Time-ordering Dependence of Measurements in Teleportation

0.39 MB·English

by  Reinhold A. Bertlmann

Time-ordering Dependence of Measurements in Teleportation
Reinhold A. Bertlmann,∗ Heide Narnhofer,† and Walter Thirring‡
University of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Vienna, Austria
The phenomenon of “delayed-choice entanglement swapping” as it was realized in
a recent experiment we trace back to the commutativity of the projection operators
which are involved in the corresponding measurement process. We also propose an
experimental set-up which depends on the order of successive measurements corre-
2
1 sponding to noncommutative projection operators. In this case entanglement swap-
0
ping is used to teleport a quantum state from Alice to Bob, where Bob has now the
2
possibility to examine the noncommutativity within the quantum history.
t
c
O
PACS: 03.65.Ud, 03.65.Aa, 02.10.Yn

0
Keywords: entanglement, entanglement swapping, delayed-choice, quantum teleportation
2

]
h
p
- I. INTRODUCTION
t
n
a
Recently the Zeilinger group [1] has performed an experiment which has been stimulated
u
q byAsherPeres[2]. Itwasnamed“delayed-choiceentanglementswapping”andwasperceived
[
as a quantum mechanical paradox if – as Peres pointed out – a simple quantum rule is

1 forgotten “it is meaningless to assert that two particles are entangled without specifying in
v
which state they are entangled ... or if we attempt to attribute an objective meaning to the
6
4 quantum state of a single system”. Then quantum effects may mimic an influence of future
6 actions on past events.
5
However, it always depends on the point of view what will be considered as a paradox. In
.
0
this Article we want to interpret, on one hand, the experimental set-up such that the result
1
2 appears quite naturally even to a classically educated physicist. The paradox arises when
1
the point of view is changed within an interpretation, in particular, when classical probabil-
:
v ities are interpreted as quantum correlations that arise due to a coherent superposition of
i
X quantum states. On the other hand, it is essential in the interpretation of the experiment
r that a measurement in quantum theory always causes a collapse of the quantum state (or
a
wave function), see von Neumann [3]. In contrast, a measurement in classical physics we
have to interpret as a kind of filtering. By filtering we sort out those states from the set of
states, which do not possess our desired properties (or boundary conditions). Of course, we
also can consider a measurement in quantum theory in such a way. The difference, however,
is that in classical physics due to the commutativity of multiplication the order of successive
measurements does not matter at all. Whereas in quantum physics (formulated by quan-
tum histories [4, 5]) we can only be sure that the results do not depend on the order of
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
2
the measurements if the corresponding projection operators commute. But precisely such
(commuting) projection operators enter in the experiment of the Zeilinger group [1].
Entanglement swapping offers the possibility to an external observer, called Victor, who
has access to a Hilbert space which is tensorized with an other Hilbert space, to change
a quantum state in the other Hilbert space by performing a measurement in his space.
The quantum state which previously appears separable for Alice and Bob, but not pure, is
changed into a new state which is now pure and entangled for Alice and Bob. This new
(entangled) state is generated by a collapse of the quantum state. That’s only possible for
states where Victor is entangled with the total system of Alice and Bob.
However, if this entanglement with Victor gets destroyed by a measurement of Alice
and/orBob–asitisthecaseintheexperimentofRef.[1]–thenalsoVictor, byperforminga
measurement, hasnochancetodeliveranentangledstatetoAliceandBob. Thecorrelations
that remain can be explained by a mixed state with only classical correlations.
We are interested in entanglement swapping since it gives Alice and Bob the chance for
quantum teleportation. This possibility is automatically supplied when Victor performs
entanglement swapping. In this case a kind of delayed-choice entanglement swapping is
indeed possible as we will explain in this Article. Alice may perform first her required
measurements and only afterwards Victor establishes the connection between Alice and
Bob. However, the measurements Alice performs and compares with those of Bob are in
this delayed-choice experiment not so much related to entanglement swapping but rather to
double teleportation. In any case, there is a sequence of measurements where the order of
successive measurements does not matter.
