Resource cost results for oneway entanglement distillation and state merging of compound and arbitrarily varying quantum sources
Abstract
We consider oneway quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelaković et. al. [Universal quantum state merging, J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum sources (AVQS). In the AVQS model, the source state is assumed to vary in an arbitrary manner for each source output due to environmental fluctuations or adversarial manipulation. We determine the oneway entanglement distillation capacity for AVQS, where we invoke the famous robustification and elimination techniques introduced by R. Ahlswede. Regarding quantum state merging for AVQS we show by example, that the robustification and elimination based approach generally leads to suboptimal entanglement as well as classical communication rates.
I Introduction
Investigations on communication tasks involving bipartite (or multipartite) sources within the local operations and classical
communications (LOCC) paradigm made a substantial contribution to the progress in quantum
Shannon theory which took place over the past two decades.
Especially the role of shared pure entanglement as a communication resource was clarified and substantiated by
establishment of LOCC protocols interconverting shared entanglement with optimal rates.
Two prominent tasks, entanglement distillation and quantum state merging are considered in this work.
Quantum state merging was introduced by Horodecki, Oppenheim, and Winter horodecki07 . In this setting
a bipartite
quantum source described by a quantum state shared by communication parties (sender) and
(receiver) is required to be merged at the receivers site by local operations and classical communication
together with shared pure entanglement as resource, such that in the limit of large blocklengths, the source
is approximately restored on s site. The optimal asymptotic net entanglement cost was determined in
Ref. 14
to be
ebits of shared entanglement per copy of the state, which was shown to be achievable with optimal classical cost
bits of classical side communication per copy ( is the quantum mutual information of
A with an environment E purifying ). This result allows interpretation the negative values of
.
For states with being negative, quantum state
merging is possible with net production of shared maximal entanglement which may serve as a credit for
future quantum communication.
Entanglement distillation is in some sense a task subaltern to quantum state merging, since entanglement
distillation protocols are readily derived from quantum state merging protocols horodecki07 .
In this task, a
given bipartite quantum source has to be transformed into shared maximal entanglement by LOCC in the
limit of large number of outputs. The optimal entanglement gain was determined in Ref. 12,
where a connection to secret key distillation from bipartite
quantum states was exhausted.
However, these results were shown under strong idealizations of the sources. It was assumed, that the
sources where tasks are performed on, are memoryless and perfectly known.
Since source uncertainties, may they be present due to hardware
imperfections of the preparation devices and/or manipulation by adversarial communication parties, are
inherent to all reallife communication settings, this assumption seems rather restrictive.
The contribution of this work is, to partly drop these conditions. We consider entanglement distillation
and quantum state merging in presence of compound and arbitrarily varying quantum sources.
A compound memoryless source models a preparation device which emits systems, uncorrelated from output
to output, all described by the same given density matrix, which in turn is not perfectly known to
the communication
parties, but identified as a member of a certain set of quantum states.
Consequently, the communication parties
have to use protocols which are of sufficient fidelity for each member of the set of states generating
the compound source.
In the arbitrarily varying source (AVQS) model, the source state can vary from output
to output over a generating set of states. This variation can be understood as a natural fluctuation as well
as a manipulation of an adversarial communication party changing the source state from output to output in an
arbitrary manner. Consequently, the parties are forced to accomplish the tasks with protocols,
which are robust in the sense, that work with sufficient fidelity for each possible state sequence.
In this work, we contribute the following.
Regarding oneway quantum state merging for compound sources, we
answer a question left open in the preceding work bjelakovic13 . We derive protocols which beside
being optimal regarding their entanglement cost, also approximate the lowest classical oneway
communication requirements allowed by corresponding converse theoremshorodecki07 ; bjelakovic13
which lower bound the resource requirements for asymptotically faithful merging schemes.
We use the results on oneway entanglement distillation for compound sources established
earlierbjelakovic13 together with the famous elimination and robustification techniques introduced
by R. Ahlswede ahlswede78 ; ahlswede80 to
determine the capacity for oneway entanglement distillation from AVQS generated by a set
of states.
We show, that the oneway entanglement
distillation capacity in this case, can be expressed by the capacity function of the compound source
generated by the convex hull of the set generating the AVQS.
Considering quantum state merging under the AVQS model, we encounter unexpected behavior.
