Entanglement-Assisted RACs

• Entanglement-Assisted Random Access Codes (EARACs) are protocols where classical messages are augmented by shared quantum entanglement to boost the probability of retrieving desired data bits.
• They employ both non-adaptive and adaptive measurement strategies, leveraging quantum correlations to outperform classical schemes and standard QRACs in various settings.
• Recent studies show that adaptivity, entanglement, and measurement incompatibility jointly enhance performance in distributed and multiparty communication tasks.

Searching arXiv for primary and closely related papers on entanglement-assisted random access codes. Entanglement-assisted random access codes (EARACs) are random access code protocols in which pre-shared entanglement supplements constrained communication so that a receiver can recover a requested component of distributed or locally held classical data with higher success probability than is achievable classically under the same communication budget. In the formulations most commonly studied, Alice holds a classical string, Bob receives a query specifying which bit or function value is to be recovered, and Alice sends only a short classical message while both parties exploit correlations generated by a shared entangled state. Across the literature, EARACs appear in several closely related guises: standard bipartite $n\to 1$ RACs, distributed and multiparty RACs on communication networks, generalized RACs in which Bob requests Boolean functions rather than individual bits, and adaptive entanglement-assisted communication protocols in which Bob’s measurement is chosen after receiving Alice’s message. These variants reveal that entanglement can improve RAC performance relative to classical RACs, but also that the precise advantage depends on message alphabet, network topology, query structure, and whether Bob’s measurements are non-adaptive or adaptive (Pauwels et al., 2022, Hameedi et al., 2017, M et al., 2021, Saha et al., 2019).

1. Core definitions and operational models

A standard random access code asks Alice to encode a classical input into a communication-constrained message such that Bob, after receiving a query, can recover a designated component with some success probability. In the $2\to1$ RAC emphasized in adaptive EARAC work, Alice holds $x=(x_1,x_2)\in\{00,01,10,11\}$, Bob receives $y\in\{1,2\}$, and Bob outputs $b\in\{0,1\}$ intended to equal $x_y$. With message alphabet size $D$, the average success probability under uniform inputs is

$$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$

For the unassisted classical benchmark, the optimal values recalled in the literature are $p_{\text{win}^{\text{bit}} = \frac34$ for a single bit ($D=2$) and $2\to1$0 for a single trit ($2\to1$1) (Pauwels et al., 2022).

In an EARAC, Alice and Bob share an entangled state $2\to1$2. The most general one-way entanglement-assisted classical communication model used for RACs is specified by Alice’s POVM $2\to1$3, where the outcome $2\to1$4 is the classical message, and Bob’s POVM $2\to1$5, which may depend on both his query $2\to1$6 and the received message $2\to1$7. The induced correlations are

$2\to1$8

For the explicit $2\to1$9 EARAC constructions, the shared state is the Bell state

$x=(x_1,x_2)\in\{00,01,10,11\}$0

with qubit observables represented on the Bloch sphere (Pauwels et al., 2022).

A different but standard EARAC model appears in generalized RACs and Boolean-function RACs. There, Alice sends a single classical bit $x=(x_1,x_2)\in\{00,01,10,11\}$1, generated by measuring her half of a shared state $x=(x_1,x_2)\in\{00,01,10,11\}$2, and Bob performs a binary POVM depending on both the query and $x=(x_1,x_2)\in\{00,01,10,11\}$3. For $x=(x_1,x_2)\in\{00,01,10,11\}$4-generalized RACs, where Bob wants a Boolean function $x=(x_1,x_2)\in\{00,01,10,11\}$5, the optimal entanglement-assisted success probability is defined by maximizing

$x=(x_1,x_2)\in\{00,01,10,11\}$6

over shared states and local POVMs (M et al., 2021).

These definitions isolate two invariant features of EARACs. First, the communicated object is classical, unlike prepare-and-measure QRACs, where Alice sends a qubit or qudit. Second, the entanglement resource induces nonclassical conditional ensembles on Bob’s side, so that a short message can be interpreted as steering information that helps select a decoding measurement (Pauwels et al., 2022, Doriguello et al., 2020).

