j a†j .
(3.2.1)
Quantum fields in curved spacetime
ds2 = gµν dxµ dxν .
(3.1.5)
µ
dΣ ,
(3.1.3)
where dΣµ = nµ dΣ, with nµ a timelike unit
vector orthogonal to the spacelike hypersurface
Σ , and dΣ a volume element of Σ.
6
MPhys Project | AL
4
We can find explicit expressions for α and β
using the scalar product (3.1.3), and the orthonormality of each set of modes. We find
αij = (ui , uj )
βij = −(ui , u∗j ).
We now have the tools necessary to describe
the effects of acceleration on a quantum field.
In this section I will detail the exact physical
situation considered, first described in [12]. I
will then calculate what happens to the field,
showing that particle creation occurs inside the
cavity, in all of the field modes. Scenarios of
this sort have previously been studied in, for
example, [11, 12, 13] and a recent review can be
found in [2]. In all of the following calculations
c = ~ = 1.
(3.2.5)
Equating the two definitions of φ, (3.1.4) and
(3.2.1), and using (3.2.5), we can also express
one set of creation and annihilation operators
in terms of the other
X
∗
∗ †
aj =
αji
ai − βji
ai .
(3.2.6)
i
Note that if we have an infinite set of modes
{ui } then α and β are also infinite. Requiring that the commutators (3.1.5) are preserved
by the transformation (3.2.4) leads to two very
useful identities for α and β
4.1
(3.2.7)
Using these, we can find the inverse to (3.2.4)
†
α
−β T
−1
M =
.
(3.2.8)
−β † αT
It can be shown that if the vector of modes
(u, u∗ )T transforms via M , then the corresponding vector of creation and annihilation
operators transforms via (M −1 )T .
An interesting feature of these transformations is that they imply the vacuum is not
unique. Taking the vacuum in one set of modes
{ui }
an |0i = 0, ∀n,
Introduction and Setup
I consider a cavity of length δ, living in (1+1)
dimensional Minkowski space. Inside the cavity is a scalar, massless, bosonic field, obeying Dirichlet boundary conditions at the cavity
walls.
The cavity is initially inertial (Region I in
Fig 1). It then undergoes a period of constant
acceleration (II) before reaching a constant velocity (III).
αα† − ββ † = I
αβ T − βαT = 0.
The Accelerated Cavity
Figure 1: A spacetime diagram of the cavities
motion. The cavity is initially stationary (Region I). It then undergoes a period of constant
acceleration from t = η = 0 until η = η1 (II),
and then moves with a constant velocity (III).
t
(3.2.9)
we find that, for example, it is not neccesarilly
anihilated by the ai
X
∗ †
αji aj + βji
aj |0i 6= 0. (3.2.10)
ai |0i =
η = η1
III
II
j
a
The vacuums only coincide if β, the matrix associated with particle creation operators, is 0.
But in general the vacuum in one co-ordinate
system is a bath of particles in another and, as
we shall see, is entangled. It is also interesting
to note that, if β = 0, the Bogoliubov identities (3.2.7) reduce to the unitarity condition,
and the matrix M becomes block diagonal - the
modes decouple from one another.
7
I
b
x
MPhys Project | AL
4.2
Minkowski and Rindler Modes
from (4.2.5). During the acceleration, χ remains constant, and the time evolution appears
in the phase eiΩη . To propagate through II, we
allow this phase to evolve until η = η1 (Fig.1),
corresponding to the time at which the cavity
stops accelerating. Defining the matrix
D 0
E=
,
(4.2.6)
0 D∗
In Regions I and III the co-ordinate system
used to describe the spacetime is the usual
(t, x) system. In these regions, the field evolves
according to the massless Klein-Gordon equation in flat spacetime
η µν ∂µ ∂ν φ = 0,
(4.2.1)
the solutions of which give the field modes
1
un = √
eiωn t sin(ωn x),
ωn δ
where D is a diagonal matrix of phases D =
diag(eiΩ1 η1 , eiΩ2 η1 , . . .), this evolution can be
expressed as
u(η1 )
D 0
u(0)
=
.
(4.2.7)
u∗ (η1 )
0 D∗
u∗ (0)
(4.2.2)
ωn only takes on the discrete values ωn = nπ
δ .
In region II, the cavity experiences acceleration. The correct co-ordinate system to describe the spacetime experienced by an accelerated observer is known as Rindler co-ordinates
[9], (η, χ) . These are related to (t, x) by
4.3
We now know how to convert from Minkowski
modes to Rindler modes using (3.2.4) , and
how to evolve the Rindler modes using (4.2.6).
Combining these, the relation of the modes in
I to those in III can be written
u
u
−1
,
(4.3.1)
= M EM
u∗
u∗
t = χ sinh η
x = χ cosh η,
(4.2.3)
