A paradigm for entanglement theory based on quantum communication

0.2 MB·English

by  Jonathan Oppenheim

A paradigm for entanglement theory based on quantum communication
Jonathan Oppenheim1
1Department of Applied Mathematics and Theoretical Physics, University of Cambridge U.K.
Here it is shown that the squashed entanglement has an operational meaning – it is the fastest
rateatwhichaquantumstatecanbesentbetweentwopartieswhosharearbitraryside-information.
Likewise, the entanglement of formation and entanglement cost is shown to be the fastest rate at
which a quantum state can be sent when the parties have access to side-information which is max-
imally correlated. A further restriction on the type of side-information implies that the rate of
state transmission is given by the quantum mutual information. This suggests a new paradigm for
understanding entanglement and other correlations. Different types of side-information correspond
todifferenttypesofcorrelationswiththesquashedentanglementbeingoneextreme. Theparadigm
also allows one to classify states not only in terms of how much quantum communication is needed
to transfer half of it, but also in terms of how much entanglement is needed. Furthermore, there is
8 adualparadigm: ifonedistributestheside-informationasmaliciouslyaspossiblesoastomakethe
0 sending of the state as difficult as possible, one finds maximum rates which give interpretations to
0 knownquantities(suchastheentanglementofassistance),aswellasnewones. Theinfamousaddi-
2 tivity questions can also be recast and receive an operational interpretation in terms of maximally
correlated states.
n
a
J There are two main paradigms for understanding en- |ψ(cid:105) a pure state, I(A : B|E˜) = S(AE˜)+S(BE˜)−
ABE
3 tanglement between two parts of a system. We say that S(ABE˜)−S(E˜) the conditional mutual information of

a system is in an entangled state if measurements on it a tripartite state ρ and S(A) the entropy of the re-
] ABE˜
h cannot be simulated by a local hidden variable model. ducedstateTr ρ . Here,wefindthatthesquashed
BE ABE˜
p Another (and perhaps inequivalent way) to understand entanglement is an extreme case: it is the rate at which
-
t entanglement is to say that we do not know what entan- a share of the state can be sent when the sender and
n
glementis,butwedoknowthatitisatypeofcorrelation receiver are given the best possible side information.
a
u which cannot increase under local operations and classi- The squashed entanglement is a remarkable entangle-
q cal communication (LOCC) [1]. Entanglement measures mentmeasurebecauseofit’sadditivity,eleganceandsim-
[ are thus monotones (they must go down under LOCC). ple expression. We now see that it has a very intuitive

Here, we will show that two of the most prominent and operational meaning as well.
1
v entanglement measure and one more general correlation
This inspires the introduction of a third paradigm in
8 measure have an operational meaning. They give the
which to understand entanglement and other correla-
5 rate at which one share of a state can be sent to a re-
4 tions. Wecaningeneralconsidertherateatwhichstates
ceiver – broadly speaking the more entangled a state is,
0 can be sent to a receiver given different types of restric-
the harder it is to send it. Here, the rate is how much
1. tions on the side-information states. This gives a rela-
quantum communication is needed to send each state,
0 tionshipbetween setsofstatesandcorrelation measures.
and which correlation measure corresponds to this rate
8 Furthermore,thereisadualparadigm. Insteadofconsid-
0 isdeterminedbywhatresourceisgiventothesenderand eringthebestrateatwhichinformationcanbesent,one
: receiver. In particular, the resource that is given to the
v canfindtheworstpossiblerate. I.e. theside-information
sender and receiver is side-information – i.e. we give the
i is distributed as maliciously as possible in order to make
X sender and receiver quantum states which contain infor-
theinformationasunhelpfulaspossibletothesenderand
r mation about the state which is being sent. Different
a receiver. This leads us to discover a set of operational
restrictionsonthestateswhichcontainthesideinforma-
quantities, some of which had already been known to be
tion give different rates at which quantum states can be
of significance, and some of which were previously un-
sent, and these rates turn out to correspond to different
known. We also are able to recast well known additivity
correlation measures.
questions [7] concerning the entanglement of formation
Twoofthecorrelationmeasures(E theentanglement
c and other quantities into an operational question about
cost [2, 3] and I(A : B) the quantum mutual informa-
the utility of certain types of side information.
tion) had other operational meanings, but the squashed
entanglement[4,5]E hadthusfarbeenapurelyformal Let us imagine that Alice and Bob share many copies
sq
quantity as the quantum analog of the intrinsic mutual ofaquantumsysteminstateρAB,andweaskhowmany
information [6]: qubitsarerequiredforAlicetosendhershareofthestate
to a third party, Charlie. Everything we discuss will be
Esq ≡inΛf 12I(A:B|Λ(E)) (1) symmetric under exchanging the roles of Alice and Bob.
