Channel Simulation and Coded Source Compression
This work establishes connection between channel simulation and coded source compression. First, we consider classical source coding with quantum side-information where the quantum side-information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct part of our result is proved via the measurement compression theory by Winter, a quantum to classical channel simulation. Our result reveals that a helper’s scheme that separately conducts a measurement and a compression is suboptimal, and the measurement compression is fundamentally needed to achieve the optimal rate region. We then study coded source compression in the fully quantum regime. We characterise the quantum resources involved in this problem, and derive a single-letter expression of the achievable rate region when entanglement assistance is available. The direct coding proof is based on a combination of two fundamental protocols, namely the quantum state merging protocol and the quantum reverse Shannon theorem. Our work hence resolves coded source compression in the quantum regime.
Source coding normally refers to the information processing task that aims to reduce the redundancy exhibited when multiple copies of the same source are used. In establishing information theory, Shannon demonstrated a fundamental result that source coding can be done in a lossless fashion; namely, the recovered source will be an exact replica of the original one when the number of copies of the source goes to infinity . If representing the source by a random variable with output space and distribution , lossless source coding is possible if and only if the compression rate is above its Shannon entropy:
Redundancy can also exist in the scenario in which multiple copies of the source are shared by two or more parties that are far apart. Compression in this particular setting is called distributed source coding, which has been proven to be extremely important in the internet era. The goal is to minimise the information sent by each party so that the decoder can still recover the source faithfully. Shannon’s lossless source coding theorem can still be applied individually to each party. However, it is discovered that a better source coding strategy exists if the sources between different parties are correlated. Denote and the sources held by the two distant parties, where the joint distribution is and the output spaces are and , respectively. Slepian and Wolf showed that lossless distributed source coding is possible when the compression rates and for the two parties satisfy :
where is the conditional Shannon entropy. This theorem is now called the classical Slepian-Wolf theorem . In particular, when source is directly observed at the decoder, the problem is sometimes called source coding with (full) side-information.
Another commonly encountered scenario in a communication network is that a centralised server exists and its role is to coordinate all the information processing tasks, including the task of source coding, between the nodes in this network. Obviously, the role of the server is simply as a helper and it is not critical to reproduce the exact information communicated by the server. This slightly different scenario results in a completely different characterisation of the rate region, as observed by Wyner  and Ahlswede-Körner . Consider that the receiver wants to recover the source with the assistance of the server (that we will call a helper from now on) holding , where the distribution is . Wyner and Ahlswede-Körner showed that the optimal rate region for lossless source coding of with a classical helper is the set of rate pairs such that
for some conditional distribution , and is the classical mutual information between random variables and . When there is no constraint on (i.e. can be as large as it can be), this problem reduces to source coding with (full) side-information.
The problem of source coding, when replacing classical sources with quantum sources, appears to be highly nontrivial in the first place
Schumacher’s quantum source coding theorem bears a close resemblance to its classical counterpart. One will naturally expect that the same will hold true for the distributed source coding problem in the quantum regime. Consider that Alice, who has the quantum system of an entangled source , would like to merge her state to the distant party Bob. The quantum distributed source coding theorem (also known as quantum state merging) aims to answer the optimal rate at which quantum states with density matrix can be communicated to a party with quantum side information faithfully. As it turns out, the optimal rate is given by the conditional von Neumann entropy , a quantum generaliztion of classical conditional Shannon entropy. While the quantum formula to the distributed source coding problem is also of the form of conditional entropy, this result has a much deeper and profound impact in the theory of quantum information as it marks a clear departure between classical and quantum information theory. It is rather perplexing that the rate is quantified by the conditional entropy , which can be negative. This major piece of the puzzle was resolved with the interpretation that if the rate is negative, the state can be merged, and in addition, the two parties will gain amount of entanglement for later quantum communication [8, 9, 10]. The distributed quantum source coding problem was later fully solved [11, 13] where the trade-off rate region between the quantum communication and the entanglement resource is derived. The result is now called the fully quantum Slepian-Wolf theorem (FQSW).
Source coding with hybrid classical-quantum systems with representing a classical system and a quantum state is also considered in quantum information theory, and one of our main results falls into this category. In , Devetak and Winter considered classical source coding with quantum side information at the decoder, and showed that the optimal rate is given by . This result can be regarded as a classical-quantum version of the source coding with (full) side-information.
