# Genuine activation of nonlocality: From locally available to locally hidden
information

Somshubhro Bandyopadhyay<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Bose Institute, EN 80, Sector V, Bidhannagar, Kolkata
700091, India  Saronath Halder<EMAIL_ADDRESS>Harish-Chandra
Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India

###### Abstract

Quantum nonlocality has different manifestations that, in general, are
revealed by local measurements of the parts of a composite system. In this
paper, we study nonlocality arising from a set of orthogonal states that
cannot be perfectly distinguished by local operations and classical
communication (LOCC). Such a set is deemed nonlocal, for a joint measurement
on the whole system is necessary for perfect discrimination of the states with
certainty. On the other hand, a set of orthogonal states that can be perfectly
distinguished by LOCC is believed to be devoid of nonlocal properties. Here,
we show that there exist orthogonal sets that are locally distinguishable but
without local redundancy (i.e., they become nonorthogonal on discarding one or
more subsystems) whose nonlocality can be activated by local measurements. In
particular, a state chosen from such a set can be locally converted, with
certainty, into another state, the identity of which can now only be
ascertained by global measurement and no longer by LOCC. In other words, a
locally distinguishable set without local redundancy may be locally converted
into a locally indistinguishable set with certainty. We also suggest an
application, namely, local hiding of information, that allows us to locally
hide locally available information without losing any part. Once hidden, the
information in its entirety can only be retrieved using entanglement.

_Introduction._ Quantum systems consisting of two or more subsystems may have
nonlocal properties that, in general, are revealed by local measurements of
the parts. Perhaps the most well-known manifestation of quantum nonlocality,
_viz_ , Bell nonlocality (Brunner-et-al-2014, ; Bell-1964, ; CHSH-1969, )
arises from entangled states (Entanglement-horodecki, ) through their
violation of Bell-type inequalities (Freedman-Causer-1972, ; Aspect+-1981, ;
Aspect+1982, ; Hensen+2015, ; handsteiner+2017, ; Rosenfeld+, ; BIG-BELL, ).
The latter implies that the predictions of quantum theory cannot be explained
by any local theory. Bell nonlocality is of particular importance in quantum
foundations (Brunner-et-al-2014, ), quantum information (Brunner-et-al-2014, )
and applications thereof. For example, Bell nonlocality tests are routinely
used to certify device-independent quantum protocols (Barrett+2005, ;
Acin+2006, ; Brunner+2008, ; Pironio+2010, ; Colbeck-Renner2012, ).

Bell nonlocality, however, is not the only kind of nonlocality of interest. In
this paper, we focus on the nonlocality that arises in the task of
discrimination of quantum states by LOCC (Peres-Wootters-1991, ; Massar-
Popescu-1995, ; ben99, ; ben99u, ; Walgate-2000, ; Virmani-2001, ; Ghosh-2001,
; grois01, ; walgate-2002, ; divin03, ; HSSH, ; Ghosh-2004, ; rin04, ;
Watrous-2005, ; fan-2005, ; Nathanson-2005, ; Wootters-2006, ; Hayashi-
etal-2006, ; nis06, ; Duan2007, ; Duan-2009, ; feng09, ; Calsamiglia-2010, ;
Bandyo-2011, ; BGK-2011, ; Yu-Duan-2012, ; BN-2013, ; Cosentino-2013, ;
Cosentino-Russo-2014, ; B-IQC-2015, ; Halder+-2019, ). Recall that LOCC
protocols are where local observers perform quantum operations on their
respective subsystems and communicate via classical channels but cannot
exchange quantum information. Now suppose that two or more observers share the
parts of a quantum system prepared in one of several known orthogonal states,
the identity of which they do not know. Their goal is to determine which state
the system is in without error. But because they are separated from each
other, they can only perform measurements realized by LOCC. So the question
here is: Can they perfectly distinguish the orthogonal states by LOCC as is
always possible by a joint measurement on the whole system? The answer,
however, turns out to be no in general. While two orthogonal-pure states can
be perfectly distinguished by LOCC (Walgate-2000, ), entangled orthogonal
bases, such as the Bell basis, cannot be (Ghosh-2001, ; HSSH, ; Ghosh-2004, ).
We say that a set of orthogonal states is _locally distinguishable_ if the
constituent states can be perfectly distinguished with certainty by a LOCC
protocol; otherwise, _locally indistinguishable_.

A locally indistinguishable set of states is nonlocal in the sense that a
suitable joint measurement on the whole system always yields more information
about the state of the system than any sequence of LOCC (ben99, ; grois01, ;
HSSH, ; Halder+-2019, ; Chitambar-2013, ; childs13, ). This new kind of
nonlocality has its own share of counterintuitive results. For example,
entanglement is neither necessary nor sufficient for local
indistinguishability. The former is proved by the existence of orthogonal
product states that are locally indistinguishable (ben99, ; ben99u, ; nis06, ;
Halder+-2019, ; Yang13, ; zhang14, ; wang15, ; zhang15, ; Yang15, ; xu16-1, ;
Xu-16-2, ; zhang16, ; zhang16-1, ; Xu-17, ; Wang-2017-Qinfoprocess, ; Zhang-
Oh-2017, ; zhang17-1, ; halder, ). They give rise to the so-called “quantum
nonlocality without entanglement” (ben99, ) and the recently discovered
stronger version of the same (Halder+-2019, ). That entanglement is not
sufficient follows from the result that any two mutually orthogonal pure
states are locally distinguishable (Walgate-2000, ).