Opposite to the intuition gathered by previous results from classical as well as quantum Shannon theory,
the entanglement as well as classical communication resource costs for oneway merging of an AVQS do
not match the costs known for the corresponding compound source generated by the
convex hull of in general. We demonstrate this fact giving a simple example.
i.1 Related Work
The task of entanglement distillation was subject to several investigations in case of perfectly known
memoryless quantum sources over the past fifteen years. In this work, we generalize a result from
Ref. 12,
where the entanglement distillation capacity with oneway LOCC for perfectly known
memoryless bipartite quantum sources where determined. Quantum state merging was first considered in
Ref. 14, where the authors determined the entanglement as well as classical cost of quantum state
merging for the scenario with perfectly known density matrix. Both results where partly generalized to the
case of compound memoryless sources in Ref. 7 within the oneway LOCC scenario.
In this work we continue and complete considerations made therein by determining the optimal classical cost of
oneway merging for compound quantum sources.
Communication tasks involving arbitrarily varying channels and sources where considered in classical
information theory from the late 60’s. Here we especially mention the robustification
ahlswede80 ; ahlswede86 and elimination ahlswede78 techniques developed by Ahlswede in the 70’s, which are crucial ingredients of our proof
of the oneway entanglement distillation capacity for AVQS.
Arbitrarily varying channels where also considered in quantum Shannon theory. The first result was
by Ahlswede and Blinovsky ahlswede07 , who determined the capacity for transmission of classical messages
over an arbitrarily varying channel with classical input and quantum output. A treatment of arbitrarily
varying quantum channels was done by Ahlswede, Bjelaković, the first author and Nötzel
published in 2013 ahlswede13 . There, they determined the quantum capacity of an arbitrarily varying quantum channel
for entanglement transmission, entanglement generation as well as strong subspace transmission.
i.2 Outline
We set up the notation used in this paper in Section II, where we also state some conventions
and preliminary facts we use freely in our considerations. The basic concepts relevant for this paper are
concisely stated and and explained in Section III.
In Section IV, we conclude the investigations on quantum state merging for compound
sources begun in Ref. 7. Explicitly, we show existence of universal oneway LOCCs which
are
asymptotically optimal regarding the entanglement as well as classical communication cost.
For the proof, we use protocols derived in Ref. 7, which are optimal regarding their
entanglement cost but require overmuch classical side communication in some cases. These are refined in a
sufficient way by combination with an entropy estimating instrument used by the sender, where we utilize
methods from representation theory of the symmetric groups from Refs. 15 and
10.
Section V is devoted to determination of the capacity for
entanglement distillation from an AVQS under restriction to oneway LOCC.
We first prove an achievability result in case that the AVQS is generated by a finite set
of bipartite states.
Here we use entanglement distillation schemes with fidelity going to one exponentially fast
for the compound source generated by the convex hull of from Ref. 7,
together with Ahlswede’s robustification and elimination techniques. Afterwards, we extend this
result to the general case approximating the AVQS generating set by suitable finite AVQS.
We also consider the issue of quantum state merging for AVQS and discover a strange feature of the quantum
state merging task in this scenario. We show in Section VI, that in general, the entanglement
as well as classical
cost of merging an AVQS generated by a set of bipartite state are strictly lower than the costs
of
merging the corresponding compound source generated by . In Section VII, we
discuss the results obtained.
Ii Notation and Conventions
All Hilbert spaces appearing in this work are considered to be finite dimensional complex vector spaces.
is the set of linear maps and the set of states (density matrices) on a Hilbert
space in our notation. We denote the set of quantum channels, i.e. completely positive and trace
preserving (c.p.t.p.) maps from to by and the set
of tracenonincreasing cp maps by for two Hilbert spaces , .
Regarding states on multiparty systems, we freely make use of the following convention for a system consisting
of some parties , for instance, we denote , and denote
the marginals by the letters assigned to subsystems, i.e. for and so on. For a bipartite pure state on a Hilbert space ,
we denote its Schmidt rank (i.e. number of nonzero coefficients in the Schmidt representation of ) by
. We define
(1) 
for any two positive semidefinite operators on (this is the quantum fidelity in case that and are density matrices). If one of the arguments is a pure state, the fidelity is linear in the remaining argument, explicitly takes the form of an inner product,
(2) 
Relations between and the trace distance are well known, we will use the inequalities
(3) 
for a matrix and state , and
(4) 
for states . The von Neumann entropy of a quantum state is defined
(5) 
where we denote by and the base two logarithms and exponentials throughout this paper. Given a quantum state on , we denote the conditional von Neumann entropy of given by
(6) 
the quantum mutual information by
(7) 
and the coherent information by
(8) 
A special class of channels mapping bipartite systems, which is of crucial importance for our considerations, are oneway LOCC channels, for which we give a concise definition in the following. For more detailed information, the reader is referred to the appendix on oneway LOCCs given in Ref. 7 and references therein. A quantum instrument on a Hilbert space is given by a set of trace nonincreasing cp maps, such that is a channel. In this paper, we will only admit instruments with . With bipartite Hilbert spaces and , a channel is an (oneway) LOCC channel, if it is a combination of an instrument and a family of channels in the sense, that it can be written in the form
(9) 
The cardinality of the message set for classical transmission from to within the application of
is (the number of measurement outcomes of the instrument).