In this work, we first revisit the classical coded source compression [3, 4]. In particular, we provide a simpler achievability proof based on the idea of channel simulation (Theorem 2). The proof indicates that channel simulation is a general subroutine employed between the helper and the decoder in the task of coded source compression.
Next, we consider classical source coding with a quantum helper. In our setup, the quantum side-information is observed by the helper, and the decoder will only have a classical description from the quantum helper. Although our problem can be regarded as a classical-quantum version of the classical helper problem studied in [3, 4], in contrast to its classical counterpart, our problem does not reduce to source coding with quantum side-information studied in  even if there is no constraint on rate . However, when the ensemble that constitutes the quantum side-information is commutative, our problem reduces to the classical helper problem [3, 4]. We completely characterize the rate region of the classical source coding with the quantum helper problem. In fact, the formulae describing the rate region (cf. Theorem 3) resembles its classical counterpart (cf. (9) and (10)). However, the proof technique is very different due to the quantum nature of the helper. In particular, we use the measurement compression theory by Winter  in the direct coding theorem. One of interesting consequences of our result is that a helper’s scheme that separately conducts a measurement and a compression is suboptimal; measurement compression is fundamentally needed to achieve the optimal rate region.
Next, we extend the classical distributed source coding problem [3, 4] and its classical-quantum generalisation to the fully quantum version; namely compression of a quantum source with the help of a quantum server. Moreover, we consider a general setting where entanglement assistance between sender-decoder and helper-decoder is available. Our direct coding proof combines two fundamental quantum protocols; the state merging protocol [8, 9] and the quantum reverse Shannon theorem . The current progress of coded source compression is summarized in Table 1. We remark that the quantum source compression with a classical helper is a very subtle task, which is left open even when the decoder has full classical-side information  (instead of partial side-information from the classical helper).
|Classical Source||Quantum Source|
|Classical Helper||Refs. [3, 4]|
|Quantum Helper||Theorem 3||Theorem 5|
The quantum source compression with a quantum helper is treated in a different scenario in Ref. . Overall there, classical communication is allowed from the helper to the receiver, and limited entanglement resource is considered. As a result, their formula requires regularization. In contract, our result resorts to quantum reverse Shannon theorem, and has the appealing single-letter expression.
There are a huge amount of work devoted to both classical and quantum lossy source coding [15, 16, 17, 18, 19, 20]. We will restrict ourselves to only noiseless source coding in this work. However, as it turns out, channel simulation simplifies both rate distortion theory and coded source compression.
Notations. In this paper, we will use capital letters etc. to denote classical random variables, and lower cases to denote their realisations. We use to denote the sample spaces. We denote .
A quantum state is a positive semi-definite matrix with trace equal to one. We will use or to denote a quantum states in this paper. In case we need to specify which party the quantum state belongs to, we will use a subscript description , meaning that the quantum system is held by A(lice). Letting be a set of orthonormal basis vectors, a classical-quantum state is written as
so that copies of it is
where we denote for the sequence . A positive-operator valued measure (POVM), , is a quantum measurement whose elements are non-negative self-adjoint operators on a Hilbert space so that .
Various entropic quantities will be used in the paper. The von Neumann entropy of a quantum state , where the subscript represents the quantum state is held by A(lice), is . The conditional von Neumann entropy of system conditioned on of a bipartite state is . The quantum mutual information between two systems and of is . The conditional quantum mutual information .
We will also employ the framework of Resource Inequality (RI) [12, 13]. The RIs are a concise way of describing interconversion of resources in an information-processing task. Denote by and an ebit (maximally entangled pairs of qubits) and a noiseless qubit channel, respectively. Then a quantum channel that can faithfully transmit qubits per channel use with an unlimited amount of entanglement assistance can be symbolically represented as
where is an asymptotic noisy resource that corresponds to many independent uses, i.e. . Schumacher’s noiseless source compression  can be similarly expressed
which means that a rate of noiseless qubits asymptotically is sufficient to represent the noisy quantum source .