Besides exhibiting a different kind of nonlocality, locally indistinguishable
states also imply the existence of locally hidden information: information
encoded in locally indistinguishable states cannot be fully accessed by the
local observers, so part of it remains hidden. For example, one can encode two
classical bits in four Bell states, but only one bit can be extracted locally
simply because the Bell states cannot be perfectly distinguished by LOCC. The
only way to avail the complete information is by using additional quantum
resources such as entanglement. Quantum cryptography primitives such as data
hiding (Terhal2001, ; DiVincenzo2002, ; Eggeling2002, ; MatthewsWehnerWinter,
) and secret sharing (Markham-Sanders-2008, ) rely on this particular fact.

For the reasons outlined above, it is not surprising why locally
indistinguishable states have received all the attention so far. That includes
finding new classes (ben99u, ; Watrous-2005, ; Duan-2009, ; BGK-2011, ;
Cosentino-2013, ; Cosentino-Russo-2014, ; B-IQC-2015, ), extending the
formalism to density matrices (Calsamiglia-2010, ; Bandyo-2011, ), obtaining
new techniques to prove local indistinguishability (Ghosh-2001, ; HSSH, ;
Ghosh-2004, ; fan-2005, ; Nathanson-2005, ), understanding the presence of
entanglement (Nathanson-2005, ; Hayashi-etal-2006, ) or lack of it (ben99, ;
ben99u, ; walgate-2002, ; childs13, ), and more recently, finding the
entanglement cost of distinguishing locally indistinguishable states
(B-IQC-2015, ). A set of locally distinguishable states, on the other hand, is
generally understood to be neither interesting nor important. The reasons
being, it is neither nonlocal because the constituent states can be perfectly
distinguished by LOCC nor useful in ways a locally indistinguishable set can
be. In this paper, however, we will show that this long-held understanding is,
at best, incomplete, for there exist orthogonal states that are locally
distinguishable but deserve consideration on par with their locally
indistinguishable cousins.

In this paper, we will consider a specific class of LOCC measurements, namely,
orthogonality-preserving-local-measurements (OPLM) (walgate-2002, ;
Halder+-2019, ). The OPLMs are local measurements that keep the post-
measurement states mutually orthogonal, but, on the other hand, they might
eliminate one or more states. Indeed, a LOCC protocol that distinguishes
orthogonal states is a sequence of OPLMs (LOCC-protocol, ). We will, of
course, leave out the trivial OPLMs, those that do not change the states and
consider only the nontrivial ones, those where not all the measurement
operators are proportional to the identity.

Let us now consider the following problem. Suppose Alice and Bob share a state
chosen from a known set $S$ of orthogonal states. They do not know the
identity of the state. They now perform an OPLM $\mathbb{M}$. Then, for a
given outcome $\mu$ of this measurement, they end up with a state that belongs
to a new orthogonal set $S_{\mu}^{\prime}$ whose cardinality
$\left|S_{\mu}^{\prime}\right|\leq\left|S\right|$. Note that the action of an
OPLM does not lead to loss of information (cardinality, ). Now, if $S$ is
locally indistinguishable, then, by definition, so is $S_{\mu}^{\prime}$ for
all $\mu$. But if $S$ is locally distinguishable instead, can
$S_{\mu}^{\prime}$ be locally indistinguishable? We will make this question
more precise in a moment but before we proceed, let us briefly discuss the
issue of local redundancy (Manik, ).

We say that local redundancy exists in an orthogonal set that remains
orthogonal if we discard one or more subsystems. It may be present provided at
least one of the local dimensions is composite. An example would make this
clear. Let $\left\\{\left|\Psi_{i}\right\rangle\right\\}_{i=1}^{4}$ and
$\left\\{\left|\Phi_{i}\right\rangle\right\\}_{i=1}^{4}$ be the two-qubit Bell
basis and the computational basis respectively. Consider the set

$\left\\{\left|\Psi_{i}\right\rangle_{\text{AB}}\otimes\left|\Phi_{i}\right\rangle_{\text{A}^{\prime}\text{B}^{\prime}}:i=1,\dots,4\right\\},$

where $\text{A},\text{A}^{\prime}$ and $\text{B},\text{B}^{\prime}$ are the
qubit-pairs held by Alice and Bob respectively. First, note that the above set
is locally distinguishable because one can locally measure
$\text{A}^{\prime}\text{B}^{\prime}$ in the computational basis and correctly
learn about the identity of the given state. Now observe that if we trace out,
for example, AB or $\text{A}^{\prime}\text{B}^{\prime}$ the resulting states
remain mutually orthogonal. This is the redundancy we are talking about,
which, however, has consequences. In particular, discarding
$\text{A}^{\prime}\text{B}^{\prime}$ makes the set locally indistinguishable
because Alice and Bob would then share one of the four Bell states. But, on
the other hand, discarding AB keeps it locally distinguishable. This situation
arises only because of the local redundancy present in the set. For our
analysis, therefore, we will consider orthogonal sets that do not have this
redundancy.

So now we suppose $S$ is locally distinguishable and does not have local
redundancy. As noted earlier, the action of an OPLM $\mathbb{M}$ achieves the
following set transformation: $S\rightarrow S_{\mu}^{\prime}$ for the outcome
$\mu$, where for every $\mu$ it holds that
$\left|S_{\mu}^{\prime}\right|\leq\left|S\right|$. This brings us to the
question that motivated this work: Do there exist an $S$ and $\mathbb{M}$ such
that for any outcome $\mu$, $S_{\mu}^{\prime}$ is locally indistinguishable?
In other words, does there exist an $S$ such that for any given state chosen
from $S$, Alice and Bob can convert it, with certainty, into a state, the
identity of which can now only be ascertained by a global measurement and not
by LOCC?