On the contrary, there is the possibility that Alice and Victor perform certain mea-
surements which imply, dependent on the measurement results, that Bob receives some
well-defined quantum state. Which one, however, depends on the order of successive mea-
surements which now correspond to noncommutative projection operators. By inspection
of Bob’s quantum state the experimenter has therefore the possibility – at least in principle
– to examine in a precise way (that means we do not average over the probabilities of the
outcomes of the measurements) this noncommutativity within the quantum history.
To demonstrate this noncommutative feature of measurements within the quantum his-
tory we use the mathematical formalism of isometries (introduced already in Ref. [6]) as
mappings from one factor to the other in the tensor product of Hilbert spaces. It turns
out that this generalizing formalism, being valid in any dimensions, is quite powerful and
convenient to handle.
II. PHYSICAL SETTINGS AND FORMALISM
The physical settings we consider are that of quantum teleportation and entanglement
swapping. Botharerelatedtoakindofquantum transport betweensubsystems. Weconsider
a tensor product of two Hilbert spaces H ⊗H of equal dimensions. Then a projection in
1 2
H reduces a given pure state in H ⊗H
1 1 2
d
(cid:88)
|ψ(cid:105) = a |ϕ (cid:105) ⊗|κ (cid:105) (1)
12 i i 1 i 2
i=1
3
to a pure separable state
d
(cid:0)|χ(cid:105)(cid:104)χ|(cid:1) ⊗1 |ψ(cid:105) = |χ(cid:105) ⊗(cid:88) a (cid:104)χ|ϕ (cid:105) |κ (cid:105) (2)
1 2 12 1 i i 1 i 2
i=1
that by appropriate choice of |χ(cid:105) and |ψ(cid:105) satisfies some desirable properties.
1 12
Let us first briefly recall the experimental set-ups and our theoretical point of view, the
formalism, for the description of quantum teleportation and entanglement swapping.
A. Quantum teleportation
John Bell with his famous inequalities [7, 8] was the first to demonstrate the nonlocal
feature of quantum mechanics. Mathematically, it is the entanglement of quantum states
determined by analogous inequalities, the entanglement witness inequalities [9–12], which
provides the basis of quantum information processing, particularly in processes like quantum
teleportation.
Quantumteleportation[13]togetherwithitsexperimentalverification[14,15]isanamaz-
ing quantum feature which lives from the fact that in all finite dimensions several qudits
can be entangled in different ways. Usually three two-dimensional qubits are considered
together with the associated Bell states.
In order to demystify the “hocus pocus” [20] of the phenomenon we first want to fix the
rules. The quantum states, described by vectors, are elements of a tensor product of three
Hilbert spaces H ⊗ H ⊗ H of equal dimensions, the corresponding matrix algebras are
1 2 3
denoted by A ⊗ A ⊗ A . In the popular terminology A ,A belong to Alice and A to
1 2 3 1 2 3
Bob. Suppose that Alice gets on her first line H an incoming message given by a vector
1
|φ(cid:105) which she wants to transfer to Bob without direct contact between the algebras A and
1 1
A , but she knows how the vectors correspond to each other by a given isometry. To achieve
3
this goal she uses the fact that the three algebras can be entangled in different ways. Alice
also knows that her second line H is entangled with Bob by a source of entangled photons
2
(called EPR source with reference to the work of Einstein, Podolsky and Rosen [16]), such
that the total state restricted to H ⊗H is maximally entangled. A maximally entangled
2 3
˜
state |ψ(cid:105) ∈ H ⊗ H defines a map being an antilinear isometry I between the vectors
23 2 3 23
˜
from one factor to the other [6]. More precise, the antilinear isometry I is a bijective map
23
from an orthogonal basis {|κ (cid:105) } in H into a corresponding orthogonal basis {|ϕ (cid:105) } in
i 3 3 i 2
H such that the components of the entangled state
2
d
1 (cid:88)
|ψ(cid:105) = √ |ϕ (cid:105) ⊗|κ (cid:105) (3)
23 i 2 i 3
d
i=1
are related by
|I˜ κ (cid:105) = |ϕ (cid:105) and |I˜ U κ (cid:105) = |U∗I˜ κ (cid:105) = U∗|ϕ (cid:105) , (4)
23 i3 2 i 2 23 3 i3 2 2 23 i3 2 2 i 2
for every unitary operator U . Notice our choice of notation for the transpose, adjoint and
complex conjugate of an operator: (U )(cid:62) = U , (U )† = (U )∗ and (U )∗ = U∗ .