Here, Bob will be completely passive, his only role being
with Λ a completely positive trace preserving map, to hold his part of the state ρ . However, for Alice to
B
2
successfullysendhershareofthestate,onerequiresthat
her protocol work for all possible pure states she may
hold. Here, Alice knows the statistics of the state (i.e.
she knows ρ ), but not which particular pure state she
AB
holds.
We are interested in maximising the rate at which the
state can be sent, and so we allow Alice and Charlie to
have as much help as possible in the form of ancillas.
I.e we give them as much additional information as is
allowed by the laws of quantum mechanics. In the clas-
sical world, this would be a lot, since there is nothing
to prevent Charlie from knowing exactly what Alice has FIG. 1: Redistribution: ρ is sent to C retaining it’s corre-
A
andthereforenothingwouldneedtobesentfromAliceto lations with ρA(cid:48)BC. By optimising over how we split ρA(cid:48)C
Charlie. However,quantumly,theamountofinformation weisolatetheentanglementsharedbetweenAandB (dashed
Charlie can have is limited by the fact that Alice’s state line)
is entangled with Bob’s. In particular, if we imagine the
purification of ρ , i.e. a pure state |ψ(cid:105) such that
AB ABE
The minimum amount of quantum communication re-
Tr |ψ (cid:105)(cid:104) ψ| = ρ , then the most information
E ABE ABE AB
quired is given by the optimal cost pair
that Charlie could possible have would be to possess the
share ρ . This might not be much information – for ex-
E Q = 1I(A:B|C) (2)
ample,ifAliceandBobshareapurestate,thanCharlie’s A(cid:48)C 2
E = 1I(A:A(cid:48))− 1I(A:C). (3)
state must be completely uncorrelated from Alice’s. 2 2
Now if Charlie has ρ then the amount of quantum
E with E being the amount of entanglement that is re-
communication that is required when no classical com-
quired. A simple explanation for this rate in terms of
municationisallowedisgivenbythemutualinformation
state merging is given in [11].
I(A:B)/2 [8], with the protocol for doing this given by
Now observe that because ρ is a splitting of ρ into
a coherent version of merging [9]. Note that the usual C E
two parts, ρ and ρ , we can treat the system A(cid:48) as
convention is for Bob to be a reference system R and for C A(cid:48)
an environment for a channel Λ (i.e. completely trace
thereceivertobecalledBob, butherethisconflictswith
preserving positive map) which acts on ρ and produces
the convention of quantifying the entanglement between E
the output ρ . Now, because we give Alice and Char-
two parties called Alice and Bob. C
lie all side information they require, this includes shared
NowtheratemightnotbemaximisedbygivingCharlie
entanglement,andthereforetherateofEquation(2)can
all the side information – it may be beneficial for the
be obtained. If we now minimise the amount of quan-
sender Alice to have some as well. One may therefor
tumcommunicationrequiredasafunctionofallpossible
want to split ρ into two shares, some of which is given
E splittingsoftheancillas, wehavethatthisminimalcom-
to the receiver Charlie (call this ρ ), and some of which
C munication Q ≡infQ is given by
isgiven toAlice(callthis ρ ). Whatisthebestrate we all A(cid:48)C
A(cid:48)
can achieve if we optimally give side information to the Q = inf 1I(A:B|C)
sender and receiver? all ρA(cid:48)C 2
It is simple to show that the answer is the squashed = inf 1I(A:B|Λ(E))
entanglement. Given a noiseless quantum channel and Λ 2
= E (A:B) . (4)
arbitrary shared entanglement between a sender and re- sq
ceiver let Q be the rate of quantum communication
A(cid:48)C Here, the last line follows from the definition of the
that is required to send ρ to a receiver who holds state
A squashed entanglement.
