Quantum correlations for anonymous metrology
A. J. Paige
1
, Benjamin Yadin
2
, and M. S. Kim
1
1
QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK.
2
Department of Atomic and Laser Physics, Clarendon Laboratory, University of Oxford
August 23, 2019
We introduce the task of anonymous
metrology, in which a physical parame-
ter of an object may be determined with-
out revealing the object’s location. Alice
and Bob share a correlated quantum state,
with which one of them probes the object.
Upon receipt of the quantum state, Char-
lie is then able to estimate the parameter
without knowing who possesses the object.
We show that quantum correlations are re-
sources for this task when Alice and Bob
do not trust the devices in their labs. The
anonymous metrology protocol moreover
distinguishes different kinds of quantum
correlations according to the level of de-
sired security: discord is needed when the
source of states is trustworthy, otherwise
entanglement is necessary.
1 Introduction
Quantum correlations can provide significant ad-
vantages over classical resources in performing
non-local tasks. Teleportation of an unknown
quantum state [1], super dense coding [2], and
quantum key distribution [3], are a few examples
amongst other contexts [47]. This has notably
included their relevance to the power of quan-
tum computation [812], their uses in metrol-
ogy [1317], and their roles in non-local infor-
mation tasks like quantum state merging [1820].
Quantum correlations can also hide information
non-locally [2125], with recent studies investi-
gating non-locally hiding computations [26], and
when it is possible to mask quantum informa-
tion [27].
In this work, we introduce the task of anony-
mous metrology, which involves encoding an ini-
tially unknown continuous parameter in a state
A. J. Paige: a.paige16@imperial.ac.uk
Benjamin Yadin: benjamin.yadin@gmail.com
whilst hiding where the encoding happened. We
identify the quantum states that enable the task
and separately treat the two cases of having a
trustworthy or untrustworthy source of states.
We term the resourceful states as weakly anony-
mous (WA) and strongly anonymous (SA) respec-
tively, and give physical intuition for the distinc-
tion by demonstrating how SA states allow the
encoding’s location to be not just hidden but
quantum mechanically delocalised.
We derive general forms for the WA and SA
states, using modes of translational asymme-
try [28, 29] for the former, and for the latter show-
ing equivalence to the entangled “maximally cor-
related states” [30] extended by degeneracy. We
then determine the nature of their quantum cor-
relations. In general, quantum correlations ex-
ist in different “strengths”, from discord [5] to
full Bell nonlocality [31], and understanding their
respective utilities remains to be fully explored.
Whilst there are several works that demonstrate
an operational use for discord [15, 17, 19, 32], our
results additionally reveal an operational distinc-
tion between different types of quantum correla-
tions. We find that WA states require a form of
discord that we term aligned discord, while SA
states require a stronger type of correlation, cor-
respondingly termed aligned entanglement.
2 Defining anonymous metrology
We introduce the task of anonymous metrology
with an example, illustrated in Fig. 1. Alice and
Bob are in spatially separated laboratories, and
one of them receives a system, the location of
which they must keep hidden (e.g. some valuable
diamond). Charlie wants them to probe it with a
(finite-dimensional) quantum system to give him
information about some initially unknown con-
tinuous parameter θ (such as a refractive index).
We make the following assumptions:
Accepted in Quantum 2019-08-19, click title to verify 1
arXiv:1812.04374v3 [quant-ph] 22 Aug 2019
Figure 1: Illustration of the anonymous metrology task.
Initially A and B share the state ρ
AB
, and one of them
is given the system to hide. They engineer U
A
, or V
B
,
to encode θ into the shared state, then both halves are
sent to C.
If the object is in Alice’s lab, it interacts with
her system such that the latter undergoes
a unitary transformation U
A
(θ) = e
iθH
A
with some parameter-independent Hamilto-
nian H
A
. (Otherwise, Alice’s system is un-
changed.) Similarly, Bob’s system undergoes
V
B
(θ) = e
iθG
B
.
Alice and Bob may use classical communica-
tion freely, but this is not secure. Charlie is
also allowed to know the set-up of their labs.
Their devices are unsecure, in that any mea-
surement outcomes obtained are assumed to
be available to Charlie. Note that they may
implement any parameter-independent local
unitaries without loss of security. Hence it is
possible to effectively change the eigenbasis
of H
A
, G
B
arbitrarily.
The task is to enable Charlie to estimate θ,
but prevent him from learning where the hidden
system is, i.e. the one who actually encoded θ into
the quantum system must remain anonymous.
At first glance this task may seem impossible,
since θ is initially unknown and Alice and Bob
can only learn its value via untrusted local mea-
surements. With Charlie accessing their measure-
ment results we might fear that he will always be
able to determine where the encoding is happen-
ing and thus the system’s location. However it
turns out that quantum resources allow them to
succeed in this task.
We formalize this statement later, but we shall
start by illustrating with a simple example us-
ing an entangled state. We allow Alice and Bob
to request copies of a bipartite quantum state
ρ
AB
. Consider the situation when they choose
the Bell state |ψ
+
i
AB
=
1
2
(|00i
AB
+ |11i
AB
).
Alice can apply U
A
(θ) = e
|1i
A
h1|
, to produce
|ψ(θ)i
AB
=
1
2
(|00i
AB
+ e
|11i
AB
), but simi-
larly Bob can apply V
B
(θ) = e
|1i
B
h1|
, and for
all θ he produces the same state. This indicates
the solution to their problem. The one who has
the hidden system interacts their half of ρ
AB
with
it, to realize the relevant encoding unitary (they
may need a rescaling such that θ [0, 2π)), and
then they both send their halves of ρ
AB
to Char-
lie. Given multiple copies Charlie can determine
θ to arbitrary precision but cannot tell if it was
U
A
or V
B
that changed the state, so cannot learn
the system’s location
1
. Clearly the Bell state is
a resource for this task, but we shall show that a
number of quantum states are, and in some cases
they are not entangled.
