One-shot entanglement assisted classical and quantum communication over noisy quantum channels: A hypothesis testing and convex split approach
Capacity of a quantum channel characterizes the limits of reliable communication through a noisy quantum channel. This fundamental information theoretic question is very well studied specially in the setting of many independent uses of the channel. An important scenario, both from practical and conceptual point of view, is when the channel can be used only once. This is known as the one-shot channel coding problem. We provide a tight characterization of the one-shot entanglement assisted classical capacity of a quantum channel. We arrive at our result by introducing a simple decoding technique which we refer to as position-based decoding. We also consider two other important quantum network scenarios: quantum channel with a jammer and quantum broadcast channel. For these problems, we use the recently introduced convex split technique  in addition to position based decoding. Our approach exhibits that the simultaneous use of these two techniques provides a uniform and conceptually simple framework for designing communication protocols for quantum networks.
A classical description of our world entails several limitations on what can be achieved physically. Law of conservation of energy prevents energy to be created out of nothing, thermodynamics disallows machines with efficiency beyond the Carnot’s limit, inertia restrains motion when there is no force as a motive. These limitations have been so pivotal in the scientific revolution that they can now be found even in the laws of information (Landauer’s principle , Shannon’s capacity theorem ) and computation (Turing’s halting theorem , P vs NP conjecture ). Their knowledge allows us to optimize our efforts as we seek the best possible results.
The theory of quantum information and computation, aided with the power of entanglement, opens up new possibilities. Bell’s landmark theorem  tells us that quantum systems possess correlations that go beyond those achievable by classical means. Shor’s algorithm  shows how a quantum computer can perform integer factoring exponentially faster than known classical algorithms. Quantum cryptography offers protocols which achieve information theoretic security in the task of key distribution . As these results begin to point to a physical reality that surpasses some well known boundaries in classical physics and computing, a fundamental technological limitation is brought upon us, quite ironically, by quantum entanglement itself. This is the limitation imposed by quantum noise.
Quantum noise, also known as a quantum channel, describes the process by which a quantum particle (possessed by an experimenter) gets correlated or entangled with the environment (upon which experimenter has no control). This can be particularly unsuitable when two experimenters wish to send messages to each other and the intermediate channel has noisy behavior. Efforts to understand and mitigate quantum noise have largely developed on two fronts: communication through a quantum channel (starting from the work of Holevo  and Schumacher and Westmoreland ) and quantum error correction (starting from the work of Shor ).
Here, we consider the case of communication through a quantum channel, and more specifically, the entanglement assisted classical capacity of a quantum channel. Entanglement assistance, a widely used terminology for communicating with the help of entanglement shared between Alice (sender) and Bob (receiver), leads to two very important protocols in quantum information theory. Quantum teleportation  allows Alice to send a qubit to Bob using two bits of classical communication and superdense coding  allows Alice to send two bits of message with one qubit. These two protocols strongly suggest that the presence of entanglement (upon which Alice and Bob have full control) can play important role in the process of reliable communication through a quantum channel.
Bennett et al.  characterized the limits of classical communication over a noisy quantum channel when the sender and receiver share entanglement between them. They studied the case where Alice was allowed to use the channel arbitrarily many times and after each use, the channel had no memory of this use. In practice however there could be several issues, for example the channel between Alice and Bob may not be memoryless and Alice may even be forced to use the channel only once (this has been a driving force behind the emerging field of one-shot information theory). Often there are more than one sender and receiver. For example, a quantum satellite may be beaming back information simultaneously to different base stations on earth, and these base stations may have no way of reliable collaboration between themselves. Sometimes the receiver may not have a complete knowledge of the channel characteristics, such as in the case of a quantum communication channel with an adversary or a jammer.