ρ with ρ being held by the sender. Then
C A(cid:48) Thisgivesanoperationalmeaningtothesquasheden-
Theorem 1 Given a state ρ and arbitrary side- tanglement. The amount of pure state entanglement, E,
AB
information ρ , infQ = E (A : B) where the in- thatisgainedgiventheancillasplittingwhichminimises
A(cid:48)C A(cid:48)C sq
fimum is taken over states ρA(cid:48)C and Esq is the squashed QA(cid:48)C givesusasecondparameterE whichwouldappear
entanglement. to characterise the state. Using the purity of the total
state, and the fact that S(F) = S(G) for a total pure
Theproblemoffindingtherateforstatemergingwhen
state ψ , we can convert Equation (3) into
FG
both the sender and receiver have side information was
found in [10] and was called state redistribution. Namely E =S(A|C)−Q (5)
Alice only wants to send part of her state to Bob who
shares part of her state. The situation is depicted in with S(A|C) the conditional entropy S(AC)−S(C) and
figure 1. E being the entanglement that is consumed during the
3
protocol (negative values indicate that entanglement is WeconsiderthequantityQO tobetheminimumqubit
S
gained). Not surprisingly, the total amount of shared rate for sending ρ to a receiver, using side information
A
states i.e. QA(cid:48)C +E gives the amount of entanglement at the encoder and decoder ρA(cid:48)C chosen from the set
required to merge [9] ρA with the ancilla ρC. Even a S, and the allowed class of operations being O. In this
state with zero squashed entanglement (thus Q = 0) paper we will mostly consider O to be local operations
all
may consume S(A|C) bits of pure state entanglement to and the sending of qubits which is quantified, so we will
share (or may result in a gain of ebits). Minimising this thusdropthesuperscript. However,onemightalsowant
quantity as well toconsiderOwhichincludefreeentanglementorclassical
communication.
E =infS(A|C)−Q (6)
all all
C Definition 2
where the infimum is taken over all C which minimise Q ≡ inf Q (A:B) (7)
S A(cid:48)C
Qall givesawaytodifferentiateevenbetweenstateswith ρA(cid:48)C∈S
the same squashed entanglement. We will see that this However, in general one can consider any class of opera-
quantity is zero for separable states, although there may tions, and one can even count some other resource (e.g.
be some other states for which the squashed entangle- privatecommunication)ratherthansentqubits. Wehave
ment is zero, but for which Eall is non-zero. We will sofarconsideredthecasewhenthesetS isunlimitedand
also find that this quantity is zero for any state where contains all possible states, including possible extensions
the squashed entanglement is equal to the entanglement of ρ .
AB
cost. Let us now consider the case where the set S is re-
It is worthwhile trying to understand why entangle- strictedtothesetofmaximallycorrelatedstates(MCS).
ment between Alice and Bob should effect the rate at I.e. states of the form
which Alice’s share can be sent to a receiver. We noted
(cid:88)
previously that monogamy of entanglement plays a cru- ρA(cid:48)C = σij|ii(cid:105)(cid:104)jj| . (8)
cial role here. If Alice’s system is entangled with Bob, ij
then it will be more difficult to find side-information on
With such a restriction on the side information ρ , we
A(cid:48)C
the ancillas A(cid:48) and C which can be correlated with ρ
A find that the maximum rate for sending ρ is given by
A
in such a way as to make the task of sending ρ easier.
A the entanglement cost E .
c
Thereisanotherreasonwhytheprocedureabovequanti-
fiescorrelation. SendingρAisdeemedsuccessfulifAlice’s Theorem 3 Given a state ρAB, QS = Ec(A : B) when
shareofthestateisatthereceiver,andthefidelityofthe S is the set of MCS.
totalpurestate|ψ(cid:105) iskeptduringtheprotocol. This
ABE E is defined as the amount of pure state entanglement
c
is equivalent to the protocol working for any pure state
whichisrequiredtocreateacopyofthestateinthelimit
decomposition of ρ (since measurements on ρ will
AB E that many copies of the state are created, and it is given
induce a pure state ensemble on ρ . To preserve the
AB by
overall pure state, Alice must thus send her state while
keeping it correlated with Bob’s. The more correlation (cid:88) S(ρi )
E = lim inf p A (9)
there is with Bob, the more she has to send. We can c n→∞ i n
i
think of Bob as being the referee who checks to make
sure that Alice has actually sent her state to Charlie (he where the infimum is taken over decompositions
would check this by bringing his share together with the (cid:88)
ρ⊗n = p |ψi (cid:105)(cid:104)ψi | (10)
other shares and measuring the total state). The more AB i AB AB
i
correlation Bob has with Alice, the more she must send.