So what we require is the following: For any given input $\rho_{i}\in S$ and
any outcome $\mu$ of an OPLM $\mathbb{M}$,
$\rho_{i}\rightarrow\sigma_{i}\left(\mu\right)$ such that the orthogonal set
$S_{\mu}^{\prime}=\left\\{\sigma_{i}\left(\mu\right)\right\\}$ is locally
indistinguishable. Note that the requirement cannot be satisfied if Alice and
Bob perform an OPLM that reveals the identity of the input state (which is
possible because the set is locally distinguishable), or an OPLM whose every
outcome results in a definite output.

An affirmative answer to our question therefore looks improbable and more so
because LOCC operations have inherent limitations (Limitations, ). So it seems
safe to conjecture that $S$ should remain locally distinguishable under OPLMs.
However, we will show that this is not the case, in general.

In particular, we present examples of orthogonal sets from
$\mathbb{C}^{2}\otimes\mathbb{C}^{4}$, $\mathbb{C}^{4}\otimes\mathbb{C}^{4}$,
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$, and
$\mathbb{C}^{5}\otimes\mathbb{C}^{5}\otimes\mathbb{C}^{5}$ with the following
properties:

  1. 1.

Locally distinguishable and without local redundancy. Note that the local
redundancy question does not arise in $\mathbb{C}^{5}\otimes\mathbb{C}^{5}$.

  2. 2.

There exists an OPLM that converts the set with certainty into a locally
indistinguishable orthogonal set such that the cardinality remains unchanged.

We will also show that not all orthogonal sets have the above two properties.
Therefore, those that do not are genuinely local. Our result can be viewed as
activation of nonlocality by local measurements in the scenario of quantum
state discrimination by LOCC. The activation is genuine for the sets do not
suffer from local redundancy.

A simple application of our result is the local hiding of locally available
information. This can be understood as follows. We know the information
encoded in a locally distinguishable set is always locally available. Now
suppose it exhibits activable nonlocality. Then it can be converted into a
locally indistinguishable orthogonal set of the same cardinality by LOCC with
certainty. So the information is no longer completely available locally. This
local hiding of information is irreversible, and to retrieve it in its
entirety, one must now use entanglement.

Let us first recall a couple of fundamental results in local
distinguishability. We will need them frequently in our proofs.

######  Theorem 1.

(Walgate-2000, ) Two multipartite orthogonal pure states are locally
distinguishable.

The next tells us which sets of orthogonal pure states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are locally distinguishable and which
are not.

######  Theorem 2.

(walgate-2002, ) (a) Three orthogonal pure states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are locally distinguishable iff at least
two of those states are product states. (b) Four orthogonal states in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are locally distinguishable iff all of
them are product states.

We begin by considering the simple cases.

######  Proposition 1.

Two multipartite orthogonal pure states $\left|\varphi_{1}\right\rangle$ and
$\left|\varphi_{2}\right\rangle$ remain locally distinguishable under OPLMs.

By Theorem 1, the states are locally distinguishable. Therefore, an outcome of
an OPLM either distinguishes them or converts them into another orthogonal set
which must also contain two pure states in which case Theorem 1 applies.

Next, consider a locally distinguishable set from
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$.

######  Proposition 2.

Let $S$ be an orthogonal set of locally distinguishable states
$\left|\varphi_{1}\right\rangle,\dots,\left|\varphi_{n}\right\rangle$ in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, where $2\leq n\leq 4$. Then, under an
OPLM, the transformed set remains locally distinguishable.

Because of Proposition 1, it suffices to consider only the cases: $n=3,4$. Let
$n=3$. As the states are locally distinguishable, at least two of them must be
product states [Theorem 2(a)]. First, suppose that all are product states.
Then, for any given outcome of an OPLM, the new orthogonal states must also be
product states because LOCC cannot convert product states into entangled
states. Then, according to Theorem 2(a), they must be locally distinguishable.
Now suppose two are product states, and the other is entangled. By a similar
argument, the new orthogonal set consists of either two product states and an
entangled state, or three product states. In both cases, Theorem 2(a) tells us
they are locally distinguishable. Now consider $n=4$. By Theorem 2(b), all of
them must be product states because they are locally distinguishable, and the
previous arguments carry over.

The above results suggest that, perhaps, locally distinguishable states do not
give up their local distinguishability under OPLMs. Although compelling, this
turns out not to be the case in the higher dimensions.

_Orthogonal sets with genuine activable nonlocality._ Note that, the
cardinality of a set of pure states with activable nonlocality is at least
three (follows from Proposition 1). We now discuss the examples.

Let
$\left\\{\left|0\right\rangle,\left|1\right\rangle,\dots,\left|d-1\right\rangle\right\\}$
be an orthonormal basis in $\mathbb{C}^{d}$, where $d\geq 2$. Then, rank-2 and
rank-3 projection operators (projectors) are defined as

$\displaystyle P_{ij}$ $\displaystyle=$
$\displaystyle\left|i\right\rangle\left\langle
i\right|+\left|j\right\rangle\left\langle j\right|,i\neq j,$ $\displaystyle
P_{ijk}$ $\displaystyle=$ $\displaystyle\left|i\right\rangle\left\langle
i\right|+\left|j\right\rangle\left\langle
j\right|+\left|k\right\rangle\left\langle k\right|,i\neq j\neq k,$

respectively, where $i,j,k\in\left\\{0,1,\dots,d-1\right\\}$. For example,
$P_{01}=\left|0\right\rangle\left\langle
0\right|+\left|1\right\rangle\left\langle 1\right|$ and
$P_{012}=\left|0\right\rangle\left\langle
0\right|+\left|1\right\rangle\left\langle
1\right|+\left|2\right\rangle\left\langle 2\right|$.