Sometimes, the RI only applies to the relative resource, , which means that the asymptotic accuracy is achieved only when uses of are fed an input of the form . For detailed treatment of combining two RIs and cancellations of quantum resources, see Ref. .
This paper is organised as follows. In Sec 2, we revisit classical source compression with a helper using the channel simulation idea. In Sec 3, we formally define the problem of source coding with a quantum helper, and present the main result as well as its proof. In Sec 4, we treat the source coding with a helper in the fully quantum regime. We conclude in Sec 5 with open questions.
2 Classical Coded Source Compression
Consider two classical channels and . We define an channel simulation code consisting of the encoding and decoding pair , that takes uses of the first channel to simulate instances of the channel output so that,
The channel simulation capacity of the first channel needed to simulate the second channel is defined as:
The Shannon’s noiseless channel coding theorem can be seen as a special case of that is the identity channel .
Another special case is the following classical reverse Shannon Theorem where is the identity channel .
Theorem 1 (Classical Reverse Shannon Theorem )
where is the Shannon capacity of the channel .
Using the noiseless resource to simulate a noisy one seems a useless task to explore at first glance. However, it turns out that such a task can be used to simplify coded source compression conceptually (among others).
The optimal rate region for lossless source coding of with a classical helper is the set of rate pairs such that
for some conditional distribution , and is the classical mutual information between random variables and .
Proof. In the proof, the strategy that the classical helper does can be conceptually viewed as assisting the decoder to simulate the local channel . For sufficiently large, the classical communication rate that the helper needs to send is and
where is the joint distribution induced by the simulation of .
Thus, the full side information about is possessed by the decoder, and the source compression with full side information can be carried on. Since the helper’s local channel can be simulated at the decoder whose inaccuracy is at most , the overall error for classical source compression with a classical helper can be achieved with this additional error.
3 Classical Source Compression with a Quantum Helper
As shown in Figure 1, the protocol for classical source coding with a quantum helper involves two senders, Alice and Bob, and one receiver, Charlie. Initially Alice and Bob hold copies of a classical-quantum state . In this case, Alice holds classical random variables while Bob (being a helper) holds a quantum state that is correlated with Alice’s message. The goal is for the decoder Charlie to faithfully recover Alice’s message when assisted by the quantum helper Bob.
We now proceed to formally define the coding procedure. We define an code for classical source compression with a quantum helper to consist of the following:
Alice’s encoding operation , where and ;
Bob’s POVM , where and ;
Charlie’s decoding operation
so that the error probability satisfies
A rate pair is said to be achievable if for any and all sufficiently large , there exists an code with rates and . The rate region is then defined as the collection of all achievable rate pairs. Our main result is the following theorem.
Given is a classical-quantum source . The optimal rate region for lossless source coding of with a quantum helper is the set of rate pairs such that
The state resulting from Bob’s application of the POVM is
where denotes complex conjugation in the standard basis. Furthermore, we can restrict the size of POVM as , where is the dimension of Bob’s system.
where is the accessible information for the ensemble . Unless the ensemble commutes , the minimum rate is larger than the rate , which is the optimal rate in the source coding with quantum side-information . To achieve , it suffices to have , which is smaller than in general. This means that the following separation scheme is suboptimal: first conduct a measurement to get and then compress . For more detail, see the direct coding proof.
Let be Alice’s encoder, and let be Bob’s measurement. Alice sends to the decoder, and Bob sends the measurement outcome to the decoder. The Fano’s inequality states that for some as .
First, we have the following bound:
where follows from Fano’s inequality: for some as ; in (b), we use chain rule and denote ; in , we denote ; in , we introduce a time-sharing random variable that is uniformly distributed in the set .
Next, we have
where (a) follows from
Following from Eq. (34), we can again introduce a time-sharing random variable that is uniformly distributed in the set ,
where the last equality follows because . To get single-letter formula, define , , and and let :
Here, we note that the distribution of can be written as
Thus, we can get as a measurement outcome of by first generating , then by appending and to ancillae systems, and finally by conducting the measurement .
Finally, the bound on can be proved via Carathódory’s theorem (cf. [26, Appendix C]).
Direct Coding Theorem. Fix a POVM measurement . It induces a conditional probability , and joint probability distribution
The crucial observation is the application of Winter’s measurement compression theory .