We denote the set of classical probability distributions on a set by . The fold Cartesian
product of will be denoted and will be a notation for elements of
. For positive integer , the shortcut is used to abbreviate the set .
For two probability distributions on a finite set , the relative entropy of
with respect to is defined
(10) 
where means . We denote the Shannon entropy of a probability distribution by . For a set we denote the convex hull of by . If is a finite set of states on a Hilbert space , it holds
(11) 
By , we denote the group of permutations on elements, in this way
for each and permutation .
For any two nonempty sets , of states on a Hilbert space , the Hausdorff
distance between and (induced by the trace norm ) is defined by
(12) 
Iii Basic Definitions
In this section, we define the underlying scenarios, considered in the rest of this paper.
Given any set
of states on a Hilbert space , the
compound source generated by (or the compound source ,
for short) is given by the family of states.
The above definition models a memoryless quantum source under uncertainty of the statistical parameters.
The source outputs each system according to a constant density matrix, while the density matrix
itself is not known perfectly by the communication parties. It only can be identified as a member of
.
The arbitrarily varying quantum source (AVQS) generated by
(or the AVQS )
is given by the family , where we use the definition
(13) 
for each member of . In the AVQS model, the source density matrix can be chosen from the set independently for each output. The variation in the source state models hardware imperfections, where the source is subject to fluctuations in the state on one hand. On the other hand, this definition also can be understood as a powerful communication attack, where the statistical parameters of the source are, to some extend, perpetually manipulated by an adversarial communication party.
iii.1 Quantum State Merging
We first give a concise notion of the protocols we admit for quantum state merging. We are interested in
the entanglement as well as classical resource costs of quantum state merging.
A quantum channel is an merging
for bipartite sources on , if it is an LOCC channel
(according to the definition from (9))
(14) 
with , where we assume (), and
(15) 
where constitutes an instrument and is a set of channels depending on the parameter . The spaces are understood to represent bipartite systems shared by and , which carry the input and output entanglement resources used in the process. As a convention, we will incorporate the maximally entangled states , into the definition of the protocol, it holds
(16) 
We define the merging fidelity of given a state by
(17) 
Here, is a purification of with an environmental system described on an additional Hilbert space (usually with some space ), and is a state identical to but defined on completely under control of . It was shown in Ref. 7 (Lemma 1), that the r.h.s. of (17) does not depend on the chosen purification (which justifies the definition of ), and that the function is convex in the first and linear in the second argument. For the rest of this section, we assume to be any set of bipartite states on .
Definition 1.
A number is called an achievable entanglement cost for merging of the compound source with classical communication rate , if there exists a sequence of mergings, such that the conditions
are satisfied.
In the following definition, priority lies on the optimal entanglement consumption (or gain) of merging processes, while the classical communication requirements are of subordinate priority. However, the classical communication is required to be rate bounded in the asymptotic limit. Since the classical communication requirements are of interest as well, we also determine the optimal classical communication cost in Section IV.
Definition 2.
The merging cost of the compound source is defined by
(18) 
We recall the following theorem proven in Ref. 7
Theorem 3 (cf. Ref. 7).
(19) 
Definition 4.
A number is called an achievable entanglement cost for merging of the AVQS with classical communication rate if there exists a sequence of mergings satisfying



.
Definition 5.
The merging cost of the AVQS is defined by
(20) 
iii.2 Entanglement Distillation
Concerning entanglement distillation, we are interested in the asymptotically entanglement gain of oneway LOCC distillation procedures. We use the following definitions.
Definition 6.
A nonnegative number is an achievable entanglement distillation rate for the AVQS generated by a set with classical rate , if there exists a sequence of LOCC channels,
(21) 
such that the conditions
are fulfilled, where is a maximally entangled state shared by and for each .