Given this intuition, we can see if it leads to a more (the entanglement of formation is defined as the entan-
generalparadigm–quantifyingastate’scorrelationwith glement cost, but for single copies of ρAB, i.e. n=1).
aanothersystembyhowmuchhastobesenttotransfer To prove Theorem 3, we note that for any decomposi-
the state and maintain the correlation. If we allow ar- tion (not just the one which minimises the entanglement
bitrary side-information at the sender and receiver, then of formation), we can write the purification as
the best rate of sending is given by the squashed entan- (cid:88)√
|ψ(cid:105)⊗n = p |ψi(cid:105) |ii(cid:105) . (11)
glement Esq. We can generalise this to consider more ABA(cid:48)C i AB A(cid:48)C
restricted types of side-information. E thus represents
sq One can verify that the reduction of this state on A(cid:48)C is
one possible extreme. We will see that another correla- √ √
a maximally correlated state with σ = p p (cid:104)ψi|ψj(cid:105).
tion measure, the mutual information, is the other ex- ij i j
Furthermore for MCS we find,
treme. In between, we have other measures such as en-
tanglement cost. We will also see that there is a dual I(A:B|C)/2=(cid:88)p S(ρi ) . (12)
i A
paradigm. Table 1 gives a summary of the results.
i
4
side-information infimum rate supremum rate
no restriction E (squashed entanglement) E (puffed entanglement)*
sq pu
maximally correlated states (MCS) E (entanglement cost) E (entanglement of assistance)
c ass
all of it at the receiver or sender I(A:B)/2 (mutual information) I(A:B)/2 (mutual information)
TABLE I: Summary of relationships between the restrictions on side-information and the rate at which half the state can be
senttothereceiver. NotethatE =E =min{S(A),S(B)}. *E isthesupremumrateifpureentanglementisanallowable
pu ass pu
resource.
ThisquantityisequaltotherateforsendingAlice’sshare Definition 4 the dual to Q is
S
toBobwhenentanglementisallowedasaresource,how-
Q ≡ sup Q (A:B) (13)
ever,becausepurestateentanglementcanbeintheform S A(cid:48)C
of a MCS, it gives the rate in this case as well. Taking ρA(cid:48)C∈S
the infimum of this expression over decompositions of Here, the distributor of the side information is forced
Eq. (11) gives infI(A : B|C)/2 = nE . Since all max- to distribute all the side information, so the only choice
c
imally correlated states can be written as the reduction they have is how to arrange it between Alice and Char-
of a purification of the form in Eq. (11) we have that lie. Whenwedothis,wewillfindthattheratesforstate
theinfimumoverdecompositions(10)isequivalenttoan transfer are given by dual correlation measures. For ex-
infimumovermaximallycorrelatedstates. Wethushave ample, the entanglement of assistance [12] is the max-
anotherinterpretationoftheentanglementcost: itisthe imum amount of pure state entanglement that can be
fastest rate at which ρ can be sent to a receiver us- created between Alice and Bob given a measurement on
A
ing side-information composed of maximally correlated the purification of their state. Here, we find another in-
states. terpretation–itistheamountofinformationwhichmust
besentiftheside-informationρ ismaliciouslychosen
Note that the entanglement of formation has a similar A(cid:48)C
from the set of maximally correlated states.
interpretation if we restrict ourselves to side-information
which is an extension of each single copy. While this
Theorem 5 Givenastateρ , Q =E (A:B)when
doesn’t matter for the squashed entanglement which is AB S ass
S is the set of MCS.
additive, we do not know whether it matters for the en-
tanglementofformation. Here,theadditivityquestionis The proof is similar to that of the entanglement cost
whether using maximally correlated states which are the except that one needs to use the fact that E = 0 for
purification of two states is no better for state transfer, MCS since a malicious distributor of the side informa-
than maximally correlated states which are the purifica- tion is unlikely to give the parties any entanglement.
tionofeachsinglestate. Wethushaveanotheradditivity The entanglement of assistance is known to be equal to
question related to the others – one that is operational. min{S(A),S(B)}[9,13]. Anoptimalprotocol[9]forcre-
ating entanglement between Alice and Bob is to perform
Also, the entanglement cost E when the side-
a random measurement of dimension S(E|minA,B) on
informationisrestrictedtoMCSisalwayszero. Thiscan
the purification. This implies that the worst possible
be seen from expression (3) and the fact that MCS are
maximally correlated state is one where there are just
symmetric. ItisforthesereasonthatE iszerowhenever
underS(E|min{A,B})basiselements|i(cid:105) ,andtheyare
the squashed entanglement is equal to the entanglement C
chosen at random over the typical space of ρ⊗n.
cost. E
It is tempting to look at maximising the amount
The most simple example of the paradigm is when the of communication required over all possible side-
set S are states such that all the side information is at informationstates. Thisquantitywouldbeadualtothe
the sender or the receiver. In such case, it is trivial to squashedentanglementbutwiththeinfimumchangedto
see that the communication rate is the mutual informa- a supremum.
tion I(A : B)/2 which is a measure of total correlations
Definition 6 The puffed entanglement[17] of the state
(both classical and quantum). Note that if we bend the
ρ isE =sup1I(A:B|E˜), withthesupremumtaken
paradigmslightly,andallowfortherestrictiononS toal- AB pu 2
low for no distribution of side-information then the best over all extensions E˜ of ρAB.
