Quantum Random Access Codes (QRACs)

• QRACs are protocols that encode n classical bits into m qubits, enabling the retrieval of any bit with a probability greater than 1/2.
• They leverage the structure of Bloch space by optimizing measurement directions to maximize bit recovery success, constrained by geometric limits.
• Known optimal constructions for small parameters (e.g., (3,2)-QRACs) validate the tight upper bounds, guiding future designs for quantum communication.

A Quantum Random Access Code (QRAC) is a communication-theoretic protocol in which $n$ classical bits are encoded into $m$ qubits so that any individual bit can be retrieved with a success probability $p > \tfrac{1}{2}$ by measuring the encoded state. QRACs provide an operational setting to leverage the compressive and non-commutative structure of quantum state space for information retrieval beyond what is achievable classically with the same physical resources. When sender and receiver are allowed to share a classical random string, the protocol is called a QRAC with shared randomness. The geometry of Bloch space plays a central role in understanding the limits and optimal constructions of such codes.

1. Formal Definition and Protocol Structure

A $(n, m, p)$-QRAC encodes $n$ classical bits into $m$ qubits such that each bit can be retrieved, at any choice of index, with probability at least $p > 1/2$. Formally, for every bit $i$ there exists a measurement (POVM or projective) that, when applied to the encoding of any $x \in \{0,1\}^n$, yields $x_i$ as outcome with probability at least $p$. In the shared randomness variant, Alice and Bob may both use a pre-shared random string to coordinate encoding/decoding strategies, but no further communication or resources.

The average success probability is typically considered with respect to the uniform distribution over all $n$-bit input strings and bit indices: $$p = \mathbb{E}_{x,i} \left[ \Pr(\text{Bob recovers } x_i \text{ from Alice's quantum state for } x) \right]$$ For quantum encoding, Alice maps $x$ to a pure or mixed quantum state $\rho_x$ on $m$ qubits. To recover $x_i$, Bob performs a pre-specified measurement $\{M_{b|i}\}$; the protocol is optimized jointly over the choice of encodings and measurements.

2. Geometric Formulation and Upper Bound

The structure of QRACs, particularly for the case with shared randomness, can be understood in terms of the Bloch vector representation of quantum states. Any $m$-qubit pure state can be represented by a vector in the $(2^{2m}-1)$-dimensional real vector space known as Bloch space. The ability to recover a given bit $x_i$ amounts to the distinguishability of certain subsets of the encoding states with respect to measurement axes.

The central result is a tight upper bound on the achievable $p$ for any $(n,m,p)$-QRAC with (possibly) shared randomness: $$p \leq \frac{1}{2} + \frac{1}{2} \sqrt{\frac{2^{m-1}}{n}}$$ For $m=2$ (two qubits), this specializes to

$p \leq \frac{1}{2} + \frac{1}{\sqrt{2n}}$

This establishes a general constraint for all code constructions: as more information is packed into a fixed quantum memory ($n$ increases for fixed $m$), the achievable reliability per bit decreases.

Derivation via Bloch Sphere Geometry

The proof exploits the high-dimensional geometry of Bloch space:

• Alice’s possible input strings are mapped to pure states (unit Bloch vectors).
• To ensure each bit can be retrieved reliably, the Bloch vectors of all encodings must have sufficiently large projections onto a collection of measurement axes—at least one per bit to be read out.
• Since the number of orthogonal axes in Bloch space is limited by its dimension, the average squared projection onto all directions is constrained by the Cauchy-Schwarz inequality and the parallelogram law.
• The optimal configuration (with mutually orthogonal measurement axes) yields the stated bound.

Thus, the geometric limit is imposed by the maximum sphere-packing of $n$ directions in $2^{2m}-1$ dimensions, which translates into the above expression for $p$.