We consider each of the scenarios mentioned above and provide a unified approach for designing communication protocols for them. We use two ingredients in our protocols: the technique of position based decoding that we introduce for the protocol described in Figure 2, and the technique of convex split (introduced in , discussed in our context for the protocols described in Figures 4 and 6). Position based decoding (where the term decoding refers to the strategy performed by the receiver) allows the receiver to accomplish the task of quantum hypothesis testing. In a communication protocol between Alice and Bob, as Alice sends messages to Bob through the channel, different quantum states are formed on Bob’s side as a function of the message Alice has sent. Bob, who does not know the message, should be able to distinguish between these quantum states in order to learn the message. A simplification of this problem is the task of quantum hypothesis testing, where Bob should be able to distinguish between two possible quantum states with small error. Position based decoding allows Bob to distinguish between many possible quantum states that may arise from Alice’s messages, if he is able to distinguish between two given quantum states.
Point to point case: The first protocol we design concerns point to point quantum channels, where there are two parties Alice (sender) and Bob (receiver). Alice, who is allowed to use the channel only once, wants to communicate message chosen with some a priori distribution from the set to Bob across the quantum channel such that Bob is able to guess the correct message with probability at least ( is a small constant). This we refer to as an entanglement assisted code for the quantum channel . The goal here is to determine largest possible value of (the amount of reliable communication in bits between Alice and Bob). Figure 2 gives a schematic of our protocol for this scenario. We show that the largest possible value of is quantified in terms of the hypothesis testing divergence. Given two quantum states and , the hypothesis testing divergence captures the probability that an experimenter, who only wishes to accept , ends up accepting . Formally, it is defined as , where is a positive operator. Using this quantity, our main theorem is as follows, which is discussed in detail in Section 3.
Let be the quantum channel and let . Let be a purifying register. Then, for any smaller than
there exists an entanglement assisted code for the quantum channel
Outline of the protocol: Fix a quantum state Alice and Bob share independent copies of the state where the register is held by Alice and the register is held by Bob. Each of these copies are uniquely assigned to a message This assignment is known to both Alice and Bob. To send the message Alice transmits her part of the -th copy of the shared state over the channel. Notice that at the end of this transmission the joint state between the -th register of Bob and the channel output is and the joint state for every other register and the channel output is Thus, if Bob is equipped with a binary measurement (obtained from the definition of ) which can differentiate the state from , then he can design his (multiple outcome) decoding measurement as follows. His measurement operator corresponding to the outcome is , where acts only on the channel output and the -th copy of and is the identity operator on the rest of Bob’s registers. We term this decoding strategy as position based decoding. Our protocol discussed above guarantees that Alice can communicate with Bob bits. This rate is also near optimal, owing to the converse bound shown in .
Resource utilization: The number of qubits of entanglement required in the above one-shot protocol is quite large, and in the asymptotic and i.i.d. setting it can grow exponentially in the number of channel uses. In order to reduce the number of qubits of the shared entanglement, we make two observations. First is that our one-shot protocol consumes only one copy of the shared entanglement and returns the rest with very small error. Thus, a large part of the shared entanglement serves as a catalyst. The second observation, motivated by the work  and made precise in Theorem 3 (Section 4) below, is that the entanglement can be efficiently consumed by encoding the messages in the sets of positions (instead of just one position). Both observations are used in Section 4 to construct an appropriate asymptotic and i.i.d. version of the position-based decoding. This leads to a protocol that has the same rate of communication and the rate of required entanglement as the protocol constructed in .
Gel’fand-Pinsker Channel: Our second protocol concerns communication in the presence of a malicious jammer, where Alice is aware of this jammer, whereas Bob has no information about this jammer. This model was analyzed in the classical case by Gel’fand-Pinsker in their seminal work . The formal setting in the quantum case is as follows (see, for example, ): Alice shares an entangled state with the channel itself, where the register is held by Alice and the register is held by the channel. Unlike in the point to point case, the channel (represented by ) takes as input both and . Alice wants to communicate message chosen from the set to Bob across the quantum channel It is quite natural to expect that because of the absence of the knowledge of register at Bob’s side, the value of (the amount of reliable communication in bits between Alice and Bob) will be smaller than the one achieved for the point to point channel. A schematic of our protocol for this task is presented in Figure 4 and details appear in Section 5.