ij ji ij ji ij ij
This metamorphosis of a vector first into an operator and then into an isomorphism (an
isometry in our case) can be understood by a partial scalar product. The scalar product
4
(cid:104)|(cid:105) occurs in the multiplication of the tensor product. If, however, the two vectors have a
different number of factors (e.g. into the decomposition of basis vectors) there will be some
(cid:0) (cid:1)
left over (cid:104)χ| |ϕ(cid:105) ⊗|κ(cid:105) = (cid:104)χ|ϕ(cid:105) |κ(cid:105) . In the above case of decomposition (3) when
2 2 3 2 3
considering the partial scalar product we obtain
d
(cid:0) (cid:1) 1 (cid:88)
(cid:104)χ| |ψ(cid:105) = √ (cid:104)χ|ϕ (cid:105) |κ (cid:105) = |φ(cid:105) . (5)
2 23 i 2 i 3 3
d
i=1
In this way a maximally entangled state |ψ(cid:105) becomes a map, a quantum transporter from
23
H → H .
2 3
The possibility to transfer the incoming state of H at Alice into a state of H at Bob
1 3
uses the fact that an isometry I between this two algebras was taken for granted. Now
13
˜
Alice chooses an isometry I , which corresponds to choosing a maximally entangled state
12
|ψ(cid:105) ∈ H ⊗H , such that the following isometry relation holds
12 1 2
˜ ˜
I ◦I = I . (6)
12 23 13
We denote the antilinear isometry by tilde, the linear one without, and the composition of
two maps by a circle ◦. Expressed in {ϕ }, an orthonormal basis (ONB) of one factor, the
i
state vector can be written as
d
1 (cid:88)
|ψ(cid:105) = √ |ϕ (cid:105) ⊗|I˜ ϕ (cid:105) . (7)
12 i 1 21 i1 2
d
i=1
A measurement by Alice in H ⊗H with the outcome of this entangled state produces the
1 2
desired state vector in H . The outcome of other maximally entangled states, orthogonal
3
to the first one, corresponds to a unitary transformation of the form U = U ⊗ 1 in
12 1 2
H ⊗H , which produces a unique unitary transformation U in H that Bob can perform
1 2 3 3
to obtain the desired state. Thus Alice just has to tell Bob her measurement outcome via
some classical channel. The measurements of Alice produce the following results for Bob:
1
(cid:0)|ψ(cid:105)(cid:104)ψ|(cid:1) ⊗1 |φ(cid:105) ⊗|ψ(cid:105) = |ψ(cid:105) ⊗|φ(cid:105) , (8)
12 3 1 23 d 12 3
1
U (cid:0)|ψ(cid:105)(cid:104)ψ|(cid:1) U† ⊗1 |φ(cid:105) ⊗|ψ(cid:105) = U |ψ(cid:105) ⊗U |φ(cid:105) , (9)
12 12 12 3 1 23 d 12 12 3 3
with the unitary transformation U = U ⊗1 in H ⊗H .
12 1 2 1 2
˜
The state vectors can be expressed by an ONB for a fixed isometry, e.g. choosing I in
23
the following way
d
1 (cid:88)
|ψ(cid:105) = √ |ϕ (cid:105) ⊗|ϕ (cid:105) , (10)
23 i 2 i 3
d
i=1
d
(cid:88)
|φ(cid:105) = α |ϕ (cid:105) . (11)
1or3 i i 1or3
i=1
˜ ˜
The antilinear isometries I and I corresponding to the two given maximally entangled
12 23
˜ ˜
states combine to the linear isometry I = I ◦I defining the map in Eq. (11). Changing
13 12 23
to another Bell state in Alice’s measurement corresponds to
I˜ ◦I˜ U = I˜ ◦U∗I˜ = U I˜ ◦I˜ . (12)
32 21 1 32 2 21 3 32 21
5
Alice Bob
Victor
BSM
1 2 3 4
EPR I EPR II time
FIG. 1: Entanglement swapping. Two pairs of entangled photons are emitted by the sources
EPR I and EPR II. When Victor entangles two photons, i.e. performs a Bell state measurement
in channels (2,3), the remaining two photons in channels (1,4) become instantaneously entangled
into the same entangled state – the entanglement swapped – which is measured by Alice and Bob.