2.1 Anonymity and encoding conditions
For anonymous metrology there are two relevant
sets of useful states, and the choice between them
depends on who provides the states for Alice and
Bob.
First consider the situation where a trustwor-
thy fourth party, who will never share informa-
tion with Charlie, is sending quantum states to
them. Alice and Bob should request copies of
a state ρ
AB
, that satisfies two conditions. First
that they can find continuously parametrised uni-
taries U
A
(θ), V
B
(θ) such that, for θ in some inter-
val,
U
A
(θ)ρ
AB
U
A
(θ) = V
B
(θ)ρ
AB
V
B
(θ). (1)
This is termed the weak anonymity condition,
since it ensures that for a given parameter the
same state is produced no matter who encoded
it. The second condition is that from the same
interval, different phases produce different states,
i.e. given θ 6= φ we have
U
A
(θ)ρ
AB
U
A
(θ) 6= U
A
(φ)ρ
AB
U
A
(φ). (2)
We term this the encoding condition since it en-
sures that different parameters are mapped to dif-
1
Note instead of sending states to Charlie they could
perform a set of informationally complete local measure-
ments, tomographically reconstruct the state and extract
θ from it, but this is unnecessarily complicated and would
generally be less efficient for learning θ.
Accepted in Quantum 2019-08-19, click title to verify 2
ferent states, so in principle Charlie learns at least
some information about the parameter.
If however it is Charlie who is sending the
states (or the fourth party is untrustworthy) then
anonymity with these states is not assured. The
most dangerous situation is when Charlie holds a
third system and knows the pure state |ψi
ABC
,
but Alice and Bob only know the state ρ
AB
=
Tr
C
(|ψi
ABC
hψ|). They can verify that they have
been sent copies of ρ
AB
by using a subset of the
states they receive to perform metrology, but by
doing this they cannot learn what Charlie holds.
We also note that using a finite number of copies
for metrology will inevitably lead to some finite
error, and we address this later in section 5.1
where we show robustness for the protocol.
To maintain anonymity when Charlie could
be holding a purification, they must be able
to find unitaries such that U
A
(θ)|ψi
ABC
=
V
B
(θ)|ψi
ABC
, up to an irrelevant global phase.
We use this to derive a condition on the
states ρ
AB
that they can choose. We expand
using the Schmidt decomposition |ψi
ABC
=
P
j
λ
j
|φ
j
i
AB
|χ
j
i
C
(in terms of an orthogo-
nal product basis), and projecting onto |χ
j
i
C
,
we have U
A
(θ)|φ
j
i
AB
= V
B
(θ)|φ
j
i
AB
. Writing
ρ
AB
=
P
j
|λ
j
|
2
|φ
j
i
AB
hφ
j
|, and acting from the
left with U
A
(θ), we arrive at
U
A
(θ)ρ
AB
= V
B
(θ)ρ
AB
. (3)
This is termed the strong anonymity condi-
tion. Reversing the argument is straightforward.
Hence the state |ψi
ABC
has unitaries that sat-
isfy U
A
(θ)|ψi
ABC
= V
B
(θ)|ψi
ABC
, if and only if
ρ
AB
= Tr
C
(|ψi
ABC
hψ|) has unitaries that sat-
isfy Eq. (3). It is clear that Eq. (3) implies
Eq. (1), but not vice versa, so the condition (3)
is stronger. With this we now formally establish
appropriate terminology.
Definitions: A state ρ
AB
is a weakly anony-
mous (WA) state if there exist unitaries U
A
(θ) =
e
iθH
A
, and V
B
(θ) = e
iθG
B
that satisfy the con-
ditions given by Eq. (1) and (2). The subset of
these that also satisfy the condition of Eq. (3) are
strongly anonymous (SA) states.
For pure states the WA and SA conditions co-
incide, and furthermore a pure state is WA/SA
if and only if it is entangled. To prove suffi-
ciency we use the Schmidt decomposition to write
|ψi
AB
=
P
j
λ
j
|φ
j
i
A
|χ
j
i
B
. Entangled states
have a Schmidt number of at least two, so without
loss of generality we take λ
0
6= 0 and λ
1
6= 0. Now
the unitaries U
A
(θ) = e
|φ
1
ihφ
1
|
A
, and V
B
(θ) =
e
|χ
1
ihχ
1
|
B
, satisfy the conditions. Hence all
pure entangled states are WA/SA. To prove en-
tanglement is necessary consider a separable state
|ψi
AB
= |φi
A
|χi
B
. The anonymity condition
becomes U
A
(θ)|φi
A
|χi
B
= |φi
A
V
B
(θ)|χi
B
.
Applying the ket hφ|
A
we get hφ|U
A
(θ)|φi|χi
B
=
V
B
(θ)|χi
B
, and projecting this equation onto it-
self we arrive at |hφ|U
A
(θ)|φi| = 1. Hence U
A
(θ)
only imparts an unobservable global phase, so vi-
olates the encoding condition. Therefore entan-
glement is necessary for pure states.
For mixed states things are more complicated
and we address this shortly, after presenting a
different non-local task that illustrates the true
distinction between the WA and SA cases.