Outline of the Protocol: Fix a state such that At the start of the protocol Alice and Bob share independent copies of the state where the register is held by Alice and by Bob ( is a purification of ). These copies are subdivided into bands of equal size . There is a unique band for each message To send the message , Alice creates the state (close to) in the register in her possession, the register with the jammer and a random register in the band , using the convex split technique (along with Uhlmann’s theorem) . Alice then transmits the register over the channel. Now, using position based decoding, Bob is able to decode the correct message with high probability. Thus, Alice is able to communicate bits to Bob.
Quantum Broadcast Channel: The final case that we consider is that of quantum broadcast channel studied in the classical case (among others) by Marton in her seminal work . Here, Alice wishes to communicate message to Bob and message to Charlie simultaneously. While Bob and Charlie are not allowed to collaborate with each other, the noisy channel may give correlated output to them, which makes the setting different from two independent cases of point to point channel. The channel takes input from Alice and produces outputs (with Bob) and (with Charlie).
Our protocol for this task is again based on similar framework of using convex split technique and position based decoding. Convex split technique is used by Alice to establish an appropriate correlated state between Bob, Charlie and the channel output, following which Bob and Charlie perform position based decoding on their respective shares of this correlated state. A schematic of our protocol is discussed in Figure 6 and details appear in Section 6
Outline of the Protocol: Fix a state At the start of the protocol Alice and Bob share
independent copies of the state where the register is held by Alice and by Bob ( is a purification of ). These copies are subdivided into bands of equal sizes, where each band is uniquely assigned to a message Similarly, Alice and Charlie share
independent copies of the state where the register is held by Alice and by Charlie ( is a purification of ). These copies are subdivided into bands of equal sizes where each band is uniquely assigned to a message The constraint on the band size is that for every we have where is the band corresponding to the message and likewise for the message . To send the message pair Alice uses the convex split technique (along with Uhlmann’s theorem) to prepare the state , where register is held by Alice, register is a random register in held by Bob and register is a random register in held by Charlie. Alice transmits her share of the state over the quantum channel On receiving their respective shares of the channel output, Bob and Charlie employ the position based decoding to output their respective messages.
Comparision to previous works
These tasks have been studied previously in classical and quantum one-shot and asymptotic settings. The works [20, 18, 21] obtained a bound for point-to-point entanglement assisted quantum channel. However, their bounds do not match the converse result obtained in . The quantum Gel’fand-Pinsker channel and quantum broadcast channel were studied in  where they obtained one-shot bounds different from ours (their bounds and our bounds converge in the asymptotic i.i.d case). An important feature of our one-shot bounds is that their forms bear close resemblance to the known results in the classical and classical-quantum settings, for example, for the point-to-point channel , broadcast channel  and Gelf’and-Pinsker channel [24, 25]. Such is not the case with the bounds obtained in the aforementioned works on one-shot entanglement assisted quantum capacities. Another important point is that most of the previous works including [20, 18] used the technique of decoupling through random unitaries to obtain their bounds, which is different from our techniques.
Classical analogues of our proof techniques of convex-split and position-based decoding have recently been presented in . Using these, we can obtain analogous results for classical versions of all the tasks considered in this paper. In the classical case, it is in fact possible to remove shared randomness by standard derandomization arguments (in the setting of average error for a prior distribution over the messages, instead of worst case error).
Consider a finite dimensional Hilbert space endowed with an inner product (in this paper, we only consider finite dimensional Hilbert-spaces). The norm of an operator on is and norm is . A quantum state (or a density matrix or a state) is a positive semi-definite matrix on with trace equal to . It is called pure if and only if its rank is . A sub-normalized state is a positive semi-definite matrix on with trace less than or equal to . Let be a unit vector on , that is . With some abuse of notation, we use to represent the state and also the density matrix , associated with . Given a quantum state on , support of , called is the subspace of spanned by all eigen-vectors of with non-zero eigenvalues.