2. Non-adaptive and adaptive entanglement assistance

A central structural distinction is between non-adaptive and adaptive EARAC protocols. In the non-adaptive paradigm, which corresponds to the usual “Bell test followed by classical communication” picture, Bob must measure his subsystem before learning Alice’s message. Operationally, the protocol has two stages: Alice and Bob each choose measurements from their local inputs, then Alice sends her measurement outcome or a function of it, and Bob post-processes his already obtained result. In the $x=(x_1,x_2)\in\{00,01,10,11\}$7 setting this is formalized by requiring that Bob’s effective measurements for different possible messages arise from a single prior measurement plus classical post-processing. The induced correlations can be written as

$x=(x_1,x_2)\in\{00,01,10,11\}$8

which is equivalent to the commutativity constraint

$x=(x_1,x_2)\in\{00,01,10,11\}$9

The commutativity condition expresses joint measurability of Bob’s “virtual” measurements associated with different messages (Pauwels et al., 2022).

Adaptive EARACs remove this restriction. Bob waits for Alice’s classical message and only then chooses a POVM $y\in\{1,2\}$0. No commutativity requirement is imposed between measurements corresponding to distinct $y\in\{1,2\}$1. The operational consequence is that Bob may use incompatible measurements conditioned on the branch of the protocol, thereby tailoring his decoding to the message-conditioned ensemble prepared by Alice. This possibility does not correspond to a standard Bell-test scenario because Bob’s measurement basis is influenced by Alice’s outcome-dependent communication (Pauwels et al., 2022).

The sharpness of this distinction becomes visible in RAC performance. For $y\in\{1,2\}$2, adaptivity is provably irrelevant in the $y\in\{1,2\}$3 RAC: the CHSH-based EARAC already attains the optimum

$y\in\{1,2\}$4

and this value is optimal over all entanglement-assisted bit protocols, adaptive or not (Pauwels et al., 2022). For $y\in\{1,2\}$5, however, adaptivity is strictly advantageous. This marks a qualitative transition in the resource theory of EARACs: the timing of Bob’s measurement relative to Alice’s message is itself a nontrivial operational resource (Pauwels et al., 2022).

3. Canonical $y\in\{1,2\}$6 EARACs and the adaptive trit separation

The canonical non-adaptive $y\in\{1,2\}$7 EARAC with one classical bit is CHSH-based. Alice measures the observable

$y\in\{1,2\}$8

obtains $y\in\{1,2\}$9, and sends $b\in\{0,1\}$0. Bob measures $b\in\{0,1\}$1 if $b\in\{0,1\}$2 and $b\in\{0,1\}$3 if $b\in\{0,1\}$4, then outputs $b\in\{0,1\}$5, where $b\in\{0,1\}$6 is his measurement outcome. This achieves

$b\in\{0,1\}$7

strictly above the classical one-bit value $b\in\{0,1\}$8 (Pauwels et al., 2022).

The trit case $b\in\{0,1\}$9 exhibits richer behavior. A simple non-adaptive EA-trit protocol augments the message on the branch $x_y$0: if $x_y$1, Alice sends $x_y$2; if $x_y$3, she sends $x_y$4. Bob uses the same quantum measurement as in the EA-bit protocol, except that when $x_y$5 and $x_y$6, he ignores the quantum outcome and outputs the $x_y$7 value explicitly contained in the trit. This gives

$x_y$8

Optimizing Alice’s observable to

$x_y$9

with $D$0, yields an improved non-adaptive protocol with

$D$1

Using NPA-type semidefinite relaxations incorporating the non-adaptive commutativity constraints, all non-adaptive EA-trit protocols are bounded by

$D$2

Thus non-adaptive EA trits outperform both classical trits $D$3 and EA bits, but only up to a strict ceiling (Pauwels et al., 2022).

The adaptive separation is realized by an explicit EA-trit protocol that reuses the CHSH measurements but changes the message branching. Alice again measures

$D$4

obtains $D$5, and sends $D$6 if $D$7, while on the $D$8 branch she sends

$D$9

If $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$0, Bob behaves exactly as in the EA-bit protocol. If $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$1, Bob chooses a measurement depending on both $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$2 and $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$3; specifically, there exists a measurement direction $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$4 such that, on this branch, he can perfectly infer Alice’s relevant bit. The resulting success probability is

$$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$5

A second NPA optimization without non-adaptive commutation constraints shows this value is optimal over all EA-trit protocols (Pauwels et al., 2022).