Hiding the location of a system from Charlie
somewhat resembles hiding which-path informa-
tion. Now consider Alice and Bob are tasked with
measuring a system that is put in a superposition
of going to Alice and Bob. Can they perform
measurements without acquiring which-path in-
formation and thus without decohering the spa-
tial superposition?
Formalizing this, we consider quantizing the
path degree of freedom P of the system to
be measured; it is put into some superposition
a|Li
P
+ b|Ri
P
of going left to Alice and right
to Bob. They probe the system with a shared
state ρ
AB
, described by the controlled unitary
W (θ) = |Li
P
hL| U
A
(θ) + |Ri
P
hR| V
B
(θ).
Requiring the final state of P to factor out un-
changed, recovers the SA condition Eq. (3) see
Appendix A for details.
This emphasises the fact that the SA condi-
tion ensures no information exists on where the
interaction took place. The measurement was de-
localised by the correlations. We now move from
operational considerations to investigate the form
of the resourceful states.
3 Form of useful states
3.1 Form of WA states
In order to arrive at a form for the WA states,
it is useful to employ modes of translational
asymmetry [28, 29]. Given our unitary action
U
A,θ
(.) = U
A
(θ)(.)U
A
(θ), a given state can be de-
composed into modes as ρ
A
=
P
ω
ρ
(ω)
A
, where
Accepted in Quantum 2019-08-19, click title to verify 3
U
A,θ
(ρ
(ω)
A
) = e
θ
ρ
(ω)
A
. This is akin to Fourier de-
composition of a function. We can select out
modes with the twirling superoperator
P
ω
A
= lim
θ
0
→∞
1
2θ
0
Z
θ
0
θ
0
e
θ
U
A,θ
. (4)
One can verify that P
ω
A
(ρ
A
) = ρ
(ω)
A
. We similarly
define P
ω
B
, using V
B
(.) = V
B
(θ)(.)V
B
(θ). The
twirling operators satisfy the relation U
A,θ
P
ω
A
=
e
θ
P
ω
A
, and a completeness relation
P
ω
P
ω
A
= 1.
We can rewrite the WA condition of Eq. (1)
in terms of superoperators by simply multiplying
by
e
iωθ
2θ
0
, integrating
R
θ
0
θ
0
, and taking the limit
θ
0
, to get
P
ω
A
ρ
AB
= P
ω
B
ρ
AB
. (5)
We prove the converse by acting on this equa-
tion with U
A,θ
V
B
, to get e
θ
V
B
P
ω
A
ρ
AB
=
e
θ
U
A,θ
P
ω
B
ρ
AB
. The e
θ
terms cancel, and then
summing over ω using the completeness relations
we return to V
B
ρ
AB
= U
A,θ
ρ
AB
. Hence Eq (5)
captures the weak anonymity condition. Note
that the encoding condition Eq. (2) becomes that
there has to be some ω 6= 0 for which P
ω
A
ρ
AB
6= 0.
From Eq. (5) we can explicitly write the form
of the WA states. First we define
ρ
(ω
1
2
)
AB
=
X
i,i
0
,j,j
0
,
E
i
0
=E
i
+ω
1
,
E
j
0
=E
j
+ω
2
c
ii
0
jj
0
|ii
A
hi
0
| |ji
B
hj
0
|, (6)
where H
A
|ii = E
i
|ii, with H
A
the Hamiltonian
generator of U
A,θ
, and similarly for B. Then the
WA states are of the form
ρ
AB
=
X
ω
ρ
(ω)
AB
, (7)
where we require non-zero terms for ω 6= 0 so that
encoding is possible.
This shows that WA states are those with cor-
related modes of asymmetry, which indicates a
connection with the resource of quantum coher-
ence [33]. We can view WA states as having corre-
lated coherence in the eigenbasis of the unitaries.
There is a formal similarity with the correlated
coherence defined in [3437].
3.2 Form of SA states
We now derive the form of SA states. Here, work-
ing with modes of asymmetry is not as straight-
forward (see Appendix B for more discussion), so
we use a different approach.
Rearranging the anonymity condition of
Eq. (3) to (U
A
(θ) V
B
(θ))ρ
AB
= 0, and tak-
ing matrix elements in the eigenbasis of the local
unitaries, we get (u
i
(θ) v
i
0
(θ))hii
0
|ρ
AB
|jj
0
i = 0.
The non-zero matrix elements are those for which
u
i
(θ) = v
i
0
(θ). Initially it is simplest to con-
sider the non-degenerate case so that u
i
(θ) 6=
u
j
(θ), i 6= j and similarly for the v
i
. With
this the largest set of non-zero matrix elements
is achieved by pairing every u
i
(θ) with a v
i
0
(θ)
such that u
i
(θ) = v
i
0
(θ). Since relabeling is phys-
ically irrelevant we can write the non-zero matrix
elements as hii|ρ
AB
|jji, and so we write the state
as ρ
AB
=
P
i,j
ρ
ij
|iiihjj|. This is the form of so-
called “maximally correlated" states [30]. Note
that we need at least one non-zero off-diagonal
ρ
ij
= ρ
ji
, to ensure that the encoding condition
of Eq. (2) is satisfied.
We can lift the non-degeneracy restriction by
introducing a new label, such that we write states
that are degenerate under U
A
as |i. We then
write the form of SA states as
ρ
AB
=
X
i,j,λ,λ
0
,µ,µ
0
ρ
ijλλ
0
µµ
0
|iλ,
0
i
AB
hjµ, jµ
0
|.
(8)
Hence the SA states are a generalisation of the
“maximally correlated" states, where we only in-
clude the entangled ones. Having established the
forms of useful states (see Appendix C for the
generalisation to multipartite cases), we now dis-
cuss the quantum correlations.