Summarizing, if Alice measures the same Bell state in the sense of relation (6) between
H and H as there was between H and H , which was given by the EPR source, she
1 2 2 3
knows that her measurement left Bob’s H in the state |φ(cid:105) which is Alice’s incoming state.
3 3
If Alice finds, on the other hand, a different Bell state, which is given by U |ψ(cid:105) (and
12 12
U = U ⊗ 1 ) since all other Bell states are connected by unitary transformations, then
12 1 2
Bob will have the state vector U |φ(cid:105) , where the unitary transformation U is determined
3 3 3
by U .
12
B. Entanglement swapping
Closely related to teleportation of single quantum states is an other striking quantum
phenomenon called entanglement swapping [17]. It can be interpreted as teleportation of an
unspecified state (with no well-defined polarization properties) in channel 2 onto the photon
in channel 4, or of the unspecified state in channel 3 onto the photon in channel 1, see
Fig. 1. Altogether, the entanglement of the photons created by a Bell state measurement
in channels (2,3) swaps onto the photons in channels (1,4). Experimentally entanglement
swapping has been demonstrated in Ref. [18] and is nowadays a standard tool in quantum
information processing [19]. The set-up is described by a tensor product of four Hilbert
spaces H ⊗H ⊗H ⊗H of equal dimensions. The entanglement swapping phenomenon
1 2 3 4
illustrates the different slicing possibilities of a 4-fold tensor product into (1,2) ⊗ (3,4) or
(1,4)⊗(2,3), where e.g. (1,2) denotes entanglement between the subsystems H and H ,
1 2
see Fig. 1.
The same way of reasoning as in Sec. IIA can be applied to entanglement swapping,
i.e. we consider a maximally entangled state as an isometry between the vectors of one
factor to the other. In case of entanglement swapping the starting point are two maximally
entangled pure states combined in a tensor product |ψ(cid:105) ⊗|ψ(cid:105) . Expressed in an ONB
12 34
of one factor the entangled states are given by Eq. (7). They describe usually two pairs
6
of EPR photons. These propagate into different directions and at the interaction point of
two of them, say photon 2 and 3, Victor performs a Bell state measurement (BSM), i.e.
a measurement with respect to an orthogonal set of maximally entangled states. In the
familiar case they are given by the Bell states |ψ±(cid:105) = √1 (|↑(cid:105) ⊗|↓(cid:105) ± |↓(cid:105) ⊗|↑(cid:105) )
23 2 2 3 2 3
and |φ±(cid:105) = √1 (|↑(cid:105) ⊗|↑(cid:105) ± |↓(cid:105) ⊗|↓(cid:105) ). But our results are more general, the four
23 2 2 3 2 3
factors just have to be of the same dimension d which is arbitrary.
In complete analogy to the case of teleportation, discussed before, the effect of the pro-
jection corresponding to the measurement on the state is
1
(cid:0)|ψ(cid:105)(cid:104)ψ|(cid:1) ⊗1 |ψ(cid:105) ⊗|ψ(cid:105) = |ψ(cid:105) ⊗|ψ(cid:105) , (13)
23 14 12 34 d 23 14
˜
where |ψ(cid:105) is a maximally entangled state corresponding to the antilinear isometry I
14 14
which satisfies the relation
˜ ˜ ˜ ˜
I = I ◦I ◦I . (14)
14 12 23 34
˜ ˜ ˜
The other isometries I ,I ,I correspond to the other maximally entangled states.