In this paper, we will be primarily interested in the entanglement gain of oneway entanglement distillation. Regarding the classical communication cost of entanglement distillation, no general cost results are known even in case that the source is memoryless with perfectly known source state devetak05c .
Definition 7.
The entanglement distillation capacity for the AVQS generated by is defined
(22) 
The corresponding definitions for achievable rates and entanglement distillation capacity of compound sources can be easily guessed (see Ref. 7). To introduce some notation we use in this paper, we state the a theorem from Ref. 12, where the entanglement distillation capacity of a memoryless bipartite quantum source with perfectly known density matrix was considered.
Theorem 8 (Ref. 12, Theorem 3.4).
Let be a state on . It holds
(23) 
with
(24) 
where is the set of finitevalued quantum instruments on ’s site, i.e.
(25) 
For each state and quantum instrument on ’s site and definitions
(26) 
for each with .
Remark 9.
It is known devetak05c , that the limit in (23) exists for each state, and maximization over instruments in this formula is always realized by an instrument with and the operation described by only one Kraus operator for .
In order to obtain a compact notation for the capacity functions arising in the entanglement distillation scenarios we consider in this paper, we introduce a oneway LOCC for each instrument with domain and an orthonormal system in a suitable space assigned to , it holds
(27) 
in (24) for each given state .
Iv Quantum State Merging for Compound Quantum Sources
In this section, we derive, for any given bipartite compound source , asymptotically faithful state merging protocols, which are approximately optimal regarding their entanglement as well as classical communication cost given the corresponding converse statement bjelakovic13 . While the merging cost was determined in Ref. 7 before (see Theorem 3 above also), the protocols used there, are suboptimal, in general, regarding their classical communication requirements. However, it was shown there (see Section V in Ref. 7), that
(28) 
(supremum of the quantum mutual information between and a purifying environment ) is a lower bound
on the classical communication cost for merging a compound source by
by protocols which have fidelity one in the limit of large blocklengths.
Proposition 13 below states, that this bound actually is achievable, and thus
together with results from Ref. 7 provides a full solution of the quantum state merging
problem for compound quantum sources.
The assertions proved in this section will be utilized in Section VI, where we compare
the merging as well as the classical communication cost of a certain AVQS merging protocol
for a set with the optimal costs of state merging protocols for the compound source generated
by .
The preliminary Proposition 10 below is a slight generalization of Theorem 6 in
Ref. 7. It states existence of protocols achieving the optimal entanglement cost, but
with generally suboptimal classical communication rates. However, these protocols will be utilized to
derive protocols suitable for the proof of Proposition 13.
Proposition 10 (cf. Ref. 7, Theorem 6).
Let . For each , there is a number , such that for each blocklength there is an merging , such that
(29) 
with a a constant ,
(30) 
and
(31) 
Proof.
The assertion to prove includes both, a strengthening of the fidelity convergence rates in Ref.
7,
Theorem 4 to exponentially decreasing tradeoffs, and a generalization of Theorem 6 in
Ref. 7 to arbitrary (not necessary finite or countable) sets of states.
Approximating by a net
for each
blocklength (see Ref. 7
for details) and using the result for finite sets, we infer by careful observation of the merging fidelities
in Ref. 7 (see eqns. (36), (37), and (58) therein), that for given and large enough blocklength , there exists a
merging , where
(32) 
is valid for the merging fidelities with a constant , and
(33) 
(see (57) in Ref. 7). Moreover, we can bound the number of messages for the classical communication (see (101) in Ref. 7) by,
(34)  
(35) 
where the summand fannes73 , i.e. follows from threefold application of Fannes’ inequality
(36) 
Due to the bound given in Ref. 7, Lemma 9, it is known, that the nets can be chosen with cardinality bounded by
(37) 
for each . Choosing net parameter with for each , we infer
(38) 
with a constant , and
(39) 
from (35) if is large enough, to satisfy . Collecting the bounds in (33), (38), and (39), we are done. ∎
Before we state and prove Proposition 13, we collect some results from representation
theory of the symmetric groups, which we utilize in the proof.
We denote by
the set of young frames with at most rows and boxes for . A young frame is determined by a tuple of nonnegative integers summing to .
The boxlengths of define a probability
distribution on in a natural way
via the definition
for each .
To each Young frame , there is an invariant subspace of ,
and we denote by the projector onto the subspace belonging to .
Theorem 11 below allows, to asymptotically estimate the spectrum of a density operator
by projection valued measurements on i.i.d. sequences of the form , and is an
important ingredient of our proof of Proposition 13.