4 Quantum correlations required
For the anonymity tasks, we shall show that in-
formation is being hidden by the quantum cor-
relations of the states. The main candidates
are entanglement and discord, which can be de-
fined mathematically by specifying forms for the
correlated states. A bipartite state is entan-
gled if it cannot be written in the separable
form ρ
AB
=
P
i
p
i
ρ
(i)
A
ρ
(i)
B
. A bipartite state
is discordant if, for some local basis, it cannot
be written in any of the three forms ρ
AB
=
P
i,j
p
ij
|iihi|
A
|jihj|
B
, ρ
AB
=
P
i
p
i
|iihi|
A
ρ
B|i
,
and ρ
AB
=
P
j
p
j
ρ
A|j
|jihj|
B
, termed Classical-
Classical (CC), Classical-Quantum (CQ), and
Quantum-Classical (QC) respectively. Entangled
states are a subset of discordant states.
As shown below, the WA and SA states form
subsets of the known sets of correlated states.
Accepted in Quantum 2019-08-19, click title to verify 4
We therefore use the terms aligned discord and
aligned entanglement for the WA and SA re-
sources, respectively. For aligned discord we es-
tablish that discord is necessary but not suffi-
cient, and entanglement is neither necessary nor
sufficient. For aligned entanglement we show en-
tanglement is necessary but not sufficient. This
is illustrated in Fig. 2.
Figure 2: Schematic summary of the relations between
all quantum states (Q), and those that are discordant
(D), aligned discordant (AD), entangled (E), and aligned
entangled (AE).
4.1 WA Hamiltonian condition
Before proving these results, we recast the WA
conditions in terms of Hamiltonians, to describe
families of unitaries that encode a continuous
parameter. If we were to demand Eq. (1) and
(2) without enforcing the continuous parame-
ter requirement, then an anonymous encoding
would be given by the classically correlated state
ρ
AB
=
1
2
(|00ih00| + |11ih11|), with bit flip uni-
taries U
A
= σ
x
A
, and V
B
= σ
x
B
. Note that for the
SA case there is no such distinction.
Writing U
A
(θ) = e
iθH
A
, and V
B
(θ) = e
iθG
B
,
we see that the weak anonymity condition of
Eq. (1) is equivalent to requiring that there exist
local Hermitian operators H
A
, G
B
, for which
[H
A
G
B
, ρ
AB
] = 0. (9)
Similarly the encoding condition of Eq. (2), be-
comes
[H
A
, ρ
AB
] 6= 0. (10)
We can work with these conditions to intrinsically
restrict to continuous parameter encodings.
4.2 Aligned discord
To prove discord is necessary we start with a
CQ state ρ
AB
=
P
a
λ
a
|ψ
a
ihψ
a
| ρ
B|a
, and take
that for some choice of Hermitian operators H
A
and G
B
we have [H
A
, ρ
AB
] = [G
B
, ρ
AB
]. We
project A onto |ψ
c
ihψ
c
| and use the fact that
hψ
c
|[H
A
, |ψ
a
ihψ
a
|]|ψ
c
i = 0 to get λ
c
[G
B
, ρ
B|c
] =
0, c. From this we find [H
A
, ρ
AB
] = 0. Hence
we can only satisfy the anonymity condition if we
violate the encoding condition. Essentially the
same argument works for a QC state, so it is true
for all non-discordant states. This proves that
discord is necessary for WA states.
The fact entanglement is not necessary is
proved with the Werner state [38] ρ
W
=
a|ψ
ihψ
|+
1a
4
1. The WA conditions can always
be satisfied using e
|1ih1|
, except when a = 1,
but the state is not entangled for values of a
1
3
.
To prove that neither entanglement nor dis-
cord are sufficient we use a two-qubit example.
We construct a state that is entangled and dis-
cordant but is not of the appropriate form as
given in Eq. (7). To do this we first define ρ
1
=
(
a|00i+
1 a|11i)(
ah00|+
1 ah11|) and
ρ
2
= (
b|01i+
1 b|10i)(
bh01|+
1 bh10|).
The example is formed by taking ρ =
1
+ (1
m)ρ
2
. Picking the values a = 0.45, b = 0.4, m =
0.35, one can show that the resulting state is en-
tangled (therefore discordant), but cannot satisfy
the WA conditions (see Appendix D for details).
Proof for discord alone can be given by the sim-
pler example
1
2
(|00ih00| + | + +ih+ + |).
4.3 Aligned entanglement
The fact entanglement is necessary for SA states
follows from applying the Peres-Horodecki crite-
rion [39, 40] to states of the form presented in
Eq. (8). The fact that it is not sufficient also fol-
lows, since not all entangled states can be writ-
ten in this form, for example the Werner state
(see Appendix E for a detailed account). Note
the Werner state also reveals that SA states are
not simply the entangled WA states, but a strict
subset of them. It also proves that steerability
and Bell non-locality are not sufficient for aligned
entanglement since for a >
1
2
the state is steer-
able [41], and for a >
1
2
it is Bell non-local [42].
5 Using non-ideal states
5.1 Robustness
The anonymous metrology protocol has robust
anonymity. If Alice and Bob verify that their
Accepted in Quantum 2019-08-19, click title to verify 5
state ρ
AB
is close to a WA/SA state σ
AB
, in
terms of trace distance T (ρ
AB
, σ
AB
) , then
this bounds Charlie’s ability to correctly guess
who applied the unitary. For the WA case we
have T (U
A
ρ
AB
U
A
, V
B
ρ
AB
V
B
) 2, and for the
SA case we have T (U
A
|ψi
ABC
, V
B
|ψi
ABC
)
2
2
, (see Appendix F for derivations). This
means that given a single copy, the probabilities
for Charlie to correctly guess are bounded [43],
as P
W A
1
2
+ , and P
SA
1
2
+
2
.