######  Example 1.

$\mathbb{C}^{2}\otimes\mathbb{C}^{4}$: We assume that Alice holds a qubit and
Bob holds a pair of qubits. We represent the orthonormal basis corresponding
to Bob’s state space as follows:
$\left|00\right\rangle\equiv\left|\bm{0}\right\rangle$,
$\left|01\right\rangle\equiv\left|\bm{1}\right\rangle$,
$\left|10\right\rangle\equiv\left|\bm{2}\right\rangle$, and
$\left|11\right\rangle\equiv\left|\bm{3}\right\rangle$. Consider the following
three orthogonal states (unnormalized):

$\displaystyle\left|\psi_{1}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|0\bm{0}\right\rangle+\left|0\bm{2}\right\rangle+\left|1\bm{1}\right\rangle-\left|1\bm{3}\right\rangle,$
$\displaystyle\left|\psi_{2}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|0\bm{0}\right\rangle-\left|0\bm{2}\right\rangle-\left|1\bm{1}\right\rangle-\left|1\bm{3}\right\rangle,$
(1) $\displaystyle\left|\psi_{3}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|0\bm{1}\right\rangle-\left|1\bm{2}\right\rangle-\left|1\bm{0}\right\rangle-\left|0\bm{3}\right\rangle.$

It is easy to see (and show) that the set does not have local redundancy. In
particular, not all pairs remain orthogonal if we discard any of Bob’s qubits.
To show the states (1) are locally distinguishable we proceed as follows.
First, Alice performs a measurement on her qubit in the
$\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$ basis and tells
Bob the result. Now, each of Alice’s outcome results in a set of three
orthogonal states for Bob to distinguish. If Alice gets “$0$”, Bob
distinguishes the states
$\left|\bm{0}\right\rangle\pm\left|\bm{2}\right\rangle$ and
$\left|\bm{1}\right\rangle-\left|\bm{3}\right\rangle$, and if Alice gets
“$1$”, Bob distinguishes
$\left|\bm{1}\right\rangle\mp\left|\bm{3}\right\rangle$ and
$\left|\bm{0}\right\rangle+\left|\bm{2}\right\rangle$.

We now prove the second property. First, Bob performs a binary measurement
defined by the orthogonal projectors $P_{\bm{01}}$and $P_{\bm{23}}$ and
informs Alice of the outcome. If Bob gets $P_{\bm{01}}$ they are left with one
of $\left|0\bm{0}\right\rangle\pm\left|1\bm{1}\right\rangle$ and
$\left|0\bm{1}\right\rangle-\left|1\bm{0}\right\rangle$. Or, if Bob gets
$P_{\bm{23}}$ they are left with one of
$\left|0\bm{2}\right\rangle\mp\left|1\bm{3}\right\rangle$ and
$\left|1\bm{2}\right\rangle+\left|0\bm{3}\right\rangle$. So, in each case,
they are left with one of three mutually orthogonal pure entangled states that
can be embedded in a $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ space. But,
according to Theorem 2(a), each set is locally indistinguishable. So the
states given by (1) can always be locally converted into another set of three
orthogonal states that cannot be locally distinguished. This completes the
proof.

The following example is built on the previous one, _mutatis mutandis_. But it
is interesting in its own right.

######  Example 2.

$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$: We assume that Alice and Bob each holds
a pair of qubits. The orthonormal basis corresponding to each local state
space is represented as follows:
$\left|00\right\rangle\equiv\left|\bm{0}\right\rangle$,
$\left|01\right\rangle\equiv\left|\bm{1}\right\rangle$,
$\left|10\right\rangle\equiv\left|\bm{2}\right\rangle$, and
$\left|11\right\rangle\equiv\left|\bm{3}\right\rangle$. Now, consider the
following orthogonal states (unnormalized):

$\displaystyle\left|\psi_{1}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{00}\right\rangle+\left|\bm{02}\right\rangle+\left|\bm{31}\right\rangle-\left|\bm{33}\right\rangle,$
$\displaystyle\left|\psi_{2}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{00}\right\rangle-\left|\bm{02}\right\rangle-\left|\bm{31}\right\rangle-\left|\bm{33}\right\rangle,$
$\displaystyle\left|\psi_{3}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{01}\right\rangle-\left|\bm{32}\right\rangle-\left|\bm{30}\right\rangle-\left|\bm{03}\right\rangle.$
(2) $\displaystyle\left|\psi_{4}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{10}\right\rangle+\left|\bm{12}\right\rangle+\left|\bm{21}\right\rangle-\left|\bm{23}\right\rangle,$
$\displaystyle\left|\psi_{5}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{10}\right\rangle-\left|\bm{12}\right\rangle-\left|\bm{21}\right\rangle-\left|\bm{23}\right\rangle,$
$\displaystyle\left|\psi_{6}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\bm{11}\right\rangle-\left|\bm{22}\right\rangle-\left|\bm{20}\right\rangle-\left|\bm{13}\right\rangle.$