Let be a source state and a POVM to simulate on this state. A protocol for a faithful simulation of the POVM is achievable with classical communication rate and common randomness rate if and only if the following set of inequalities hold
where the entropies are with respect to a state of the following form:
and is some purification of the state .
Let be a random variable on , which describes the common randomness shared between Alice and Bob. Let be collection of POVMs. Let
where . The faithful simulation of copies of POVM , i.e. , implies that for any , there exists sufficiently large, such that there exist POVMs , where , with
Alice and Bob shared copies of the state , and assume that Bob performs measurement on his quantum system whose outcome is sent to the decoder to assist decoding Alice’s message. Bob’s measurement on each copy of will induce the probability distribution according to (47). Apparently, if Bob sends the full measurement outcomes to Charlie (say bits), then Charlie can successfully decode simply from Slepian-Wolf Theorem. The next strategy is to make use of classical result since after Bob’s measurement, Alice and Bob become fully classical with joint distribution . Therefore, the minimum rate for Bob is (w.r.t. some conditional distribution ). However, there is a non-trivial quantum coding strategy. Detail follows.
Bob’s coding. Instead of the measurement performed on Bob’s system and coding w.r.t. the classical channel , the decoder Charlie can directly simulate the measurement outcome using Winter’s measurement compression theorem [23, 24]. Denote Bob’s classical communication rate . Then Theorem 4 promises that by sending from Bob to the decoder Charlie, Charlie will have a local copy and the distribution between Alice and Charlie will satisfy (51).
Alice’s coding. Now Alice’s strategy is very simple since Charlie has had . She just uses the Slepian-Wolf coding strategy as if she starts with the distribution with Charlie. In fact, it is well known (cf. ) that there exists an encoder and a decoder such that and
for sufficiently large , where
is the set of correctably decodable pairs.
Now, suppose that Alice and Bob use the same code for the simulated distribution . Then, by the definition of the variational distance and (51), we have
Thus, if we can find a good code for , we can also use that code for for sufficiently large .
Derandomization. The standard derandomization technique works here. If the random coding strategy works fine on average, then there exists one realisation works fine too. Since the distribution , and
Thus, there exists one so that is small.
4 Fully Quantum Source Compression with a Quantum Helper
As shown in Figure 3, the protocol for fully quantum source coding with a quantum helper involves two senders, Alice and Bob, and one receiver, Charlie. Initially Alice and Bob hold copies of a bipartite quantum state , where Alice holds quantum systems while Bob (being a quantum helper) holds quantum systems . Moreover, there are pre-shared entangled states between Alice and Charlie, and pre-shared entangled states between Bob and Charlie. The goal is for the decoder Charlie to faithfully recover Alice’s quantum state when assisted by the quantum helper Bob.
We now proceed to formally define the coding procedure. Let be a purification of . We define an code for fully quantum source compression with a quantum helper to consist of the following:
Alice’s encoding operation , where is a quantum system and is a classical system; Alice only sends to Charlie;
Bob’s encoding operation , where is a quantum system of dimension ; Bob sends the quantum system to Charlie;
Charlie’s decoding operation that produces
so that the final state satisfies
where is a maximally entangled state.
Let . A rate pair is said to be achievable if for any and all sufficiently large , there exists an code with rates and . The rate region is then defined as the collection of all achievable rate pairs. Our main result is the following theorem.
Given is a bipartite quantum source . The optimal rate region for lossless source coding of with a quantum helper is the set of rate pairs such that
The state resulting from Bob’s application of some CPTP map is
4.1 Direct part
Relevant quantum protocols
Given a bipartite state whose purification is , the state merging protocol [8, 9, 10] is the information-processing task of distributing -part of the system that originally belongs to Alice to the distant Bob without altering the joint state. Moreover, Alice and Bob have access to pre-shared entanglement and their goal is to minimise the number of EPR pairs consumed during the protocol. The state merging can be efficiently expressed as the following RI:
where the notation denotes the state is originally shared between three distant parties Alice, Bob, and Eve, while means that the system is now together with system . This protocol involves classical communication; however, for the purpose of this paper, quantum resources are much more valuable and classical communication is considered to be free. As a result, the state merging protocol either consumes EPR pairs with rate when this quantity is positive, or generates rate of EPR pairs for later uses, if is negative, after the transmission of the system to .