In general Alice and Bob send multiple copies,
which Charlie could use to improve his guess.
However using the property of the fidelity that
F (ρ
n
1
, ρ
n
2
) = F (ρ
1
, ρ
2
)
n
, and the Fuchs-van de
Graff inequality 1 F T
1 F
2
[44], we
find that T (ρ
n
1
, ρ
n
2
)
p
1 [1 T (ρ
1
, ρ
2
)]
2n
.
Hence robustness for many copies follows.
5.2 General figure of merit
Following similar considerations, we now define a
general figure of merit for any bipartite state ρ
used for anonymous metrology. For given Hamil-
tonians H, G, we can bound the increase in Char-
lie’s guessing probability to δ by limiting the
number of copies sent to be no more than
n
δ
=
log(1 (2δ)
2
)
2 log min
θ
F (ρ
H
(θ), ρ
G
(θ))
, (11)
where ρ
H
(θ) and ρ
G
(θ) are the states Charlie is
trying to discriminate between. The minimisa-
tion over θ ensures anonymity for the whole range
of parameter values.
The usefulness of a state in parameter estima-
tion may be quantified by the quantum Fisher in-
formation (QFI) F [45], which sets a lower limit
on the uncertainty θ with which a parameter θ
can be estimated, via the quantum Cramér-Rao
bound: θ (nF)
1/2
for n measurements. For
a unitary encoding ρ
H
(θ) = U
A
(θ)ρU
A
(θ), the
QFI is parameter-independent and we denote it
as F(ρ; H
A
). Since the QFI can depend on which
party does the encoding, we define the average
¯
F(ρ; H, G) =
1
2
(F(ρ; H
A
) + F(ρ; G
B
)).
We combine
¯
F, and n
δ
to form the figure of
merit n
δ
¯
F. This captures the amount of parame-
ter information that can be transferred to Char-
lie with δ anonymity. We identify the state-
dependent part as the figure of merit
M(ρ; H, G) =
¯
F(ρ; H, G)
log min
θ
F (ρ
H
(θ), ρ
G
(θ))
. (12)
The larger M(ρ; H, G), the better a state ρ is for
anonymous metrology with Hamiltonians H, G.
For a function purely of the state, we must
maximize over all possible choices of Hamilto-
nian. In order for this to be well-defined, we need
to bound the spectra therefore we define
M(ρ) = min
H,G, kHk=kGk=1
M(ρ; H, G), (13)
where k · k is the operator norm.
6 Conclusions
We have shown that quantum mechanics enables
a metrology protocol whereby a continuous pa-
rameter may be determined whilst hiding the lo-
cation where it was encoded. We established the
nature of the quantum correlations responsible
for this phenomenon, according to the level of pri-
vacy required. With a trusted source of states,
discord is needed, while entanglement provides
privacy with an untrusted source. The useful
correlations have a particular symmetry, and are
named aligned discord and entanglement respec-
tively.
We note that the difference between the WA
and SA tasks resembles device-dependent ver-
sus device-independent cryptography, where the
former requires discord and the latter entangle-
ment [46]. However the tasks are clearly distinct,
and importantly we note that they do not pro-
duce the same sets of resourceful states. This is
most readily seen by considering the aligned dis-
cord states that are useful for the WA task. For
these states discord and entanglement are not suf-
ficient, whereas for the device-dependent quan-
tum key distribution task entanglement is suffi-
cient.
The fact the WA case only requires discord
reduces the technological challenge in realizing
protocols, with discord being relatively robust to
noise [32, 47]. The SA states are more practically
challenging, but bring greater operational power.
It is also noteworthy that there is a connection
between aligned discord/entanglement and quan-
tum coherence. This is indicated by the redef-
inition of the WA states in terms of modes of
asymmetry given in Eq. (5). This suggests a po-
tential link between the anonymity resources and
the resource of quantum coherence [4851]. The
anonymity resources should arguably be viewed
Accepted in Quantum 2019-08-19, click title to verify 6
as a hybrid of coherence and correlation. One
could describe it as correlated coherence, though
this appears distinct from the correlated coher-
ence of [3437].
Our results highlight an operational bound-
ary within the hierarchy of quantum correlations,
providing a novel nonclassical task whereby dif-
ferent correlations are at play depending on the
desired level of anonymity.
Acknowledgments
The authors acknowledge discussions with
Thomas Hebdige, and Hyukjoon Kwon. AP is
funded by the EPSRC Centre for Doctoral Train-
ing in Controlled Quantum Dynamics, BY is
funded by an EPSRC Doctoral Prize, and MSK
by the Royal Society and Samsung GRO project.
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Accepted in Quantum 2019-08-19, click title to verify 9
A Delocalised measurement
Alice and Bob want to measure a system that has been put into a spatial superposition, without
decohering it. We consider the unitary that they jointly perform. Alice sets up her lab such that if
the particle comes to her then she performs the controlled unitary U
A
, and Bob does similarly with
V
B
(we leave the θ-dependence implicit). Together this gives the full unitary as
W = |Li
P
hL| U
A
1
B
+ |Ri
P
hR| 1
A
V
B
, (14)
We act with this on the initial state
ρ = (a|Li + b|Ri)
P
(a
hL| + b
hR|) ρ
AB
. (15)
Writing the new state as a matrix in the L, R basis we have
ρ
0
=
"
|a|
2
U
A
ρ
AB
U
A
ab
U
A
ρ
AB
V
B
a
bV
B
ρ
AB
U
A
|b|
2
V
B
ρ
AB
V
B
#
. (16)
We see that if
U
A
ρ
AB
U
A
= V
B
ρ
AB
V
B
= V
B
ρ
AB
U
A
,
(17)
then we can factor out and write our state in the product form
ρ
0
= (a|Li + b|Ri)
P
(a
hL| + b
hR|) U
A
ρ
AB
U
A
. (18)
Factorisation is also possible with any phase factor on the right-hand side of Eq. (17), resulting in a
relative phase appearing in system P . This phase may be absorbed into the definition of U
A
.