It is a tedious but straightforward exercise to show that the above set does
not have local redundancy. We now show that the states are locally
distinguishable. First, Alice performs a binary measurement defined by the
orthogonal projectors $P_{\bm{03}}$ and $P_{\bm{12}}$ on her qubits. If she
gets the first outcome, they end up with one of the first three states
$\left|\psi_{1}\right\rangle,\left|\psi_{2}\right\rangle,\left|\psi_{3}\right\rangle$.
Now, these three states are in one to one correspondence with those given by
(1). This follows by inspection. Hence,
$\left|\psi_{1}\right\rangle,\left|\psi_{2}\right\rangle\left|\psi_{3}\right\rangle$
are locally distinguishable. Now, if she gets the second outcome, they end up
with one of the three states
$\left|\psi_{4}\right\rangle,\left|\psi_{5}\right\rangle,\left|\psi_{6}\right\rangle$.
Once again, these are in one-to-one correspondence with those given by (1).
Hence they are locally distinguishable. So the whole set is locally
distinguishable.

Now we prove the second property. First, Bob performs a binary measurement
defined by the orthogonal projectors $P_{\bm{01}}$and $P_{\bm{23}}$ and
informs Alice of the outcome. If Bob gets $P_{\bm{01}}$ they are left with one
of the six orthogonal states:
$\left|\bm{00}\right\rangle\pm\left|\bm{31}\right\rangle$,
$\left|\bm{01}\right\rangle-\left|\bm{30}\right\rangle$,
$\left|\bm{10}\right\rangle\pm\left|\bm{21}\right\rangle$, and
$\left|\bm{11}\right\rangle-\left|\bm{20}\right\rangle$. Or, if Bob gets
$P_{\bm{23}}$ they are left with one of another six orthogonal states:
$\left|\bm{02}\right\rangle\mp\left|\bm{33}\right\rangle$,
$\left|\bm{32}\right\rangle+\left|\bm{03}\right\rangle$,
$\left|\bm{12}\right\rangle\mp\left|\bm{23}\right\rangle$, and
$\left|\bm{22}\right\rangle+\left|\bm{13}\right\rangle$. Now, in each case,
the corresponding set contains locally indistinguishable triplets. For
example, the first set contains the triplets
$\left\\{\left|\bm{00}\right\rangle\pm\left|\bm{31}\right\rangle,\left|\bm{01}\right\rangle-\left|\bm{30}\right\rangle\right\\}$
and
$\left\\{\left|\bm{10}\right\rangle\pm\left|\bm{21}\right\rangle,\left|\bm{11}\right\rangle-\left|\bm{20}\right\rangle\right\\}$.
Each triplet is locally indistinguishable by Theorem 2(a) as the corresponding
states can be embedded in a suitable $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
space. Hence, the whole set must be locally indistinguishable. A similar
argument holds for the second case. So the states given by (2) can always be
locally converted into another set of six orthogonal states that cannot be
perfectly distinguished with certainty by LOCC. This completes the proof.

By now, the basic idea behind the LOCC protocols proving the desired
properties is clear. In the following example, each local dimensions is prime
and, therefore, there cannot be local redundancy.

######  Example 3.

$\mathbb{C}^{5}\otimes\mathbb{C}^{5}$: Consider the set of three orthogonal
states (unnormalized):

$\displaystyle\left|\phi_{1}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|00\right\rangle+\left|11\right\rangle+\left|22\right\rangle+\left|33\right\rangle+\left|44\right\rangle,$
$\displaystyle\left|\phi_{2}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|00\right\rangle-\left|11\right\rangle-\left|22\right\rangle-\omega\left|33\right\rangle-\omega^{2}\left|44\right\rangle,$
(3) $\displaystyle\left|\phi_{3}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|01\right\rangle+\left|23\right\rangle,$

where $\omega$, $\omega^{2}$ are cubic roots of unity. First we show the
states are locally distinguishable. Alice performs a binary measurement
defined by the orthogonal projectors $P_{02}$ and $P_{134}$ on her system. If
she gets the first outcome, they are left to distinguish
$\left|00\right\rangle\pm\left|22\right\rangle,\left|01\right\rangle+\left|23\right\rangle$.
Now, Bob performs a binary measurement defined by $P_{02}$ and $P_{134}$. If
he gets the first outcome they are left to distinguish the orthogonal pair
$\left|00\right\rangle\pm\left|22\right\rangle$ that we know can be locally
distinguished (Theorem (1)). If he gets the second outcome they are left only
with $\left|01\right\rangle+\left|23\right\rangle$, so the task is completed.
Now, if Alice gets the second outcome $P_{134}$ in the first round, they are
left to distinguish a pair of orthogonal states
$\left|11\right\rangle+\left|33\right\rangle+\left|44\right\rangle$ and
$\left|11\right\rangle+\omega\left|33\right\rangle+\omega^{2}\left|44\right\rangle$
and Theorem (1) applies. So we have shown that the states (3) are locally
distinguishable.

Now we prove the second property. Alice performs a binary measurement defined
by the orthogonal projectors $P_{01}$ and $P_{234}$. There are only two
possible outcomes. If she gets $P_{01}$ they are left with one the three
orthogonal states
$\left|00\right\rangle\pm\left|11\right\rangle,\left|01\right\rangle$. From
Theorem (2)(a) it follows that they are locally indistinguishable. On the
other hand, if she gets $P_{234}$ they are left with one of the three
orthogonal states
$\left|22\right\rangle+\left|33\right\rangle+\left|44\right\rangle$,
$\left|22\right\rangle+\omega\left|33\right\rangle+\omega^{2}\left|44\right\rangle$,
and $\left|23\right\rangle$. But we know such a set cannot be perfectly
distinguished with certainty by LOCC (HSSH, ). This completes the proof.