12 23 34
Thus after the Bell state measurement of photon 2 and 3 into a definite entangled state
the photons 1 and 4 become instantaneously entangled into the same state (see Fig. 1),
remarkably, the two photons originate from different noninteracting sources.
III. DELAYED-CHOICE OF ENTANGLEMENT SWAPPING
Nowwearepreparedtoturntothephenomenonofdelayed-choiceentanglementswapping
[1, 2] which is interpreted as being quite paradoxical. In this experiment the order of the
measurements is reversed as compared to entanglement swapping. Alice and Bob measure
first and record their data, then at a later time Victor is free to choose a projection of his
two states onto an entangled state or to measure them individually. The outcome of the
measurements of Alice and Bob and Victor is recorded and compared to the previous case
where Victor measured before Alice and Bob. It turns out that the joined probability for the
outcome of these two cases is the same. Particularly in case of delayed-choice, it is Victor’s
measurement that decides the context and determines the interpretation of Alice and Bob’s
data. Alice and Bob can sort their already recorded data, according to Victor’s later choice
and his results, in such a way that they can verify either the entangled or the separable
states.
A. General Discussion
To analyze this delayed-choice procedure we prefer to use the density matrix formalism
for the description of the quantum states. We start with a tensor product of four matrix
algebras A = A ⊗ A ⊗ A ⊗ A of equal dimensions; A and A belong to Alice and
tot 1 2 3 4 1 4
Bob and the subalgebra A ⊗A refers to Victor’s algebra.
2 3
At the beginning the state described by the density matrix ρ ∈ A is separable on
tot
A ⊗A and by manipulation – Bell state measurement – in A ⊗A the state over A ⊗A
1 4 2 3 1 4
becomes entangled.
(cid:0) (cid:1)
The measurements correspond to the projection operators Qi = |ψi(cid:105)(cid:104)ψi| (with i
23 23
= 1, ...,4) onto the maximal entangled Bell states |ψ1,2(cid:105) = |ψ±(cid:105) and |ψ3,4(cid:105) = |φ±(cid:105) .
23 23 23 23
Thus the operators Qi form a Bell basis in subalgebra A ⊗A .
23 2 3
7
If we project on such a basis we immediately get a new state on A ⊗A , i.e., we have a
1 4
collapse of the quantum state, which is up to normalization given by
Qi ρQi . (15)
23 23
Let us start with the case where ρ ∈ A is separable on A ⊗A and maximal entangled
tot 1 2
on A ⊗A
3 4
Qi
ρ = ρ ⊗ρ ⊗ρ −→23 ρ ⊗ρi ⊗ρi, (16)
1 2 34 1 23 4
(cid:0) (cid:1)
where ρ = |ψ(cid:105)(cid:104)ψ| corresponds to some entangled state that at its best can be a Bell
34 34
state on the algebra A ⊗A and ρi ≡ Qi . We find ρ ⊗ρi separable with a corresponding
3 4 23 23 1 4
probability depending on the projector Qi and on the initial state ρ .
23 34
Next we consider the case where ρ ∈ A is maximal entangled on A ⊗ A and on
tot 1 2
A ⊗A and we perform a BSM on A ⊗A then we have
3 4 2 3
Qi
ρ = ρ ⊗ρ −→23 Qi ρQi = ρi ⊗ρi . (17)
12 34 23 23 14 23
We now find the state ρi maximal entangled depending on the projector Qi and corre-
14 23
sponding to the antilinear isometry I˜i which satisfies – as before in Eq. (14) – the relation
14
I˜i = I˜ ◦I˜i ◦I˜ . (18)
14 12 23 34
(cid:0) (cid:1)
Thus after the projection the state Qi ρQi is pure and maximal entangled on
23 23 A1⊗A4
A ⊗A ; the entanglement has swapped.
1 4
It gives us the idea that with help of entanglement swapping we should be able to check
to what extend the collapse of a quantum state happens. Let’s consider a sequence of
measurements where the results emerge with a certain probability.
Question:
To what extend does the probability of the measurement outcome depend on the chrono-
logical order of the measurements?
Classical physics:
Consider the time-invariant states in classical physics. There clearly the measurements
are independent of the chosen time, thus independent of the order of the successive mea-
surements.