A variant of the first statement of the theorem was first proven in by Keyl and Wernerkeyl01 .
The actual bounds stated below are from Ref. 10, while the remaining statements of the
theorem are wellknown facts in group representation theory (Ref. 10 and references
therein are recommended for further information).
Theorem 11 (cf. Refs. 15 and 10).
The following assertions are valid for each .

For and , it holds
(40) where is the probability distribution given by the normalized boxlengths of , and is the probability distribution on induced by the decreasingly ordered spectrum of (with multiplicities of eigenvalues counted).

.

For , it holds if .
The following proposition is the main result of this section.
Proposition 13.
Let be a set of states on . For each , there exists a number , such that for each there is an merging with
(42) 
with a constant ,
(43) 
and
(44) 
where the quantum mutual information in (44) is evaluated on the AE marginal state of any purification of (notice, that the above abuse of notation does not lead to ambiguities, since holds for any purification of ).
Remark 14.
Regarding the classical communication cost a quite restrictive converse statement was shown to be validbjelakovic13 . Asymptotically faithful oneway state merging schemes demand classical communication at rate
(45) 
regardless of the entanglement rate achieved, i.e. even investing more entanglement resources (choosing protocols with suboptimal merging rates) does not lead to a reduction of the classical communication cost in a significant way.
Proof of Proposition 13.
One half of the above assertion was already proven (see Ref. 7 and Proposition 10 at the beginning of this section). Explicitly, it was shown there, that LOCC channels exist for each set of bipartite states, which for sufficiently large blocklengths fulfill the conditions formulated in (42) and (43). We complete the proof by demonstrating, that also the constraint (44) on the classical communication rate can be met simultaneously with (42) and (43) by certain protocols. The strategy of our proof will be as follows. We decompose into disjoint subsets , each containing only states with approximately equal entropy on the marginal system and combine an entropy estimating instrument on the system with a suitable merging scheme for each set according to Proposition 10. We fix , and assume, to simplify the argument, that
(46) 
holds (i.e. merging is possible without input entanglement resources for large enough blocklengths). Otherwise the argument below can be carried out using further input entanglement and wasting it before action of the protocol. We define and fix to be determined later. Consider the sequence
(47) 
Define Intervals and for , which generate a decomposition of into disjoint sets by definitions
(48) 
and set
(49) 
where is defined for all . In order to construct an entropy estimating instrument in the marginal systems, we define an operation by
(50) 
for each using the notation from Theorem 11. Notice, that form a projection valued measure on due to Theorem 11.3. By construction, we have for each state , ,
(51)  
(52)  
(53)  
(54) 
where (51) and (52) are valid due to construction and (54) follows from Theorem 11.1. Since the relative entropy term in the exponent on the r.h.s. of (54) is bounded away from zero for each fixed number (consult the appendix of this paper for a proof of this fact), i.e.
(55) 
with a constant , and the functions outside the exponential term are growing polynomially for (see Theorem 11.2) , we deduce
(56) 
provided that is large enough.
Define index sets and . We know from Proposition 10,
that for each sufficiently large , we find an merging
for each such that
(57) 
holds with a constant ,
(58) 
and
(59) 
for the classical communication rate. By construction of the sets , , it also holds
(60) 
for each . Taking suprema over the set on both sides of the above inequality in combination with (59) leads us to the estimate
(61) 
for each . Combining the entropy estimating instrument with the corresponding merging protocols, we define
(62) 
The maps are yet undefined for all numbers
.
Since they will not be relevant for the fidelity, they may be defined by any trivial local
operations, with
for . Moreover, we assume, that the merging rate of
for each is stuck to the the worst and each outputs approximately
the same maximally entangled resource output state . We can always achieve this by partial tracing and
local unitaries, which do not further affect the classical communication rates.
By inspection of the definition in (62) one readily verifies, that is,
in fact, an
merging, with
(63) 
and therefore, classical communication rate bounded by
(64)  
(65)  
(66) 
It remains to show, that we achieve achieve merging fidelity one with for each with exponentially decreasing tradeoffs for large enough blocklengths. Assume is a member of for any index . Then, it holds
(67)  
(68) 
The inequality above holds, because the merging fidelity is linear in the operation and all summands are nonnegative together with the definition of . The equality is by some zeroadding of terms an using the definition together with linearity of the merging fidelity in the operation again. We bound the terms in (68) separately. Beginning with the second term, we notice, that the fidelity is homogeneous in its inputs and bounded by one for states, it holds