We now give an example from a multipartite system. Recall that local
(in)distinguishability for a $k$-partite system, where $k\geq 3$, is defined
where all the $k$ parties are separated from each other. However, if a given
set is locally indistinguishable in some $k^{\prime}$-partite configuration,
where $k^{\prime}\geq 2$, it must be locally indistinguishable. But note that
the converse is false. One can find orthogonal states in a tripartite system
ABC that are locally distinguishable across all the bipartitions A|BC, B|CA,
and C|AB but not in A|B|C (ben99u, ).

######  Example 4.

$\mathbb{C}^{5}\otimes\mathbb{C}^{5}\otimes\mathbb{C}^{5}$: Consider the
orthogonal states:

$\displaystyle\left|\phi_{1}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|000\right\rangle+\left|111\right\rangle+\left|222\right\rangle+\left|333\right\rangle+\left|444\right\rangle,$
$\displaystyle\left|\phi_{2}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|000\right\rangle-\left|111\right\rangle-\left|222\right\rangle-\omega\left|333\right\rangle-\omega^{2}\left|444\right\rangle,$
(4) $\displaystyle\left|\phi_{3}\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|011\right\rangle+\left|233\right\rangle,$

The above states are a three-party generalization of those in Example 3. The
proof is similar.

_Discussions._ The examples that we discussed provide an idea of constructions
in other state spaces. The first and the second example may be suitably
generalized in $\mathbb{C}^{2n}\otimes\mathbb{C}^{4n}$ and
$\mathbb{C}^{4n}\otimes\mathbb{C}^{4n}$ respectively for $n\geq 2$. However,
proper care should always be taken to ensure there is no local redundancy
involved when the dimensions of the local subsystems are composite. The third
example could help us to find examples in other spaces where the local
dimensions are prime. However, we do not know if they could be found in state
spaces such as $\mathbb{C}^{2}\otimes\mathbb{C}^{3}$ and
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}$. We also showed that sets with genuine
activable nonlocality do not exist in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$.
Whether there are other state spaces also where they do not exist is an
interesting question.

Is there an upper bound on the size of sets with genuine activable nonlocality
in a given state-space? We do not have any particularly helpful intuition
here. But we believe other methods for obtaining such sets could help. So how
big they could be in a given state-space remains an open question. Finally, is
it possible to have activable nonlocality without entanglement? In particular,
does there exist an orthogonal product set with genuine activable nonlocality?
We suspect not. In fact, it would be surprising if it does.

_Conclusions._ In this paper, we showed that quantum nonlocality can be
genuinely activated in the scenario of quantum state discrimination by LOCC.
In particular, we considered orthogonal sets of pure states that are locally
distinguishable and without local redundancy. We gave several examples where
such a set can be locally converted, with certainty, into another orthogonal
set, which is locally indistinguishable. That is, a state chosen from an
activable set can be locally converted, with certainty, into another state,
the identity of which can be determined by a global measurement but not by
LOCC. We also discussed a potential application, namely, local hiding of the
entire locally available information. The information, once hidden, is no
longer locally available in full and to access it, one must use entanglement.
We also discussed interesting open questions.

The notion of activation of quantum nonlocality is known to hold in the
context of Bell-nonlocality (Popescu-1995, ; Gisin-1996, ; Peres-1996, ;
Hirsch+2013, ; Camlet-2017, ). Our result shows that activation of nonlocality
also appears meaningfully in local quantum state discrimination. So whether
the activation phenomenon can also be observed in other manifestations of
nonlocality (for example, Ref. (other-nonlocality, )) is an intriguing
question.

###### Acknowledgements.

We would like to thank Manik Banik, IISER Thiruvananthapuram, for his
observations that led us to address the issue of local redundancy.