3. Optimality of Known QRAC Constructions

Previously known explicit QRACs for small $n$ and $m$ have been shown to match the upper bound of the new geometric result, confirming their optimality. Notably:

• The $(3,2,p)$-QRAC with $p = \frac{1}{2} + \frac{1}{\sqrt{6}}$,
• The $(4,2,p)$-QRAC with $p = \frac{1}{2} + \frac{1}{2\sqrt{2}}$,
• The $(6,2,p)$-QRAC with $p = \frac{1}{2} + \frac{1}{2\sqrt{3}}$

are all optimal with respect to the new universal bound.

This confirmation resolves prior conjectures regarding optimality, such as the open question posed by Imamichi and Raymond (AQIS 2018) for the case $m=2$, $n$ arbitrary, and sharpens the landscape for code construction in this parameter regime.

Summary Table

$n$	$m$	Upper Bound on $p$	Known Optimal $p$
3	2	$\frac{1}{2} + \frac{1}{\sqrt{6}}$	$\frac{1}{2} + \frac{1}{\sqrt{6}}$
4	2	$\frac{1}{2} + \frac{1}{2\sqrt{2}}$	$\frac{1}{2} + \frac{1}{2\sqrt{2}}$
6	2	$\frac{1}{2} + \frac{1}{2\sqrt{3}}$	$\frac{1}{2} + \frac{1}{2\sqrt{3}}$

No better quantum code—classical, quantum, or hybrid—can outperform these values for the specified cases.

4. Bloch Space Geometry and Constraints

The Bloch representation not only informs the upper bound but also shapes code design:

• For any configuration, measurement directions (for each bit) should be as orthogonal as possible.
• The geometric lemma underpinning the bound states that the sum of squared projections of all encoding Bloch vectors onto these directions is inextricably linked to the number of directions and the Bloch space dimension.
• For large $n$, the available “angular space” per bit shrinks, implying that $p$ must decline.

Codes attaining the bound do so by configuring measurement operators corresponding to maximal sets of mutually orthogonal directions, and encoding states as evenly distributed as possible with respect to these axes.

5. Implications for Theory and Protocol Design

The geometric bound provides a definitive ceiling on quantum advantage in QRACs with shared randomness:

• It establishes a clear formula for determining the maximal performance—allowing practitioners to immediately assess feasibility for target parameters ($n, m$).
• Existing codes for small $n, m$ are proven optimal, focusing future code-design efforts on attaining the bound for other parameter sets.
• Any attempt to design improved protocols must focus on exploiting the geometric configuration outlined by the bound—mutually orthogonal arrangement of measurement axes in Bloch space.

This result generalizes and sharpens prior work (e.g., by Ambainis et al.), incorporating earlier geometric insights into a tight, closed-form upper bound.

6. Theoretical and Practical Significance

For quantum information theory:

• QRACs are central primitives in communication complexity, dimension witnessing, and quantum cryptography.
• The bound advances the understanding of quantum information packing limits and the role of high-dimensional quantum geometry.
• The techniques used—Cauchy-Schwarz and geometric optimization in Bloch space—are broadly applicable to other quantum coding and measurement optimization problems.

For implementation:

• Designers of QRAC-based protocols (for communication, dimension certification, or semi-device-independent tasks) have a precise performance target.
• Any QRAC protocol utilizing shared randomness—meaning systematic randomization or post-processing of code options—is definitively limited by this geometric criterion.

7. Key Formula and Attribution

The key bound for any $(n, m, p)$-QRAC with shared randomness is: $$p \leq \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{2^{m-1}}{n} }$$ This result confirms the conjecture of Imamichi and Raymond (AQIS 2018) and generalizes geometric methods pioneered by Ambainis et al. (0810.2937).

8. Conclusion

The geometry of Bloch space tightly governs the achievable reliability of quantum random access coding with shared randomness. The derived bound provides both a theoretical ceiling for QRAC success probability and a prescription for optimal code construction—namely, aligning measurement axes as mutually orthogonal as permitted by Hilbert space dimension and distributing encodings symmetrically. Established analytical constructions are confirmed to be optimal in this framework, and future QRAC designs must respect this geometric constraint.