B SA condition with twirling superoperators
Starting from the SA condition U
A
(θ)ρ
AB
= V
B
(θ)ρ
AB
, we act from the right with V
B
(θ). We then
multiply by
e
iωθ
2θ
0
, integrate
R
θ
0
θ
0
, and take the limit θ
0
. Defining the split twirling operator
P
ω
AB
(.) = lim
θ
0
→∞
1
2θ
0
Z
θ
0
θ
0
e
θ
U
A
(θ)(.)V
B
(θ), (19)
we arrive at
P
ω
AB
ρ
AB
= P
ω
B
ρ
AB
. (20)
To go the other way we first note that by summing over ω we have
P
ω
P
ω
AB
ρ
AB
= ρ
AB
, even though
P
ω
AB
does not satisfy a completeness relation. This allows us to perform essentially the same argument
as in the WA case. First we define the superoperator W
AB
(.) = U
A
(θ)(.)V
B
(θ), and note that
W
AB
P
ω
AB
= e
θ
P
ω
AB
. We then act on Eq. (20) with W
AB
V
B
, cancel the e
θ
terms and sum
over ω to arrive back at the original SA conditions. However, the split twirling operator is not an
established tool, and it does not appear straightforward to go from these expressions to a form for the
useful states.
C Form for multipartite states
C.1 WA case
To generalise the WA states is straightforward. We consider the case of n subsystems labeled 1, 2, ..., n
and demand that the anonymity condition holds between every pair of subsystems. Now the WA
condition in terms of the twirling superoperators becomes
P
ω
α
ρ
12...n
= P
ω
β
ρ
12...n
, (21)
Accepted in Quantum 2019-08-19, click title to verify 10
for all α, β in the set 1, 2, ..., n. In direct analogy to the bipartite case we have
ρ
(ω
1
,...,ω
n
)
1...n
=
X
i,i
0
,...,k,k
0
,
E
i
0
=E
i
+ω
1
,...,
E
k
0
=E
k
+ω
n
c
ii
0
...kk
0
|ii
1
hi
0
| ... |ki
n
hk
0
|, (22)
and now we can write the WA states as
ρ
12...n
=
X
ω
ρ
(ω,...,ω)
12...n
, (23)
where as before we require non-zero terms for ω 6= 0 so that encoding is possible.
C.2 SA case
For the SA state we generalise the condition to be U
α
ρ
12...n
= U
β
ρ
12...n
, for all α, β in the set 1, 2, ..., n.
We apply the same approach as before, taking matrix elements in the eigenbasis of the unitaries. Work-
ing in the non-degenerate case with the same approach as before we find each of these equations gives us
an expression for the allowed non-zero terms like |iikl...nihjjk
0
l
0
, , , n|, and |ikil...nihjk
0
jl
0
, , , n|. Taking
all such conditions together we see the only non-zero terms allowed are of the form |iii...iihjjj...j|.
And so we have the form
ρ
12...n
=
X
i,j
ρ
ij
|ii...iihjj...j|. (24)
One can then extend to allow for degeneracy in the same way as before by introducing degeneracy
labels.
D Proof discord and entanglement not sufficient for aligned discord
Here we prove by examples that discord and entanglement are not sufficient for a state to be a WA
state. The proof uses the explicit forms of WA states for two qubits, so we first discuss this.
For a two qubit bipartite state ρ
AB
, where the eigenvalues of ρ
A
= Tr
B
(ρ
AB
), are non-degenerate
and similarly for ρ
B
, such that they have unique eigenvectors, then the WA states are those that, when
written in the eigenbasis of their reduced density matrices, have the form
ρ =
α 0 0
0 β 0 0
0 0 γ 0
0 0 δ
or
α 0 0 0
0 β 0
0
γ 0
0 0 0 δ
, (25)
where 6= 0. (These two forms are related by a relabelling of the eigenbasis of ρ
A
.)
We can arrive at this by considering the general form given in Eq. (5), however here we present
a constructive proof from the Hamiltonian WA conditions of Eq. (9) and (10). First note the local
Hamiltonian H
A
must share an eigenbasis with ρ
A
and similarly for G
B
. Using this local eigenbasis
we take matrix elements of Eq. (9) to get (h
i
g
j
h
i
0
+ g
j
0
)hij|ρ
AB
|i
0
j
0
i = 0, where h, g are the
local Hamiltonian eigenvalues. From this equation we see that the diagonal terms hij|ρ
AB
|iji are
unconstrained. To see what other terms are free to be non-zero we need to consider when we can make
(h
i
g
j
h
i
0
+ g
j
0
) = 0.
