Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in $(0,1)$, we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an $m$-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm $\|ρ\|_{(m)}$, the sum of the $m$ largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever $m$ is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves $\log_2(e)-1$ bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to $2/e$.
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an m-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm ∥ρ∥(m), the sum of the m largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever m is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log2(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e.
Understanding the classical communication cost of simulating a quantum channel is a fundamental problem in quantum information theory, which becomes even more intriguing when considering the role of non-locality in quantum information processing. This paper investigates the bidirectional classical communication cost of simulating a bipartite quantum channel assisted by non-signalling correlations, which are permitted across the spatial dimension between the two parties. By introducing non-signalling superchannels, we present lower and upper bounds on the one-shot ε-error one-way classical communication cost of a bipartite channel via its smooth max-relative entropy of one-way classical communication, and establish that the asymptotic exact cost is given by its max-relative entropy of one-way classical communication. For the bidirectional scenario, we derive semidefinite programming (SDP) formulations for the one-shot exact bidirectional classical communication cost via non-signalling bipartite superchannels. We further introduce a channel's bipartite conditional min-entropy as an efficiently computable lower bound on the asymptotic cost of bidirectional classical communication. Our results in both one-shot and asymptotic settings provide lower bounds on the entanglement-assisted simulation cost in scenarios where entanglement is available to the two parties. Moreover, we propose a seesaw-based algorithm to compute an upper bound on the minimum simulation error via local operations and shared entanglement, which provides valuable insights into the relationship between non-signalling bipartite superchannels and more physically realizable protocols. Numerical experiments demonstrate the effectiveness of our bounds in estimating communication costs for various quantum channels, showing that our bounds can be tight in different scenarios.
This thesis consists of two parts: quantum compression and quantum learning theory. A common theme between these problems is that we study them through the lens of information theory. \n \nWe first study the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially higher communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. We show that an ensemble given by Jain, Radhakrishnan, and Sen (ICALP'03) cannot be compressed by more than a constant number of qubits without shared entanglement, while in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. \n \nNext, we study the task of quantum state redistribution, the most general version of compression of quantum states. We design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to connect quantum state redistribution and Markov chains and gives an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. \n \nIn the last part, we focus on quantum algorithms for learning Boolean functions using quantum examples. We consider two commonly studied models of learning, namely, quantum PAC learning and quantum agnostic learning. We reproduce the optimal lower bounds by Arunachalam and de Wolf (JMLR’18) for the sample complexity of either of these models using information theory and spectral analysis. Our proofs are simpler than the previous ones and the techniques can be possibly extended to similar scenarios.
The novel ways in which correlations may be encoded non-locally in quantum mechanics have led to the promise of new technologies. This, in turn, has led to the first attempts at the development of quantum networks. Nonetheless, our understanding of quantum correlations and the limits of how we may process them over a quantum network remains nascent. This thesis studies the limits of processing quantum correlations between spatially separated parties in various settings and the mathematical tools necessary to determine such limits. To do so, this thesis studies three general problems in quantum network theory. The first problem is the minimum resources to prepare a quantum correlation between non-communicative receivers, which may be viewed as a star network. This study is separated into three pieces. The first piece is a near-tight one-shot quantum generalization of a problem introduced by Wyner, distributed source simulation. Establishing our one-shot bounds builds on recent work on soft-covering without smoothing and relies on establishing a variety of properties for one-shot entropic quantities, as well as improving our understanding of the smooth entropy framework in quantum network information theory. As distributed source simulation cannot be achieved to arbitrary error for sufficiently entangled states, the second piece characterizes the minimal error of converting one entangled state to another using zero communication. This relies on establishing new properties of the fidelity measure. The third piece establishes an entanglement-assisted version of the distributed source simulation task that can be achieved to arbitrary error for arbitrary quantum systems. This result builds on recent work on multipartite state splitting and establishing new properties of one-shot multipartite max mutual information quantities. The second problem is the extraction of perfect classical correlation from a multipartite quantum source where the receivers do not communicate. This may be seen as the reverse process of distributed source simulation. We establish that this is asymptotically possible in quantum mechanics for the first time and establish the asymptotic rate for classical-quantum states. The technical barrier in establishing the asymptotic rate is establishing a limit on how much noiseless correlation can be extracted without communication. Our major technical contribution is establishing such a limit by generalizing a technical lemma of Gács and Körner to classical-quantum states. The final problem is distributed state discrimination of quantum product states. This is motivated by its relevance to both understanding non-locality in quantum mechanics and developing quantum cryptographic schemes. Our study again breaks into three key pieces. First, we study the framework of local operations and simultaneous classical or quantum communication (LOSCC and LOSQC respectively) for product states, which is relevant to quantum cryptography. We establish the first separation between quantum and classical communication in the simultaneous communication model using only product state inputs. We also establish a new uncertainty relation that implies error bounds for discriminating product states using LOSQC. Second, we study state discrimination under local operations and classical communication (LOCC). We develop a new framework for zero-error state discrimination under LOCC. Our framework provides a simple proof of orthogonal product states that cannot be distinguished using LOCC--- a property introduced by Bennett et al. Finally, we study the thermodynamics of zero-error distributed state discrimination in the framework of Faist et al. This allows us to resolve an open problem from the aforementioned paper of Bennett et al.