and cover the wedge of spacetime x > |t| as
shown in Fig.1. Lines of constant η are radial
lines, while lines of constant χ are hyperbolae,
which can be shown to be the world-lines of
accelerated observers with proper accelerations
a0 = χ1 .
Along these hyperbolae, the proper time is
related to η by
τ=
1
η.
a0
Transforming the Cavity Modes
which we identify with an overall Bogoliubov
transformation
A B
.
(4.3.2)
MI→III =
B ∗ A∗
(4.2.4)
Solving the Klein-Gordon equation in these coordinates gives us the Rindler modes [2, 14]
1
1
x ∓ t ±iΩ
i(±Ωη−Ω ln lχ )
Ω .
=√
uΩ = √
e
lΩ
4πω
4πΩ
(4.2.5)
Ω is called the Rindler frequency, and is used to
label the Rindler modes. With the dimensions
used in this report, Ω = ω = nπ
δ . lΩ is just an
arbitrary constant introduced for dimensional
reasons.
In terms of the Rindler modes, the evolution of the field in II is simple, as can be seen
8
Equating (4.3.1) and (4.3.2), we obtain expressions for the overall transformation matrices
A, B.
A = α† Dα − β T D∗ β ∗
†
T
∗ ∗
B = α Dβ − β D α
(4.3.3)
(4.3.4)
The transformation (4.3.1) can be viewed, in
some sense, as a diagonalisation. M ‘changes
basis’ between the modes, and E is the evolution of the field in this basis. It is comprised, as
we would expect, of a simple diagonal matrix
of phases.
MPhys Project | AL
4.4
Transformation of the vacuum state
product of the cavity length and the acceleration at the centre of the cavity, and assume the
Bogoliubov matrices have Taylor expansions in
h
Though we know how to relate the modes (and
hence the creation/annihilation operators) between regions I and III, we still do not know
how the quantum state of the field evolves.
The result is partially derived in [15], and I
will summarise it below. Taking the region (I)
vacuum
an |0i = 0, ∀n,
α = α(0) + α(1) + α(2) . . .
β=β
(4.4.1)
so
∗ −1 †
Bji
Aik aj |0i = 0.
(4.4.3)
ij
Defining the symmetric matrix
V = −B ∗ A−1 ,
(4.4.4)
W =
1X
Vpq a†p a†q ,
2 pq
(4.5.2)
These Matrices have several important features. Firstly, α and β are real. Secondly
it can be shown that, recursively applying
(4.4.3), the region (I) and (III) vacua are related by
|0i = N eW |0i,
(4.4.5)
where
...,
(4.5.1)
αnn = 1 + O(h2 )
(4.5.3)
m−n
√
(−1 + (−1)
)
h + O(h2 ) (m 6= n)
αmn = mn
π 2 (m − n)3
(4.5.4)
m−n
√
(1 − (−1)
)
h + O(h2 ).
βmn = mn 2
3
π (m + n)
(4.5.5)
j
X
+β
(2)
where the superscript denotes the power of h.
Note that β (0) = 0, since we wish for the vacua
to co-incide in the small h limit. As discussed
in [11], to O(h), α and β are given by
and expressing an in terms of the a† , a
X
∗ †
aj |0i = 0,
(4.4.2)
Aji aj + Bji
ak +
(1)
α(1) = −α(1)
β (1) = β
(1) T
T
,
(4.5.6)
(4.5.7)
i.e. α(1) is antisymmetric and β (1) is symmetric. Finally, we note that both α(1) and β (1)
contain zeroes for even index spacing
(4.4.6)
and N is a normalisation constant.
In principle, we can now compute the evolution of any state, by applying creation operators to both sides of (4.4.5). However, there
are several practical difficulties to overcome.
To perform explicit calculations, we must
first compute the elements of α, β using (3.2.5),
which gives them in terms of scalar products
between the Rindler and Minkowski modes on
the line t = η = 0.
These integrals cannot be evaluated exactly
- their explicit expressions are given in [2] - but
they can be computed perturbatively.
We can use these expansions to compute power
series for A, B and V , and hence give a power
expansion of the vacuum state (4.4.5). In what
follows, we work to O(h), the small acceleration
limit.
We begin by computing the O(h) terms of A
and B. Using (4.3.3), we find that
4.5
and
(1)
αnm
=0
(1)
βnm
=0
n − m ∈ 2Z.
(4.5.8)
A = A(0) + A(1) + O(h2 )
= D + Dα(1) − α(1) D + O(h2 ), (4.5.9)
Perturbative Expansions of Bogoliubov Matrices
B = B (1) + O(h2 )
= Dβ (1) − β (1) D∗ + O(h2 ).
To perform a perturbative expansion, we introduce a dimensionless parameter h = aδ, the
9
(4.5.10)
MPhys Project | AL
Using these expressions, we can compute the
power series of V
V = V (1) + O(h2 ) = D∗ β (1) D∗ − β (1) + O(h2 ).
We can also compute what happens to a
number state. Applying a creation operator
a†n to both sides of (4.6.3) to place the field in
initial state |1n i, we find
(4.5.11)
∗
|1n i = Dnn
|1n i +
We note that V (1) is symmetric, and that it has
the same structure as β (1) , i.e.