Since we have 2 qubits we have 4 eigenvalues to set: h
0
, h
1
, g
0
, g
1
. The encoding condition Eq. (10)
enforces h
0
6= h
1
and g
0
6= g
1
, and that ρ
AB
has at least one off-diagonal term, since H
A
1
B
is
diagonal and diagonal matrices commute with each other. We now have two options, choose h
0
= g
0
,
and h
1
= g
1
, or h
0
= g
1
, and h
1
= g
0
. The first case allows the terms h00|ρ
AB
|11i, and h11|ρ
AB
|00i,
to be non-zero and the second case allows h01|ρ
AB
|10i, and h10|ρ
AB
|01i. Putting this all together we
arrive at the forms stated in Eq. (25).
Accepted in Quantum 2019-08-19, click title to verify 11
The facts that entanglement and discord are not sufficient are proved by a two qubit example
that is not of the form given in Eq. (25) but is entangled (and therefore discordant). First we define
ρ
1
= (
a|00i+
1 a|11i)(
ah00|+
1 ah11|) and ρ
2
= (
b|01i+
1 b|10i)(
bh01|+
1 bh10|).
Now we can form ρ =
1
+ (1 m)ρ
2
. Choosing a, b 6=
1
2
ensures we have unique local eigenvectors.
Picking the fairly arbitrary values a = 0.45, b = 0.4, m = 0.35, we find the matrix in its local eigenbasis
is approximately
0.2 0 0 0.2
0 0.3 0.3 0
0 0.3 0.4 0
0.2 0 0 0.1
, (26)
where we are quoting values only to one significant figure. This is not in one of the viable forms given
in Eq. (25). Taking the partial transpose it has a negative eigenvalue. Thus by the Peres-Horodecki
criterion [39, 40] the state is entangled. Since entangled states are always discordant, this example
proves neither discord nor entanglement are sufficient. However if one wanted to show it just for
discord, the simpler example
1
2
(|00ih00| + | + +ih+ + |), suffices. This concludes the proof.
For completeness we present an alternative way to arrive at Eq. (25), using tools from asymmetry
theory [52]. We can write the WA anonymity condition as a symmetry constraint by using the G-
twirling superoperator. We have
G(ρ
AB
) = ρ
AB
, (27)
where we define
G(ρ
AB
) = lim
θ
0
→∞
1
2θ
0
Z
θ
0
θ
0
U
A,θ
V
B
ρ
AB
. (28)
Taking the two-qubit case we write
U
A
(θ) V
B
(θ)
=
"
1 0
0 e
iaθ
#
"
1 0
0 e
ibθ
#
=
1 0 0 0
0 e
ibθ
0 0
0 0 e
iaθ
0
0 0 0 e
i(ab)θ
. (29)
We then find
U
A,θ
V
B
ρ
AB
=
1 e
ibθ
e
iaθ
e
i(ab)θ
e
ibθ
1 e
i(a+b)θ
e
iaθ
e
iaθ
e
i(a+b)θ
1 e
ibθ
e
i(ab)θ
e
iaθ
e
ibθ
1
ρ
AB
, (30)
where denotes the entrywise product in the computational basis. When we integrate to perform the
G-twirling, the two choices of either a = b, or a = b, give the two forms of viable density matrix, as
in Eq. (25).
E Proof entanglement necessary but insufficient for aligned entanglement
Starting from the form given in Eq. (8) we find that entanglement is necessary for aligned entanglement.
To see this we write ρ
AB
=
P
i,j
α
ij
|iii
AB
hjj|, where we have dropped the degeneracy labels. In order
to not violate the encoding condition of Eq. (2) we must have a k and l for which k 6= l, and
α
kl
= e
kl
|α
kl
| 6= 0. We can apply the local unitary e
kl
|ki
A
hk|
, without affecting the entanglement,
and we absorb the phases into the α
kl
, and α
lk
, such that now they are both real and positive. We
then perform the partial transpose to get ρ
T
B
AB
=
P
i,j
α
ij
|iji
AB
hji|, and see that the state |kli |lki,
is an eigenvector, with the negative eigenvalue −|α
kl
|. Hence by the Peres-Horodecki criterion [39, 40],
aligned entangled states are always entangled. Below we also give an alternative proof that works from
the SA conditions without making use of the form from Eq. (8).
Accepted in Quantum 2019-08-19, click title to verify 12
To prove that entanglement is not sufficient we use the Werner state example ρ = a|ψ
ihψ
| +
1a
4
1, and show the only way this can satisfy the strong anonymity condition of Eq. (3) is if it
violates the encoding condition. The anonymity condition becomes aU
A
(θ)|ψ
ihψ
| +
1a
4
U
A
(θ) =
aV
B
(θ)|ψ
ihψ
|+
1a
4
V
B
(θ). We now act from the right with |ψ
i, and we get through to U
A
(θ)|ψ
i =
V
B
(θ)|ψ
i. Substituting this into the original condition we have U
A
(θ)1
B
= I
A
V
B
(θ). This implies
U
A
(θ) = 1
A
, and hence we must violate the encoding condition. This is true for 0 < a < 1, and thus
is true for values of a for which the state is entangled. Hence entanglement is not sufficient for aligned
entanglement.
Now we present a proof that entanglement is necessary for SA states without making explicit use of
the form given in Eq. (8). First we state and prove a useful lemma.
Given a probability distribution {p
j
}, and a set of complex numbers {Z
j
}, where |Z
j
| 1. Then
P
j
p
j
Z
j
= e
, iff Z
j
= e
, j where we only consider j for which p
j
6= 0.
To see this is true, note any complex number on the boundary of the unit disk cannot be expressed
as a convex combination of other complex numbers in the unit disk. This should be convincing but for
completeness we give a more formal proof.