We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.
We determine the optimal success probability for transmitting a fixed number of classical messages through a finite number of uses of the quantum erasure channel, assisted by noiseless classical feedback and without initial shared entanglement. For an input dimension d, an erasure probability p, a blocklength n, and M equiprobable messages, the optimal success probability is equal to E[min{1,dK/M}], where K is binomial with parameters n and 1−p. The converse allows arbitrary adaptive quantum encoders, quantum memories, and receiver instruments. Its main ingredient is an elementary dimension bound for noiseless quantum communication with classical feedback, proved by fixing the classical controls without conditioning the sender's state on the observed transcript. A classical protocol that transmits the base-d digits of an integer representing the message, repeating each digit until the receiver acknowledges its reception or the prescribed blocklength is reached, attains the bound for every integer M. We also establish a relative-majorization property of erasure-channel outputs and exactly evaluate a hypothesis-testing converse, recovering the same numerical bound without feedback. That converse need not be achievable without feedback: four uses of the qubit erasure channel and four messages give a strict gap. A product-state code employing tetrahedral qubit states nevertheless outperforms every classical binary erasure code with these parameters. We give an explicit message-size formula and show that the bounded-remainder normal approximation and the average-success strong-converse exponent are unchanged without feedback. We also determine the feedback-assisted error exponent below capacity, prove that it agrees with the no-feedback exponent above a critical rate, and give no-feedback bounds at lower rates.
When a thermal environment damps the two transverse components of a qubit differently, can entangling signals across many channel uses improve the classical communication rate? This article gives a complete coding answer for an explicit phase-sensitive class and determines the optimal exponential cost of transmitting above capacity. Physical model and communication permissions. The channel acts as N(ρ0X∗Xρ1)=(aρ0+bρ1kX∗+ℓXkX+ℓX∗cρ0+dρ1). All coefficients are nonnegative, a+c=b+d=1, and the main class satisfies k≥ℓ≥0,ad−bc=k2−ℓ2,bc≥ℓ(k+ℓ). The class contains actual squeezed thermal reservoirs, including nonunital channels that are not entanglement breaking. The coding model is unassisted and memoryless, without feedback; arbitrary entangled block codewords and collective decoding are allowed. The coefficient condition is a sufficient region, not a classification of every physically allowed squeezed bath. An exact capacity from one-use data. Write h2 for binary entropy in bits, q for the excited-state input population, and define t(q)=c+(d−c)q,sx(q)=h2(21+[1−2t(q)]2+4(k+ℓ)2q(1−q)). With cosx the lower convex envelope on [0,1], the unrestricted capacity is C(N)=χ(N)=max0≤q≤1{h2[t(q)]−(cosx)(q)}. At most two opposite-real-phase signal pairs, hence at most four pure signals, attain the one-use optimum. The convex envelope retains optimization over all ensembles. Holevo information is additive with every finite-dimensional channel partner, so arbitrarily long entangled encodings cannot exceed this rate. The best success exponent above capacity. Let Pn⋆(R) be the optimal average success probability over all n-use codes with at least ⌈2nR⌉ messages. For R>C(N), Esc(R,N)=limn→∞−n1log2Pn⋆(R)=supα>1αα−1[R−χα(N)]. Here χα is the sandwiched Rényi information radius, reduced in the paper to a scalar reference/population optimization. It is strongly additive at every real α>1. Product codewords with collective decoding attain the asymptotic exponent, while the converse covers arbitrary entangled codewords. The finite-block inequality Pn⋆(R)≤2−nEsc is an upper bound, not a finite-block equality. Why the reduction works. Normal and reversed positive block factors retain the same quantum source and admit a common scalar Gram bound. This yields maximum-output norm multiplicativity with arbitrary completely positive partners and unnormalized diagonal filters. Binary phase symmetry then suffices to make the information-radius reference diagonal, even though continuous phase covariance is absent. The same input-filter comparison also determines the formation cost of the specified diagonal-input Stinespring states across the output and the full purified environment. Relation to existing results. We independently obtained the additivity and strong-converse results for the axially symmetric Davies family that are also established by L. Eriksson, Capacities, strong-converse exponents and complementary simulation for qubit Davies channels, version 3, 26 October 2026. These results and their proofs are retained as a reference case. The phase-sensitive class and the full axial Davies class share the generalized-amplitude-damping boundary; neither is presented as a strict enlargement of the other. Fang's triangular norm-compression theorem and the general information-radius and coding theorems retain their attribution. The 16-page integrated article contains the full proofs, two figure groups and references. The figures separate the physical parameter region, the globally evaluated capacity formula and explicit finite-block success upper bounds.