The state merging protocol gives the first operational interpretation to the conditional von Neumann entropy. Most importantly, it provides an answer to the long-standing puzzle—the conditional von Neumann entropy could be negative, a situation that has no classical correspondence.
It is a simple exercise to show, via the resource inequalities, that the state merging protocol can be obtained by combining teleportation with the FQSW protocol [11, 13]. Moreover, the FQSW protocol can be transformed into the a version of the quantum reverse Shannon theorems (QRST) that involves entanglement assistance .
The quantum reverse Shannon theorem (QRST) addresses a fundamental task that asks, given a quantum channel , how much quantum communication is required from Alice to Bob so that the channel can be simulated. There are variants of the QRSTs depending on whether entanglement or feedback is allowed in the simulation (see [28, Theorem 3]). The QRST protocol has become a powerful tool in quantum information theory. It can be used to establish a strong converse to the entanglement-assisted capacity theorem. Moreover, it can also be used to establish quantum rate distortion theorems [18, 19, 20].
In this paper, we will use the QRST with entanglement assistance.
Theorem 6 (Quantum Reverse Shannon Theorem)
Let be a quantum channel from to so that its isometry results in the following tripartite state when inputting :
where . Then with sufficient amount of pre-shared entanglement, the channel with input can be simulated with quantum communication rate :
We use the channel simulation method. For any local channel performed by the quantum helper on his half of bipartite state , it can be simulated by the decoder using the quantum reverse Shannon theorem (QRST) (Theorem 6):
In other words, by using the pre-shared entanglement between the helper and the decoder with rate and sending quantum message from the helper to the decoder with rate , the decoder can simulate the quantum state locally with error goes to zero in the asymptotic sense.
Alice’s coding: Once the decoder has the system , then Alice and the decoder start the state merging protocol, using the pre-shared entanglement with rate .
4.2 Converse part
where we set and in the last equality, we relabel and .
To bound the quantum communication rate , we follow steps in the converse proof of the entanglement-assisted quantum rate-distortion theorem (see equation (21) in ):
Note that can be generated from via Bob’s local CPTP. In fact, Bob can first append the maximally entangled states , systems , and . Then, he can perform , and get .
5 Conclusion and Discussion
We considered the problem of compression of a classical source with a quantum helper. We completely characterised its rate region and showed that the capacity formula does not require regularisation, which is not common in the quantum setting. While the expressions for the rate region are similar to the classical result in [3, 4, 22], it requires vey different proof technique. To prove the achievability, we employed a powerful theorem, measurement compression theorem , that can decompose quantum measurement. A similar approach was recently applied to derive a non-asymptotic bound on the classical helper problem .
The rate region in our Theorem 5 bears a close resemblance to its classical counterpart. Our result also shows a helper’s strategy of simply compressing the side information and sending it to the decoder is sub-optimal with entanglement assistance. Recall the following identity:
where the state is given in (59). The QRST protocol allows us to cleverly divide the amount of quantum communication required for lossless transmission of system to the decoder into pre-shared entanglement with rate and quantum communication with rate .
We will like to point out that the definition of the fully quantum source compression with a quantum helper requires to explicitly include additional quantum systems (see Eq. (56)) for a technical purpose. The reason behind this is because when the quantum state merging is performed, the target systems to which the quantum state is merged needs to be specified. We believe that the inclusion of these additional systems in the definition is inevitable, and it signals a fundamental difference between the fully quantum source compression with a quantum helper and its classical counterpart.
Note that it is possible to replace the state merging protocol with the FQSW protocol, and derive an alternative theorem for quantum source compression with a quantum helper. It is also possible to consider the same problem without entanglement assistance between the helper and the decoder. These extensions will be treated in the future.
Finally, in the classical source coding with a helper problem, it is possible to bound the dimension for the helper’s output system. However, such a result is not unknown to be possible in the quantum regime.
MH is supported by an ARC Future Fellowship under Grant FT140100574. SW was supported in part by JSPS Postdoctoral Fellowships for Research Abroad.
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- The subscript is a label to which the quantum system belongs.
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