A quantum register is associated with some Hilbert space . Define . Let represent the set of all linear operators on . Let represent the set of all positive semidefinite operators on . We denote by , the set of quantum states on the Hilbert space . State with subscript indicates . If two registers are associated with the same Hilbert space, we shall represent the relation by . Composition of two registers and , denoted , is associated with Hilbert space . For two quantum states and , represents the tensor product (Kronecker product) of and . The identity operator on (and associated register ) is denoted . For any operator on , we denote by the subspace spanned by non-negative eigenvalues of and by the subspace spanned by negative eigenvalues of . For a positive semidefinite operator , the largest and smallest non-zero eigenvalues of are denoted by and , respectively.
Let . We define
where is an orthonormal basis for the Hilbert space . The state is referred to as the marginal state of . Unless otherwise stated, a missing register from subscript in a state will represent partial trace over that register. Given a , a purification of is a pure state such that . Purification of a quantum state is not unique.
A quantum map is a completely positive and trace preserving (CPTP) linear map (mapping states in to states in ). A unitary operator is such that . An isometry is such that and . The set of all unitary operations on register is denoted by .
We shall consider the following information theoretic quantities. Reader is referred to [27, 28, 29, 30, 31] for many of these definitions. We consider only normalized states in the definitions below. Let .
Fidelity For ,
For classical probability distributions ,
Purified distance For ,
-ball For ,
Von-Neumann entropy For ,
Relative entropy For such that ,
Relative entropy variance For such that ,
Max-relative entropy For such that ,
Smooth max-relative entropy For such that ,
Smooth min-relative entropy For ,
Information spectrum relative entropy For such that ,
Information spectrum relative entropy [Alternate definition] For such that ,
Max-information For , define
Smooth max-information For , define
Smooth max-information [Alternate definition] For , define
Restricted smooth max-information For , define
We will use the following facts.
Fact 1 (Triangle inequality for purified distance, ).
For states ,
For quantum states , , and quantum operation , it holds that
In particular, for bipartite states , it holds that
Fact 3 (Uhlmann’s Theorem, ).
Let . Let be a purification of and be a purification of . There exists an isometry such that,
Fact 4 (Pinsker’s inequality, ).
For quantum states ,
Fact 5 (Alicki-Fannes inequality, ).
Given bipartite quantum states , and , it holds that
Fact 6 (Triangle property of smooth max- relative entropy).
For , it holds that
Let , which implies that . Let be the state achieving the infimum in . Then . This implies that , which concludes the fact using the inequality .
Let be a purification of . Then is a purification of . Now, applying monotonicity of fidelity under quantum operations (Fact 2), we find
In last inequality, we have used . ∎
Fact 8 (Hayashi-Nagaoka inequality, ).
Let be positive semi-definite operators. Then
For the function and , it holds that .
Thus, , which completes the proof. ∎
Following fact says that if a collection of quantum operations do not change a given state much, then successive application of them brings limited change.
Fact 11 (Fact 21, ).
Let be a quantum state and be a collection of quantum maps. Define a series of quantum states recursively as . It holds that
We shall also need the following series of results, that are central to our achievability approach.
Fact 12 ( ).
Let be quantum states and be a probability distribution. Let be the average state. Then
Let and be quantum states. Then, for every let be an operator,
Let be such that and Thus, using monotonicity of fidelity we have
where the last inequality follows because of the following:
where the inequality above follows because The claim of the Lemma now follows from (1) and the relation between the purified distance and fidelity between two quantum states. ∎
Lemma 2 (Convex-split lemma,).
Let and be quantum states such that . Let . Define the following state
on registers , where and . Then for and ,
We have the following corollary of above lemma.
Corollary 1 (Corollary of convex-split lemma).
For an . Let and be quantum states such that . Let . Define the following state
on registers , where and . For and ,
We will use the following new version of the convex split lemma.
Lemma 3 (Bi-partite convex-split lemma).
Let be a quantum state, and . Choose integers and define the following state