0 V12 0 V14 · · ·
V12 0 V23 0 · · ·


 0 V23 0 V34 · · ·
(1)
V
=
 . (4.5.12)
V14 0 V34 0 · · ·


..
..
..
..
..
.
.
.
.
.
i
5
Perturbative Expansion of the Vacuum State
We can now combine (4.4.5) and (4.5.12) to
give a power expansion of the vacuum. Firstly,
we need to find N in (4.4.5). This is done by
requiring that the relation
†
1 = h0|eW |N |2 eW |0i,
5.1
(4.6.1)
1 X (1) 2
|V | .
4 pq pq
(4.6.2)
(5.1.1)
Taking (4.4.5), expanding the exponential, and
keeping terms to O(h), we find
|0i = |0i +
1 X (1) † †
V a a |0i.
2 pq pq p q
where h.c denotes the Hermitian conjugate.
Taking the partial trace over all but the m, m0
modes leaves us with a general two mode density matrix
(4.6.3)
We see that, to 0th order, the vacuum is unchanged, as expected. But at O(h), we have a
superposition of many states of the form
|1p 1q i = |0 . . . 01p 0 . . . 01q 0 . . .i,
The Vacuum State
Considering the vacuum state (4.6.3), we have,
to O(h) a total density matrix
!
1 X (1) † †
ρ = |0ih0| = |0ih0| +
V a a |0i h0| + h.c,
2 pq pq p q
holds to every order in h. Expanding the exponentials, it can be shown that, to O(h2 )
N =1−
Quantum Information in Accelerating Reference Frames
In this section I will apply ideas from section 2
to the above states to quantify the new entanglement present in the field.
The quantities we will be interested in are reduced density matrices, both single mode and
two mode. These are obtained from the full
density matrix corresponding to, say, (4.6.3),
by taking the partial trace over every mode
that we are not considering, leaving a mixed
state containing only the modes we are interested in.
We also note that, referring to (4.5.5), Vnm → 0
for large n, m.
4.6
1 X ∗ (1)
D V |1p 1q 1n i
2 pq nn pq
X
+
A∗in (1) |1i i. (4.6.5)
(1)
ρmm0 = |00ih00| + Vmm0 |11ih00| + h.c. (5.1.2)
Tracing out one of the remaining mode simply
leaves
(4.6.4)
ρm = |0ih0|.
with decreasing amplitudes as p, q increases.
The vacuum is now populated with a sea of
particles, spread over all frequencies, and certainly looks entangled.
(5.1.3)
i.e. though the two mode density matrices have
corrections at O(h), the single mode ones do
not. The mutual information between any two
10
MPhys Project | AL
5.2
modes can then be quantified using (2.3.2), by
computing S(ρm ) + S(ρm0 ) − S(ρmm0 ).
Unfortunately, to O(h), there is no correction to the mutual information. This can be
seen by considering the eigenvalues of (5.1.2),
which are
{0, 0,
The First Excited State
These calculations can be repeated for the first
excited state (4.6.5). As discussed above, we
will not find any O(h) correction to the mutual
information, but we we may find one for the
twist. We have an overall density matrix (for
convenience I have dropped the bar notation
from the states)
1
p
p
1
1 − 1 + 4|Vmm0 |2 ,
1 + 1 + 4|Vmm0 |2 }.
2
2
(5.1.4)
Taylor expanding(5.1.4), we see corrections occur at O(h2 ).
This result is in contrast to that of [12],
where it is found that the negativity, another
measure of entanglement, can increase linearly
with h. It seems probable that all measures
of information and entanglement based on entropies will only change at O(h2 ), whereas measures like the negativity (which is based on an
operation known as partial transposition [2])
can increase to O(h). This can be explained by
noting that the first O(h) correction can only
enter ρ on the off diagonal elements. When
computing the eigenvalues of this matrix, as required for (2.3.1), these corrections are always
squared, and thus become O(h2 ) corrections to
all entropic information measures.
We can also compute the twist for the vacuum state (4.6.3), using the reduced density
matrices (5.1.2). The corresponding two mode
correlation matrix is given by