The resulting hierarchy for the $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$6 RAC is therefore: $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$7 with the two middle inequalities interpreted more carefully as “EA bit is below classical trit” and “adaptive EA trit exceeds the non-adaptive EA-trit bound.” The strict gap $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$8 isolates a genuine adaptive advantage in entanglement-assisted RACs (Pauwels et al., 2022).

4. Distributed and multiparty EARACs

EARACs also arise in networked settings where the input bits are distributed among multiple devices. In distributed $$p_{\rm win} = \frac{1}{8} \sum_{x_1,x_2,y} p\bigl(b = x_y \,\big|\, (x_1,x_2),y\bigr).$$9 RACs, three devices are arranged linearly: device 1 receives $p_{\text{win}^{\text{bit}} = \frac34$0, device 2 receives $p_{\text{win}^{\text{bit}} = \frac34$1, and device 3 receives query $p_{\text{win}^{\text{bit}} = \frac34$2 and outputs $p_{\text{win}^{\text{bit}} = \frac34$3. Only one 2-dimensional system is sent from device 1 to device 2 and one 2-dimensional system from device 2 to device 3. In the fully classical distributed case with independent target functions $p_{\text{win}^{\text{bit}} = \frac34$4, the optimal success probability is

$p_{\text{win}^{\text{bit}} = \frac34$5

Both distributed QRACs and distributed EARACs can reach the quantum benchmark

$p_{\text{win}^{\text{bit}} = \frac34$6

for certain tasks, but not for the same tasks (Hameedi et al., 2017).

The distributed EARAC construction uses a tripartite GHZ state

$p_{\text{win}^{\text{bit}} = \frac34$7

For the task $p_{\text{win}^{\text{bit}} = \frac34$8, Alice measures her qubit in a basis determined by

$p_{\text{win}^{\text{bit}} = \frac34$9

sends $D=2$0, Bob measures in a basis with angle

$D=2$1

sends $D=2$2, and Charlie measures in one of the $D=2$3 bases depending on $D=2$4, then outputs $D=2$5. The GHZ decomposition ensures that Charlie’s conditional state is exactly one of the cube-vertex states used in the optimal $D=2$6 QRAC, so the distributed EARAC achieves $D=2$7 for this task (Hameedi et al., 2017).

A striking complementarity emerges. For four tasks associated with reflections of the cube of Bloch vectors, EARAC achieves $D=2$8 while distributed QRACs are strictly suboptimal; for instance, for $D=2$9, the QRAC success is approximately $2\to1$00, below $2\to1$01. For four other tasks associated with rotations, QRAC achieves $2\to1$02 whereas EARAC cannot. The corresponding almost-quantum upper bounds on EARAC are $2\to1$03 or $2\to1$04, depending on the task, while QRAC retains $2\to1$05 (Hameedi et al., 2017). This establishes a task-dependent complementarity rather than a global ordering between EARAC and QRAC in distributed $2\to1$06 scenarios.

Multiparty EARACs further generalize the model to chains of parties. In the multiparty $2\to1$07-RAC, the $2\to1$08 bits are distributed among $2\to1$09 encoding parties arranged linearly, and the last party $2\to1$10 must recover a requested bit. Two quantum communication scenarios are compared: entanglement-assisted classical communication and sequential quantum communication. In the EARAC scenario, the construction is based on concatenating primitive EARACs $2\to1$11 with success probability $2\to1$12. If a target bit lies at level $2\to1$13 of the concatenation tree, the overall success becomes

$2\to1$14

Using the two-bit Bell-state primitive $2\to1$15, the average success probability for $2\to1$16-EARAC is

$2\to1$17

Using the three-bit GHZ-state primitive $2\to1$18, for odd $2\to1$19 one obtains

$2\to1$20

which is strictly better than the Bell-based construction (Saha et al., 2019).

These multiparty protocols outperform classical zigzag AND/OR strategies. For $2\to1$21, the optimal classical value reported is

$2\to1$22

whereas the entanglement-assisted constructions achieve higher values. The broader implication is that entanglement-assisted classical communication remains advantageous even under severe per-link communication constraints and linear network topology (Saha et al., 2019).