Quantum physics:
In quantum physics, however, there exists a quantum history [4, 5]. Consider a quantum
state on some larger algebra and several measurements on that state. The measurements
we describe by projectors P i = (cid:0)|ψi(cid:105)(cid:104)ψi|(cid:1) and Qj = (cid:0)|φj(cid:105)(cid:104)φj|(cid:1) on some states of the
1 1 2 2
subalgebras A ,A . We perform the measurements M = {P i} and M = {Qj} and obtain
1 2 1 1 2 2
ρ −M→1 ρi = P iρP i −M→2 ρji = QjP iρP iQj. (19)
1 1 2 1 1 2
The state ρi after the first measurement M occurs with a probability wi = TrP iρP i and
1 1 1
the state ρji after the second measurement M with a probability wji = TrQjP iρP iQj =
2 2 1 1 2
TrρP iQjP i.
1 2 1
Now we interchange the order of measurements M ←→ M then the state
1 2
ρij = P iQjρQjP i (20)
1 2 2 1
8
occurs with probability wij = TrρQjP iQj. In case of commuting projection operators
2 1 2
there is no dependence on the order of measurement operations, i.e.
(cid:2)Pi, Qj (cid:3) = 0 =⇒ wij = wji, (21)
1 2
the probabilities of the measurement outcomes, which can be determined from the protocols
of the measurements, are the same! Actually, it is enough to examine whether the condition
Trρ(cid:0)QjP iQj −P iQjP i(cid:1) = 0 holds.
2 1 2 1 2 1
B. Experiment of Zeilinger’s Group
Now we turn to the experiment of the Zeilinger group [1]. Let’s begin with the case
of entanglement swapping. Thus initially we have the two Bell states ρ and ρ and first
12 34
VictorperformsaBSMM = {Qi }onsomemaximalentangledstateρi ≡ Qi onA ⊗A ,
1 23 23 23 2 3
then we obtain as above, Eq. (17),
ρ −M→1 ρi = Qi ρ ⊗ρ Qi = ρi ⊗ρi , (22)
23 12 34 23 14 23
where the maximal entangled state ρi on A ⊗ A occurs with probability wi = 1 and
14 1 4 4
satisfies the isometry relation I˜i = I˜ ◦I˜i ◦I˜ between the algebras.
14 12 23 34
The second measurement M = {P ab = P a ⊗P b} is performed by Alice and Bob, they
2 14 1 4
measure the incoming states at A and A by projecting on some states |ψa(cid:105) and |ψb(cid:105) .
1 4 1 4
(cid:0) (cid:1) (cid:0)(cid:12) (cid:11)(cid:10) (cid:12)(cid:1)
Denoting the corresponding projectors by P a = |ψa(cid:105)(cid:104)ψa| and P b = (cid:12)ψb ψb(cid:12) the
1 1 4 b
state turns into (see Fig. 1)
ρi −M→2 ρiab = P abQi ρQi P ab = ρa ⊗ρi ⊗ρb. (23)
14 23 23 14 1 23 4
The probability wiab of finding the state ρiab is determined by the isometry I˜i , i.e., it
14
depends on the chosen maximal entangled state denoted by i.
Next we study the reversed order of measurements, i.e., the case of delayed-choice entan-
glement swapping (see Fig. 2).
The first measurement is now M = {P ab = P a⊗P b} performed by Alice and Bob, they
2 14 1 4
measure their incoming states by projecting on some states |ψa(cid:105) and |ψb(cid:105) , then the total
1 4
state ρ turns into
ρ −M→2 ρab = P abρ ⊗ρ P ab = ρa ⊗ρa ⊗ρb ⊗ρb. (24)
14 12 34 14 1 2 3 4
which is separable and even pure on the subalgebra A ⊗A .
1 4
In a second measurement Victor may choose freely to project onto an entangled state or
onto a separable state on A ⊗A . Let’s discuss first the measurement M = {Qi } onto
2 3 1 23
an entangled state then the above state changes into
ρab −M→1 ρabi = Qi P abρ ⊗ρ P abQi = ρa ⊗ρi ⊗ρb, (25)
23 14 12 34 14 23 1 23 4
whichremainsseparableonA ⊗A . Theentangledstateρi clearlydependsontheisometry
1 4 23
I˜i (a,b) and on the chosen (a,b).