1
0
0
1


(1)
(1)
0 2<(Vmm0 ) 2=(Vmm0 ) 0
S(m, m0 ) = 
.
(1)
(1)
0 2=(Vmm
−2<(Vmm0 ) 0
0)
1
0
0
1
(5.1.5)
ρ =|1n ih1n |
1X ∗ 1
+
D V |1p 1q 1n ih1n | + h.c
2 pq nn pq
X
+
A∗in Dnn |1i ih1n | + h.c.
(5.2.1)
i
Again,tracing out all but the m, m0 modes, we
find two cases: m = n and m 6= n.
CASE
1 : m 6= n
ρmm0 = |00ih00|
(1)
∗
|11ih00| + h.c
+ Vmm0 Dnn
∗
+ A∗nn (1) Dnn
|00ih00| + h.c
∗
+ A∗mn (1) Dnn
|10ih00| + h.c
∗
|01ih00| + h.c.
+ A∗m0 n (1) Dnn
(5.2.2)
This can be simplified by noting that A(1) has
the same structure as α(1) , i.e.
A(1)
nm = 0
n − m ∈ 2Z,
(5.2.3)
allowing us to drop the A∗nn term. If we are
considering consecutive modes, i.e. m0 = m +
1, this expression can be further simplified by
(1)
(1)
noting that one of Amn or Am0 n will always be
0.
For definiteness, lets consider the density
matrix ρmm+1 , and set Am+1n = 0, Amn 6= 0
. Then we have a density matrix of the form


1 0 a∗ b∗
0 0 0 0 

pmm+1 = 
(5.2.4)
a 0 0 0  ,
b 0 0 0
This matrix is symmetric. Thus any loop
through the modes that we choose will have a
trivial twist, since all the correlation matrices
involved will be symmetric.
We note that, though closing the loop may
seem artificial here, since every mode is populated by particles, it is not hard to imagine an
experimental set-up in which some modes are
restricted, generating states with loops naturally built into the modes.
11
MPhys Project | AL
where we would expect to find more interesting
with a corresponding correlation matrix

 behaviour. In order to do this, it is in fact only
1
0
0
1
necessary to evaluate β (2) , since we find
2<(a) 2<(b) 2=(b) 2<(a)