5. Generalized RACs, Boolean-function retrieval, and analytic characterizations

EARACs are not limited to recovering individual input bits. One important generalization replaces the query “retrieve $2\to1$23” by “retrieve $2\to1$24” for a family of Boolean functions. In $2\to1$25-GRACs, Bob’s query is a vector $2\to1$26, and the target is

$2\to1$27

The function families are chosen to be mutually unbiased balanced sets (MUBS), meaning that each $2\to1$28 is balanced and any pair of distinct functions is mutually unbiased in the sense that exact knowledge of one yields no information about the other under the uniform input distribution. This condition mirrors the information-theoretic role of mutually unbiased bases in QRAC theory and underlies analytic upper bounds for quantum and entanglement-assisted performance (M et al., 2021).

For three input bits, the seven canonical functions are

$2\to1$29

The optimal EACC values obtained numerically via see-saw SDP and NPA relaxations satisfy

$2\to1$30

For $2\to1$31, a subtle separation appears. If the chosen four functions satisfy the parity symmetry $2\to1$32, then classical, QRAC, and EACC strategies all attain $2\to1$33. If not, the classical optimum drops to $2\to1$34, the prepare-and-measure qubit optimum becomes

$2\to1$35

and the EACC optimum remains

$2\to1$36

Thus generalized EARACs can separate from qubit QRACs already with only three input bits, unlike the standard RAC setting where the usual EACC-over-QRAC separation appears only later (M et al., 2021).

A related analytic generalization is the $2\to1$37-EARAC, where Bob requests the value of a Boolean function $2\to1$38 on any $2\to1$39-subset of Alice’s $2\to1$40 bits. In this setting, the success probability is governed by the noise stability of $2\to1$41. For $2\to1$42-correlated inputs, the noise stability is

$2\to1$43

If $2\to1$44 and $2\to1$45, there exists an $2\to1$46 $2\to1$47-EARAC with

$2\to1$48

This follows from partitioning the input into $2\to1$49 blocks and using the optimal one-bit EARAC primitive on each block, then decoding $2\to1$50 from independently noisy copies of the queried bits (Doriguello et al., 2020). The dependence on $2\to1$51 gives a Fourier-analytic characterization of EARAC performance. For parity, $2\to1$52; for majority, the bias is controlled by the majority noise-stability asymptotics. This suggests that the intrinsic noise sensitivity of the queried Boolean function, rather than only the communication rate, determines how much entanglement assistance can help (Doriguello et al., 2020).

6. Resource comparisons, contextuality, and related communication models

EARACs play an important role in comparing communication resources. A key result from adaptive entanglement-assisted communication is that once Bob may measure adaptively, an entanglement-assisted classical bit becomes strictly stronger than a qubit in prepare-and-measure scenarios. The argument has two components. First, any prepare-and-measure qubit protocol with arbitrary extremal qubit states $2\to1$53 and extremal POVMs

$2\to1$54

can be simulated by an adaptive EA-bit protocol using $2\to1$55, where Alice measures in direction $2\to1$56, sends $2\to1$57, and Bob performs the POVM

$2\to1$58

Second, there exist non-adaptive EA-bit correlations that no qubit protocol can simulate. Combining both directions yields the conclusion that an EA bit is a strictly stronger resource than a qubit in one-shot prepare-and-measure tasks (Pauwels et al., 2022).

EARAC-related protocols also appear in finite-length entanglement-assisted classical coding over noisy quantum channels. Mixed-alphabet Reed–Solomon constructions with parameters

$2\to1$59

model a channel used $2\to1$60 times, of which $2\to1$61 uses exploit superdense coding and shared maximally entangled $2\to1$62-dimensional pairs. In the optimized regime

$2\to1$63

for appropriate parameter ranges, while a quantum Singleton-type converse gives

$2\to1$64

Although these constructions are not RACs themselves, they provide explicit entanglement-assisted classical “pipes” suitable as lower layers for robust EARAC implementations over noisy quantum channels (Prasad et al., 2023). A plausible implication is that the asymptotic RAC viewpoint and the finite-blocklength coding viewpoint can be merged by treating EARACs as local-decodability layers above entanglement-assisted error-correcting transmission.