rate that Alice can achieve is through simple compres-
Onecanshowthatitisnotanentanglementmeasure,but
sion, giving a rate of S(A).
it’s operational meaning is that it is the rate required to
Thus far, we have been distributing the side informa- sendthestateρ ifthedistributorofthesideinformation
A
tion so as to minimise the amount of communication re- is as malicious as possible, and distributes ρ in as
A(cid:48)C
quired. There is a dual paradigm – instead we arrange unhelpful a way possible. However, here one does have
the side-information in the worst possible manner. I.e. to allow the parties free entanglement.
5
Remarkably, the puffed entanglement is equal to the Here the trace is taken over the Hilbert space which is
entanglement of assistance. complement to σ ,σ (i.e. the purification).
0 1
Theorem 7 Given a state ρ , Q = E (A : B) =
AB S pu TheabovetheoremisenoughtoimplythatQS isconvex.
min{S(A),S(B)} when S is the set of all possible states.
OnethenonlyneedstoshowthatQ cannotincreaseon
S
average after application of a local CPT map [15]. The
adaptation of the proof of Theorem 11.15 of [16] related
Thesecondequalitycomesbecausestandardentropicin-
in [4] is sufficient for our purposes.
equalities imply that I(A : B|C)/2 ≤ min{S(A),S(B)},
while a protocol exists (using an MCS) to get equal- In general, we have found a relationship between var-
ity. Thus the same side information which maximises ious sets of states and correlation measures. Here, the
theentanglementofassistancealsomaximisesthepuffed squashedentanglementarisesasamorequantumversion
entanglement. It is perhaps surprising that getting to of the entanglement cost. While traditionally, these cor-
choose the side information over all possible states but relationmeasuresariseasamonotoneundersomeclassof
with free entanglement for the parties, is equivalent to operations,hereeverythingisdeterminedintermsofthe
choosingthesideinformationfrommaximallycorrelated setofstates. Remarkably,thenotionofclassicalcommu-
states. One can also show that another way of select- nicationorclassicalitydoesnotenterintothediscussion,
ing the worst possible side-information is to perform a yet it gives arise to at least two entanglement measures.
random unitary on ρ⊗n and then choose ρ(cid:48) of size just There is a sense in which the paradigm is reversible:
E A
over min{nI(E :A)/2,nI(E :B)/2} qubits. This means ratesarethesameregardlessofwhogetswhatpartofthe
that ρ will be decoupled from both ρ and ρ . It is side-information state, i.e. if we swap A(cid:48) with C. This
C A B
an open question what the supremum sending rate is if meansthatthesameamountofcommunicationisneeded
entanglement is not a free resource. to send the state from Alice to Charlie as from Charlie
Both the entanglement of assistance, and puffed en- to Alice, and we can send the state back and forth using
tanglement are super-additive because they involve an and recovering the pure state entanglement.
infimum over possible extensions. However, neither are It would also be interesting to explore the significance
additive i.e. one can have oftheamountofentanglementwhichisconsumedatthe
maximal (or dual) sending rate. It may even shed some
Epu(ρ1AB ⊗ρ2AB)>Epu(ρ1AB)+Epu(ρ2AB) (14) light on the relationship between the paradigm consid-
ered here, and that of LOCC. In this case, a key ques-
a situation which can occur when, for example, S(A )<
1 tioniswhetherastatethatdoesn’tshowitsentanglement
S(B ) and S(B )>S(B ).
1 2 2 here (in the sense that it requires no quantum commu-
For the case of side-information all at the sender or
nication to send half of it to a receiver), is also unentan-
receiver, the rate is half the mutual information regard-
gled in the LOCC case. We know that if E = 0 then
less of whether we want to minimise or maximise the c
E = 0, but the converse could well be false. It would
communication rate, so in this paradigm, the mutual in- sq
alsobeinterestingtoknowwhatthedualtothesquashed
formation is self-dual.
entanglement is in the case where pure entanglement is
It would be interesting to look at the correlation mea-
not an allowable resource.
sures and rates under other restrictions of the set S. For
This work is supported by EU grants QAP and by the
example, one could consider side-information which is
Royal Society. I am grateful to Michal(cid:32) Horodecki for
separable, or whose states have positive partial trans-
inspirational discussions.
pose, or which are only classically correlated. Likewise,
onemightalsoconsiderothertypesofoperations. Asone
potential example, there appears to be a strong link be-
tween the intrinsic mutual information [6] and the max-
imum rate of sending a private distribution using only
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