For a set of non-zero complex numbers ζ
j
, we have |
P
j
ζ
j
|
P
j
|ζ
j
|, with equality iff arg(ζ
j
) =
arg(ζ
k
), j, k. This is just a restatement of the polygon inequality (generalisation of the triangle inequal-
ity) for complex numbers. We now write
P
j
p
j
Z
j
=
P
j
ζ
j
. Now we have
P
j
|ζ
j
| =
P
j
p
j
|Z
j
|
P
j
p
j
=
1, where we used |Z
j
| 1, so the equality holds when |Z
j
| = 1j. We now have |
P
j
ζ
j
|
P
j
|ζ
j
| 1,
but |
P
j
ζ
j
| = |e
| = 1, and so the three terms are all equal. The conditions on the inequalitites then
tell us every Z
j
has unit magnitude and the same argument and we see this argument must be ξ giving
us Z
j
= e
. The converse is trivial.
We can now prove entanglement is necessary for SA states. We first obtain from anonymity condi-
tion Tr(U
A
(θ)V
B
(θ)ρ
AB
) = 1. With a separable state ρ
AB
=
P
j
p
j
ρ
(j)
A
ρ
(j)
B
, the condition becomes
P
j
p
j
Tr(U
A
(θ)ρ
(j)
A
)Tr(V
B
(θ)ρ
(j)
B
) = 1. We have |Tr(U
A
(θ)ρ
(j)
A
)| 1, and |Tr(V
B
(θ)ρ
(j)
B
)| 1, via the
above lemma we have Tr(U
A
(θ)ρ
(j)
A
)Tr(V
B
(θ)ρ
(j)
B
) = 1. This can only be true if |Tr(U
A
(θ)ρ
(j)
A
)| = 1,
so Tr(U
A
(θ)ρ
(j)
A
) = e
(j)
. Using the eigendecomposition ρ
(j)
A
=
P
k
q
(j)
k
|ψ
(j)
k
ihψ
(j)
k
|, this becomes
P
k
q
(j)
k
hψ
(j)
k
|U
A
(θ)|ψ
(j)
k
i = e
A
(j)
. Again using the lemma we have hψ
(j)
k
|U
A
(θ)|ψ
(j)
k
i = e
A
(j)
.
This implies U
A
(θ)|ψ
(j)
k
i = e
A
(j)
|ψ
(j)
k
i. We now see that this violates the encoding condition since
U
A
(θ)ρ
AB
U
A
(θ) =
P
j,k
p
j
q
(j)
k
e
A
(j)
|ψ
(j)
k
ihψ
(j)
k
|e
A
(j)
ρ
(j)
B
= ρ
AB
. We have therefore shown that all
separable states cannot satisfy the SA conditions and hence entanglement is necessary.
F Error and robustness
F.1 WA case
Alice and Bob use a selection of the copies they receive to verify that the state they are using ρ
AB
is
close, in terms of trace distance, to a WA state σ
AB
. Formally we say they verify that
T (ρ
AB
, σ
AB
) . (31)
We now show that this leads to a bound on T (U
A
ρ
AB
U
A
, V
B
ρ
AB
V
B
). This quantifies Charlie’s ability
to distinguish whether Alice or Bob applied their unitary, since the maximal probability of correctly
guessing the state is P = (1 + T (U
A
ρ
AB
U
A
, V
A
ρ
AB
V
A
))/2 [43].
Starting from Eq. (31) we use the fact that trace distance is preserved under unitaries to write
T (U
A
ρ
AB
U
A
, σ
0
AB
) , (32)
T (V
B
ρ
AB
V
B
, σ
0
AB
) , (33)
where we have defined σ
0
AB
= U
A
σ
AB
U
A
= V
B
σ
AB
V
B
, using the fact σ
AB
is a WA state. We now use
the triangle inequality T (A, C) T (A, B) + T (B, C), to arrive at
T (U
A
ρ
AB
U
A
, V
B
ρ
AB
V
B
) 2. (34)
Accepted in Quantum 2019-08-19, click title to verify 13
F.2 SA case
Again Alice and Bob use some of their states to establish Eq. (31). However, for the SA
case we need to consider distinguishability for the fully purified states, so we need to bound
T (U
A
(θ)|ψi
ABC
, V
B
(θ)|ψi
ABC
).
First consider the fidelity between the two states U
A
|ψi
ABC
, and V
B
|ψi
ABC
. This fidelity is given
by
F (U
A
|ψi
ABC
, V
B
|ψi
ABC
) = |hψ|
ABC
V
B
U
A
|ψi
ABC
| = |Tr(V
B
U
A
ρ
AB
))|. (35)
Now consider
1 |Tr(V
B
U
A
ρ
AB
)| |1 Tr(V
B
U
A
ρ
AB
)|,
= |Tr(σ
AB
V
B
U
A
ρ
AB
)|,
= |Tr(V
B
U
A
(σ
AB
ρ
AB
))|,
Tr(|V
B
U
A
(σ
AB
ρ
AB
)|),
= Tr(|σ
AB
ρ
AB
|),
= 2T (ρ
AB
, σ
AB
),
where in the third line we used that fact that σ
AB
is an SA state. From this we have |Tr(V
B
U
A
ρ
AB
)|
1 2T (ρ
AB
, σ
AB
) 1 2, and using this with Eq. (35) leads to a bound on the fidelity of
F (U
A
|ψi
ABC
, V
B
|ψi
ABC
) 1 2. (36)
We now change this to an inequality in terms of the trace distance by using the fact that for pure
states T (ψ, φ) =
p
1 F (ψ, φ)
2
, to arrive at
T (U
A
|ψi
ABC
, V
B
|ψi
ABC
) 2
p
2
. (37)
Accepted in Quantum 2019-08-19, click title to verify 14