Another conceptual link connects RAC advantage to contextuality. A 2026 study formulates $2\to1$65 RACs using intraparticle entanglement between path and spin (or polarization) degrees of freedom of a single particle. In that setting, the average success probabilities satisfy linear relations

$2\to1$66

where $2\to1$67 are Bell-type noncontextuality parameters. For $2\to1$68, the maximal quantum value $2\to1$69 yields

$2\to1$70

and for $2\to1$71, $2\to1$72 yields

$2\to1$73

This reproduces the standard EARAC/QRAC optima and ties RAC advantage quantitatively to contextuality rather than exclusively to spatial nonlocality (Saha et al., 13 May 2026). Since this work uses a single-particle implementation rather than the standard bipartite EARAC model, it is best interpreted as a structurally analogous realization rather than a replacement for the canonical EARAC framework.

A distinct sequential setting considers an entanglement-assisted sequential $2\to1$74 QRAC with two decoders, Bob and Charlie, who act successively on the same subsystem of an entangled pair while sharing Alice’s one-bit message. The average success probabilities are

$2\to1$75

$2\to1$76

with $2\to1$77 and $2\to1$78. By relaxing equal-sharpness and mutually unbiasedness assumptions, the double-violation region $2\to1$79 and $2\to1$80 expands from a one-dimensional interval to a two-dimensional triangular region in $2\to1$81-space, and experimentally both values can exceed $2\to1$82 even for nearly projective measurements $2\to1$83 (Xiao et al., 2021). This setting is not a standard one-receiver EARAC, but it reinforces the role of measurement incompatibility and entanglement in RAC-like information extraction.

7. Conceptual themes, misconceptions, and research directions

Several conceptual themes recur across the EARAC literature. One is that entanglement assistance does not automatically dominate prepare-and-measure quantum communication in every RAC variant. In distributed $2\to1$84 RACs, QRAC and EARAC are complementary rather than globally ordered: EARAC excels on reflection-based tasks, QRAC on rotation-based tasks (Hameedi et al., 2017). Another is that the amount of communication alone does not determine performance. In the adaptive $2\to1$85 trit RAC, the adaptive and non-adaptive protocols use the same trit alphabet, and the performance gap arises entirely from Bob’s ability to postpone measurement and thereby choose incompatible observables conditioned on Alice’s message (Pauwels et al., 2022).

A common misconception is to identify all entanglement-assisted RACs with Bell-test-based non-adaptive schemes. That identification is too narrow. Non-adaptive EARACs do coincide with a Bell-test-first paradigm, but adaptive EARACs do not, and their stronger performance comes from exploiting message-conditioned incompatible measurements (Pauwels et al., 2022). A related misconception is that entanglement assistance and QRACs become interchangeable at small sizes. While $2\to1$86 and $2\to1$87 optima often coincide numerically between standard QRAC and standard EARAC, generalized function-retrieval tasks and distributed network tasks provide explicit early separations (M et al., 2021, Hameedi et al., 2017).

The literature also suggests several research directions. One concerns optimality and converse bounds beyond currently solved cases. Adaptive EA-trit optimality is known for the $2\to1$88 task (Pauwels et al., 2022), but broader adaptive RAC hierarchies remain largely unexplored. Another concerns higher-dimensional alphabets and qudit-based entanglement assistance. Dense-coding-like QRAC with shared entanglement has explicit lower bounds for $2\to1$89, with worst-case success probabilities $2\to1$90, $2\to1$91, and $2\to1$92, respectively, against trivial baselines $2\to1$93, $2\to1$94, and $2\to1$95, but general optimal constructions and upper bounds remain open (Sakharwade et al., 2022). A further direction concerns robust and finite-length implementations, where mixed-alphabet entanglement-assisted coding may supply transmission layers for noisy EARAC realizations (Prasad et al., 2023).

Finally, there is a broader unifying perspective. Whether formulated via Bell nonlocality, steering, generalized RAC function retrieval, finite-blocklength communication, or contextuality, EARACs expose the same underlying phenomenon: pre-shared quantum correlations reshape the geometry of accessible encodings and decodings under stringent communication constraints. The sharpest current results show that this reshaping depends not only on how much entanglement is available, but also on when measurements occur, how incompatible Bob’s effective observables may be, and which algebraic structure the target function or network task imposes (Pauwels et al., 2022, M et al., 2021, Hameedi et al., 2017).