## References

  * (1) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014); Erratum, Rev. Mod. Phys. 86, 839 (2014).
  * (2) J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964); On the Problem of Hidden Variables in Quantum Mechanics, Rev. Mod. Phys. 38, 447 (1966).
  * (3) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  * (4) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * (5) S.J. Freedman and J.F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972).
  * (6) A. Aspect, P. Grangier, and G. Roger, Experimental Tests of Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. 47, 460 (1981).
  * (7) A. Aspect, J. Dalibard, and G. Roger, Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982).
  * (8) B. Hensen _et al.,_ Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).
  * (9) J. Handsteiner _et al_., Cosmic Bell Test: Measurement Settings from Milky Way Stars, Phys. Rev. Lett. 118, 060401 (2017).
  * (10) W. Rosenfeld _et al_., Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes, Phys. Rev. Lett. 119, 010402 (2017).
  * (11) BIG Bell Test Collaboration, Challenging local realism with human choices, Nature 557, 212 (2018).
  * (12) J. Barrett, L. Hardy, and A. Kent, No Signaling and Quantum Key Distribution, Phys. Rev. Lett. 95, 010503 (2005).
  * (13) A. Acín, N. Gisin, and L. Masanes, From Bell’s Theorem to Secure Quantum Key Distribution, Phys. Rev. Lett. 97, 120405 (2006).
  * (14) N. Brunner, S. Pironio, A. Acin, N. Gisin, A. A. Methot, and V. Scarani, Testing the Dimension of Hilbert Spaces, Phys. Rev. Lett. 100, 210503 (2008).
  * (15) S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random Numbers Certified by Bell’s Theorem, Nature 464, 1021 (2010).
  * (16) R. Colbeck and R. Renner, Free randomness can be amplified, Nat. Phys. 8, 450 (2012).
  * (17) A. Peres and W. K. Wootters, Optimal Detection of Quantum Information, Phys. Rev. Lett. 66, 1119 (1991).
  * (18) S. Massar and S. Popescu, Optimal extraction of information from finite quantum ensembles, Phys. Rev. Lett. 74, 1259 (1995).
  * (19) C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59, 1070 (1999).
  * (20) C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases and Bound Entanglement, Phys. Rev. Lett. 82, 5385 (1999).
  * (21) J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local distinguishability of multipartite orthogonal quantum states, Phys. Rev. Lett. 85, 4972 (2000).
  * (22) S. Virmani, M. F. Sacchi, M. B. Plenio, D. Markham, Optimal local discrimination of two multipartite pure states, Phys. Lett. A. 288, 62 (2001).
  * (23) S. Ghosh, G. Kar, A. Roy, A. Sen (De), and U. Sen, Distinguishability of Bell states, Phys. Rev. Lett. 87, 277902 (2001).
  * (24) B. Groisman and L. Vaidman, Nonlocal variables with product state eigenstates,  J. Phys. A: Math. Gen. 34, 6881 (2001).
  * (25) J. Walgate and L. Hardy, Nonlocality, asymmetry, and distinguishing bipartite states, Phys. Rev. Lett. 89, 147901 (2002).
  * (26) M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local indistinguishability: more nonlocality with less entanglement, Phys. Rev. Lett. 90, 047902 (2003).
  * (27) D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases, uncompletable product bases and bound entanglement, Commun. Math. Phys. 238, 379 (2003).
  * (28) S. De Rinaldis, Distinguishability of complete and unextendible product bases, Phys. Rev. A 70, 022309 (2004).
  * (29) S. Ghosh, G. Kar, A. Roy, and D. Sarkar, Distinguishability of maximally entangled states, Phys. Rev. A 70, 022304 (2004).
  * (30) H. Fan, Distinguishability and indistinguishability by local operations and classical communication, Phys. Rev. Lett. 92, 177905 (2004).
  * (31) M. Nathanson, Distinguishing bipartite orthogonal states by LOCC: best and worst cases, Journal of Mathematical Physics 46, 062103 (2005).
  * (32) J. Watrous, Bipartite subspaces having no bases distinguishable by local operations and classical communication, Phys. Rev. Lett. 95, 080505 (2005).
  * (33) M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Bounds on entangled orthogonal state discrimination using local operations and classical communication, Phys. Rev. Lett. 96, 040501 (2006).
  * (34) W. K. Wootters, Distinguishing unentangled states with an unentangled measurement, Int. J. Quantum Inf. 4, 219 (2006).
  * (35) J. Niset and N. J. Cerf, Multipartite nonlocality without entanglement in many dimensions, Phys. Rev. A 74, 052103 (2006).
  * (36) R. Y. Duan, Y. Feng, Z. F. Ji, and M. S. Ying, Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication, Phys. Rev. Lett. 98, 230502 (2007).
  * (37) Y. Feng and Y.-Y. Shi, Characterizing locally indistinguishable orthogonal product states, IEEE Trans. Inf. Theory 55, 2799 (2009).
  * (38) R. Y. Duan, Y. Feng, Y. Xin, and M. S. Ying, Distinguishability of quantum states by separable operations, IEEE Trans. Inform. Theory 55, 1320 (2009).
  * (39) J. Calsamiglia, J. I. de Vicente, R. Munoz-Tapia, E. Bagan, Local discrimination of mixed states, Phys. Rev. Lett. 105, 080504 (2010).
  * (40) S. Bandyopadhyay, S. Ghosh and G. Kar, LOCC distinguishability of unilaterally transformable quantum states, New J. Phys. 13, 123013 (2011).
  * (41) S. Bandyopadhyay, More nonlocality with less purity, Phys. Rev. Lett. 106, 210402 (2011).
  * (42) N. Yu, R. Duan, and M. Ying, Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States, Phys. Rev. Lett. 109, 020506 (2012).
  * (43) A. Cosentino, Positive-partial-transpose-indistinguishable states via semidefinite programming, Phys. Rev. A 87, 012321 (2013).
  * (44) S. Bandyopadhyay, M. Nathanson, Tight bounds on the distinguishability of quantum states under separable measurements, Phys. Rev. A , 88 052313 (2013).
