No-partial erasure of quantum information
Introduction
Classical information can be stored in distinct macroscopic states of a physical system and processed according to classical laws of physics. That ‘information is physical’ is exemplified by the fact that erasure of classical information is an irreversible operation with a cost of of energy per bit operating at a temperature T [1], which is a fundamental source of heat for standard computation [2]. This is the Landauer erasure principle. In quantum information processing, a qubit cannot be erased by a unitary transformation1 and is subject to Landauer's principle. In recent years considerable effort has been directed toward investigating possible and impossible operations in quantum information theory.
Impossible operations are stated as no-go theorems, which establish limits to quantum information capabilities and also provide intuition to enable further advances in the field. For example, the no-cloning theorem [3], [4], [5] underscored the need for quantum error correction to ensure that quantum information processing is possible in faulty systems despite the impossibility of a quantum FANOUT operation. Other examples of important no-go theorems are the no-deletion theorem [6], [7], which proves the impossibility of perfectly deleting one state from two identical states, the no-flipping theorem [8], which establishes the impossibility of designing a universal NOT gate for arbitrary qubit input states, and the impossibility of universal Hadamard and CNOT gates for arbitrary qubit input states [9]. The strong no-cloning theorem states that the creation of a copy of a quantum state requires full information about the quantum state [10]; together with the no-deletion theorem, these establish permanence of quantum information. A profound consequence of the no-cloning and no-deleting theorems suggest a fundamental principle of conservation of quantum information [11].
Here we establish a new and powerful no-go theorem of quantum information, which suggests both a limitation and protection of quantum information. Our theorem shows that it is impossible to partially erase quantum information, even by using irreversible means, where partial erasure corresponds to a reduction of the parameter space dimension for the quantum state that holds the quantum information, namely the qubit or qudit. As a special case, it is impossible to erase azimuthal angle information of a qubit whilst keeping the polar angle information intact, which we show is the no-flipping principle. Our theorem adds new insight into the integrity of quantum information, namely that we can erase complete information but not partial information. This in turn implies that quantum information is indivisible and we have to treat quantum information as a ‘whole entity’. We also introduce the no partial erasure propositions for SU(2) coherent states and for continuous variable quantum information. Since the first e-print release of our work, a no-splitting theorem for quantum information [12] has been presented, which we show is a straightforward corollary to our Theorem 4.
An arbitrary qubit is expressed as with and , . Each pure state is uniquely identified with a point on the Poincaré sphere with θ the polar angle and ϕ the azimuthal angle. The states and are the logical zero and one states, respectively. Complete erasure would map all arbitrary qubit states into a fixed qubit state regardless of the input state parameters θ and ϕ, which is known to be impossible by unitary means. More generally, the d-dimensional analogue of the 2-dimensional qubit is a qudit with quantum state with , and with is the Bargmann angle between the ith orthonormal vector and the qudit state.
Each vector and is d-dimensional, but normalization of the qudit state and the unphysical nature of the overall phase reduces the number of free parameters for the qudit to . A pure qudit can be represented as a point on the projective Hilbert space which is a real -dimensional manifold. For the qubit case the projective Hilbert space is the two-dimensional Bloch sphere.
Section snippets
No-partial erasure of qubit and qudit
In this section, we prove powerful theorems that establish the impossibility of partial erasure of arbitrary qudit states but first begin with a definition of partial erasure.
Definition 1 Partial erasure is a completely positive (CP), trace preserving mapping of all pure states , with real parameters , in an n-dimensional Hilbert space to pure states in an m-dimensional Hilbert subspace via a constraint such that .
The process of partial erasure reduces the dimension of the
No-partial erasure of spin coherent state
Our theorem that no-partial erasure of qudits is possible is important because quantum information is clearly not only conserved but also indivisible. However, the ‘no partial erasure’ theorem yields another important result for erasure of spin coherent states, also known as SU(2) coherent states [15], [16], [17].
The SU(2) coherent states are a generalization of qubits, which can be thought of as spin- states, to states of higher spin j. The SU(2) raising and lowering operators are and
No-partial erasure of continuous variable state
Next, we prove the ‘no-partial erasure’ theorem for continuous variable quantum information. Ideally continuous variable (CV) quantum information encodes quantum information as superpositions of eigenstates of the position operator , namely with complex amplitude [19]. We can represent a CV state as Note that can be any complex-valued function, subject to the requirement of square-integrability and normalization. Now let us reduce to
Conclusion
In summary we have introduced a new process called partial erasure of quantum information and asked if quantum information can undergo partial erasure. We have shown that partial erasure of qubits, qudits, SU(2) coherent states, and continuous variable quantum information is impossible. These results point to the integrity principle for quantum information, namely that it is indivisible and robust even against partial erasure. This principle gives a new meaning to quantum information and nicely
Acknowledgements
A.K.P. thanks C.H. Bennett for useful discussions. B.C.S. has been supported by Alberta's Informatics Circle of Research Excellence (iCORE), the Canadian Institute for Advanced Research, and the Australian Research Council and the Canadian Network of Centres of Excellence for Mathematics of Information Technology and Complex Systems (MITACS).
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