S(m, m + 1) = 
2=(a) 2=(b) −2<(b) 2=(a) ,
(1)
(0)
V (2) = − B (1) A−1 + B (2) A−1
, (6.0.8)
1
0
0
1
(5.2.5)
where B (2) is given by
where
B (2) = Dβ (2) − β T
a=
b=
∗
A∗mn (1) Dnn
(1)
∗
Vmm0 Dnn
,
(5.2.6)
2: m=n
Tracing out all but modes n and m0 , we find
ρnm0 = |10ih10|
(1)
∗
+ Vnm0 Dnn
|21ih10| + h.c
∗
+ A∗nn (1) Dnn
|10ih10| + h.c
∗
|01ih01| + h.c.
+ A∗nm0 (1) Dnn
(5.2.7)
Here we encounter a problem. It is not possible
to define a correlation matrix, and hence define
the twist, using the density matrix (5.2.7) , because of the presence of the |21i state.
The modes of the field are not qubits, they
are simple harmonic oscillators. Any measure of entanglement defined for qubits will encounter this problem - the fact that the vacuum
state does not is a special case. We would also
find states where each mode is populated by
more than one boson if we computed the vacuum state to second order. I will discuss this
further in the next section.
6
D∗ + α(1) Dβ (1) − β (1) D∗ α(1) .
(6.0.9)
Another problem is that any measures of entanglement defined for qubits, such as the twist,
cannot always be defined on the modes of the
cavity for a bosonic field. One solution to this
problem would be to work with a fermionic
field, as in [13], where the exclusion principle
creates natural qubits. Another would be to
consider hard core bosons - a bosonic field, but
with an exclusion principle imposed on it.
A third direction, which seems promising,
could be to work with Gaussian states, and covariance matrix (CM) formalism, which I will
very briefly outline below. For more extensive
discussions see, for example, [2, 16, 17].
In this framework, states are described by a
phase space distribution, which can be specified completely by the moments of the quadrature operators
which is not symmetric. This suggests that it
should be possible to find number states with
non trivial twists in the cavity.
CASE
(2)
1
xn = √ (an + a†n )
2
1
pn = −i √ (an − a†n ),
2
(6.0.10)
(6.0.11)
i.e. hxi, hpi, hx2 i, hp2 i, hxpi, hx3 i etc. In general
there are infinitely many moments that must be
specified. However, Gaussian states are a particularly simple subset of states, which are described by the first and second moments only.
They literally have a Gaussian profile in phase
space.
The correlations are encoded in a matrix
known as the covariance matrix, Γ, and mathematical tools exist for converting between the
unitary evolution of a density matrix
Conclusions and Future Directions
This project has been rather exploratory, and
could be taken in several directions. We have
seen that the mutual information, and more
generally any entropic measure, will not show
changes to O(h). One extension would then be
to perform computations of the states to O(h2 ),
ρ0 = U ρU † ,
12
(6.0.12)
MPhys Project | AL
for non trivial twists, there should be coupling
between the x and p quadrature operators between different modes. The operator (4.4.6) is
an example of such a scenario.
and the evolution of this covariance matrix. In
particular, it can be shown that unitary transformations in the density matrix formalism correspond to symplectic transformations in CM
formalism,
Γ0 = SΓS T ,
Acknowledgements
(6.0.13)
I would like to thank my supervisor, Vlatko
Vedral, and his research group, for their advice
during the project - particularly Felix Pollock
for all his time and help. I would also like to
thank David Bruschi, Nicolai Friis and Antony
Lee at Nottingham University for explaining
their research to me, and Nicolai for Figure 1,
a slightly modified version of that found in [13].
where S is a member of the symplectic group
(see appendix A).
There are several reasons to adopt this formalism. It is not overly restrictive, since many
experimentally accessible states are gaussian
states - examples include the vacuum state, coherent states, and squeezed states - and the
tools available considerably simplify calculations. As an example, using the CM, the process of taking a partial trace reduces to simply
removing all the relevant rows and columns of
Γ.
With reference to the accelerated cavity, it
can be shown that if we begin with a Gaussian state, the evolution in the cavity is guaranteed to give another Gaussian state at the
end (this turns out to be a consequence of the
fact that (4.4.6) contains only quadratic powers of the quadrature operators). Thus CM
formalism can be applied to all of the calculations discussed in previous sections, provided
we begin with a Gaussian state. The symplectic form of the Bogoliubov transformations has
already been given in [18], and many measures
of entanglement have previously been defined
for Gaussian states [17].
In addition, a quantity analogous to the
twist has also already been defined for Gaussian states [19]. The idea behind this definition is identical to that found in the qubit definition. However, the symmetrizing operations
are now members of the Symplectic group, not
the Lorentz group, and the objects to be symmetrized are reduced covariance matrices, subsets of the overall CM, which contain the correlations between adjacent modes of the quantum
field.
Using this definition, it should be possible
to avoid the problems that came with defining
the twist in this project. I also note that in the
last section of [19], the author points out that,
References
[1] A. Peres and D. Terno, Rev. Mod. Phys.
76, 93123 (2004)
[2] P. Alsing and I. Fuentes, e-print
arXiv:1210.2223 [quant-ph] (2012)
[3] I. Fuentes and R. B. Mann, Phys. Rev.
Lett. 95 120404 (2005)
[4] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000),
ISBN 0-521-63503-9.
[5] M.S.
Williamson,
M.Ericsson,
M.Johansson, E. Sjoqvist, A. Sudbery and V.Vedral, Phys. Rev. A 84,
032302 (2011)
[6] F.Pollock, Oxford Mphys Project Report
(2011)
[7] F. Verstraete, J. Dehaene and B. De Moor,
Phys Rev. A 65, 032308 (2002)
[8] G. Munster and M. Walzl, ArXiv High
Energy Physics - Lattice e-prints (2000),
arXiv:hep-lat/0012005.
[9] N. D. Birrell and P. C. W. Davies, Quantum fields in Curved Space (Cambridge
University Press, 1982) ISBN 0-521-278589.
13
MPhys Project | AL
Using this vector, the canonical commutation
relations can be expressed as
[10] M. Maggiore, A Modern Introduction to
Quantum Field Theory (Oxford University Press 2005), ISBN 978-0-19-852074-0.
[Ri , Rj ] = iΩij ,
[11] D. E. Bruschi, I. Fuentes and J. Louko,
Phys. Rev. D 85, 061701(R) (2012)
where Ω is the symplectic matrix