  * (45) A. Cosentino and V. Russo, Small sets of locally indistinguishable orthogonal maximally entangled states, Quantum Information and Computation 14, 1098 (2014).
  * (46) S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu, Limitations on separable measurements by convex optimization, IEEE Transactions on Information Theory, 61, 3593 (2015)..
  * (47) S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Strong quantum nonlocality without entanglement, Phys. Rev. Lett. 122, 040403 (2019).
  * (48) E. Chitambar and Min-Hsiu Hsieh, Revisiting the optimal detection of quantum information, Phys. Rev. A 88, 020302(R) (2013).
  * (49) A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, A framework for bounding nonlocality of state discrimination, Commun. Math. Phys. 323, 1121 (2013).
  * (50) Y.-H. Yang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Local distinguishability of orthogonal quantum states in a $2\otimes 2\otimes 2$ system, Phys. Rev. A 88, 024301 (2013).
  * (51) Z.-C. Zhang, F. Gao, G.-J. Tian, T.-Q. Cao, and Q.-Y. Wen, Nonlocality of orthogonal product basis quantum states, Phys. Rev. A 90, 022313 (2014).
  * (52) Z.-C. Zhang, F. Gao, S.-J. Qin, Y.-H. Yang, and Q.-Y. Wen, Nonlocality of orthogonal product states, Phys. Rev. A 92, 012332 (2015).
  * (53) Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, Nonlocality of orthogonal product-basis quantum states, Phys. Rev. A 92, 032313 (2015).
  * (54) Y.-H. Yang, F. Gao, G.-B. Xu, H.-J. Zuo, Z.-C. Zhang, and Q.-Y. Wen, Characterizing unextendible product bases in qutrit-ququad system, Sci. Rep. 5, 11963 (2015).
  * (55) G.-B. Xu, Y.-H. Yang, Q.-Y. Wen, S.-J. Qin, and F. Gao, Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system, Sci. Rep. 6, 31048 (2016).
  * (56) G.-B. Xu, Q.-Y. Wen, S.-J. Qin, Y.-H. Yang, and F. Gao, Quantum nonlocality of multipartite orthogonal product states, Phys. Rev. A 93, 032341 (2016).
  * (57) Z.-C. Zhang, F. Gao, Y. Cao, S.-J. Qin, and Q.-Y. Wen, Local indistinguishability of orthogonal product states, Phys. Rev. A 93, 012314 (2016).
  * (58) X. Zhang, X. Tan, J. Weng, and Y. Li, LOCC indistinguishable orthogonal product quantum states, Sci. Rep. 6, 28864 (2016).
  * (59) G.-B. Xu, Q.-Y. Wen, F. Gao, S.-J. Qin, and H.-J. Zuo, Local indistinguishability of multipartite orthogonal product bases, Quantum Inf. Processing 16, 276 (2017).
  * (60) Y.-L. Wang, M.-S. Li, Z.-J. Zheng, and S.-M. Fei, The local indistinguishability of multipartite product states, Quantum Inf. Processing 16, 5 (2017).
  * (61) Z.-C. Zhang, K.-J. Zhang, F. Gao, Q.-Y. Wen, and C. H. Oh, Construction of nonlocal multipartite quantum states, Phys. Rev. A 95, 052344 (2017).
  * (62) X. Zhang, J. Weng, X. Tan, and W. Luo, Indistinguishability of pure orthogonal product states by LOCC, Quantu Inf. Processing 16, 168 (2017).
  * (63) S. Halder, Several nonlocal sets of multipartite pure orthogonal product states, Phys. Rev. A 98, 022303 (2018).
  * (64) B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding bits in Bell states, Phys. Rev. Lett. 86, 5807 (2001).
  * (65) D. P. DiVincenzo, D. W. Leung, and B. M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48, 580 (2002).
  * (66) T. Eggeling, and R. F. Werner, Hiding classical data in multipartite quantum states, Phys. Rev. Lett. 89, 097905 (2002).
  * (67) W. Matthews, S. Wehner, A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding, Comm. Math. Phys. 291, Number 3 (2009).
  * (68) D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Rev. A 78, 042309 (2008).
  * (69) This was pointed out by Manik Banik, IISER Thiruvananthapuram.
  * (70) The cardinality of an orthogonal set of states is related to the amount of classical information. It is obvious that if $\left|S_{\mu}^{\prime}\right|=\left|S\right|$ there is no loss of information. On the other hand, if $\left|S_{\mu}^{\prime}\right|<\left|S\right|$ then it implies that we have only extracted part of the classical information encoded in the parent set $S$.
  * (71) In a particular round, the local measurements either identify the state correctly at which point the protocol ends or failing to do so, preserve orthogonality of the post-measurement states, possibly eliminating one or more from the contention, so that the protocol can continue.
  * (72) For example, one could never convert a separable state into an entangled state, or increase entanglement on an average, or even reliably distinguish orthogonal product states prepared separately.
  * (73) S. Popescu, Bell’s Inequalities and Density Matrices: Revealing “Hidden” Nonlocality, Phys. Rev. Lett. 74, 2619 (1995).
  * (74) N. Gisin, Hidden quantum nonlocality revealed by local filters, Phys. Lett. A 210, 151 (1996).
  * (75) A. Peres, Collective tests for quantum nonlocality, Phys. Rev. A 54, 2685 (1996).
  * (76) Flavien Hirsch, Marco Tu´lio Quintino, Joseph Bowles, and Nicolas Brunner, Genuine hidden quantum nonlocality, Phys. Rev. Lett. 111, 160402 (2013).
  * (77) S. Camalet, Measure-independent anomaly of nonlocality,Phys. Rev. A 96, 052332 (2017).
  * (78) For example, the recent PBR theorem (Pusey+2012, ) and other results (Bandyopadhyay+2014, ) have shown that there exist product states for which state elimination with certainty becomes possible only by entangled measurements on the joint system. The nonlocality here is in the sense that entangled measurements are necessary for state elimination with certainty even though the underlying states are product states.
  * (79) M. F. Pusey, J. Barrett and T. Rudolph, On the reality of the quantum state, Nature Physics 8, 475 (2012).
  * (80) S. Bandyopadhyay, R. Jain, J. Oppenheim, and C. Perry, Conclusive exclusion of quantum states, Phys. Rev. A 89, 022336 (2014).