23
9
Victor
BSM
Alice Bob
1 2 3 4
EPR I EPR II time
FIG. 2: Delayed-choice entanglement swapping. There are two EPR sources I and II and we have
a reversed order of measurements. Alice and Bob measure first in channels (1,4) and record their
data and later on Victor projects in a free choice his state in channels (2,3) onto an entangled or
separable state. When comparing this case with entanglement swapping of Fig. 1 it turns out that
the joint probability for the outcome of these two cases is the same due to the commutativity of
the projection operators, i.e. there is independency of the order of successive measurements.
Since the projection operators – the measurement operators – commute
(cid:2) (cid:3)
P ab, Qi = 0 =⇒ wabi = wiab, (26)
14 23
theprobabilitywabi = TrρP abQi P ab forthe occurrenceof thefinal stateρabi is the sameas
14 23 14
before, in fact, ρabi = ρiab and the results are independent of the order of the measurements.
It is valid for all projection operators P ab independent of their chosen orientations a,b.
14
Therefore different quantum histories lead to the same result.
In case Victor projects onto a separable state on A ⊗ A we have as measurement
2 3
operators M(cid:99) = {P j}, where the projectors P j = (cid:0)|φj(cid:105)(cid:104)φj|(cid:1) project onto the separable
1 23 23 23
states |φ1,2,3,4(cid:105) = {|↑(cid:105) ⊗|↑(cid:105) ,|↓(cid:105) ⊗|↓(cid:105) ,|↑(cid:105) ⊗|↓(cid:105) ,|↓(cid:105) ⊗|↑(cid:105) }. Then again the
23 2 3 2 3 2 3 2 3
commutator of the projectors is commuting
(cid:2)P ab, P j (cid:3) = 0 =⇒ wabj = wjab, (27)
14 23
and again we find independency of the order of the measurements.
IV. TELEPORTATION AND DELAYED ENTANGLEMENT SWAPPING
In this section we are going to use entanglement swapping as a source for quantum
teleportation between Alice and Bob. As we shall see it’s not so much about entanglement
swapping – delayed or not – but rather about double teleportation. We will find two cases,
one where the order of successive measurements does not matter, and an other one where it
does corresponding to the noncommutative projection operators.
10
Bob
4
Victor
BSM
Alice
BSM
0 1 2 3 4
EPR I EPR II time
FIG. 3: Teleportation with delayed entanglement swapping. There are two EPR sources I and II
and we have double quantum teleportation of an incoming state at channel 0 to the same outgoing
stateatchannel4, whichismeasuredbyBob. InbetweenthereisfirstaBellstatemeasurementby
Alice in the channels (0,1), teleporting the incoming state from 0 to 2, and second a measurement
by Victor in channels (2,3), producing a kind of delayed entanglement swapping, which teleports
the state from 2 to 4.
We consider here a tensor product of five matrix algebras A = A ⊗A ⊗A ⊗A ⊗A
tot 0 1 2 3 4
of equal dimensions, where we have an additional (general) incoming state
(cid:0) (cid:1)
ρ = |φ(cid:105)(cid:104)φ| (28)
0 0
on the algebra A at Alice’s side, which we can write as
0
|φ(cid:105) = α|↑(cid:105) + β|↓(cid:105) and |α|2 +|β|2 = 1, (29)
0 0 0
if we are dealing with qubits. If now Alice and Victor perform a BSM the incoming state
|φ(cid:105) at Alice’s side is teleported into |φ(cid:105) = |I φ (cid:105) at Bob’s part.
0 4 40 0 4
A. Commutative Projection Operators
We consider again a physical situation similar to Sec. IIIB – a kind of delayed entangle-
ment swapping – where we have two EPR sources providing the Bell states ρ and ρ , but
12 34
now an additional particle in the quantum state (28) enters on Alice’s side, see Fig. 3.