0 1
−1 0
N

M
0 −1

..
Ω=
=
.
1 0

i=1

0
[12] N. Friis, D. E. Bruschi, J. Louko, and
I. Fuentes, Phys. Rev. D 85, 081701(R)
(2012)
[13] N. Friis, A. R. Lee, D. E. Bruschi, and J.
Louko, Phys. Rev. D 85, 025012 (2012)
[14] D.E Bruschi, PhD thesis, University of
Nottingham, (2012)

0



.

0 1
−1 0
(A.0.16)
It can be shown that an (active) unitary evolution of a quantum state is equivalent to a
(passive) operation on R
[15] A. Fabbri and J. Navarro-Salas, Modeling
Black Hole Evaporation (Imperial College
Press, London, 2005), ISBN 1-86094-527-9
R0 = SR,
[16] J.Anders, PhD thesis, University of Potsdam, (2003)
(A.0.17)
where S is a member of the symplectic group.
Elements of this group satisfy the relation
[17] G. Adesso and F. Illuminati, Phys. Rev.
A 72, 032334 (2005)
SΩS T = Ω,
[18] N.Friis and I.Fuentes, Journal of Modern
Optics 60, 22-27 (2013)
(A.0.18)
where Ω is the symplectic matrix (A.0.16). An
important feature of the transformations S is
that they preserve the commutation relations
X
[Ri0 , Rj0 ] =
Sik Sjn [Rk , Rn ]
(A.0.19)
[19] R. Hobson, Oxford Mphys Project Report
(2012)
A
(A.0.15)
k,n
Covariant Matrix Formalism and the
Symplectic Group
=i
X
Sik Sjn Ωkn = iΩij ,
(A.0.20)
k,n
The quadrature operators are typically arranged in a vector R. For a field with N modes
 
x1
 p1 
 
 
(A.0.14)
R =  ...  .
 
xN 
pN
much like the Bogoliubov transformations do.
Under (A.0.17), it can be shown the covariance
matrix transforms as
Γ0 = SΓS T .
14
(A.0.21)