Reference frames which separately store non-commuting conserved quantities

TL;DR

This paper investigates how non-commuting conserved quantities like angular momentum components can be stored separately in different parts of a reference frame, impacting quantum thermodynamics and resource theories.

Contribution

It demonstrates that non-commuting conserved quantities can be independently stored in different reference frame components, extending understanding of quantum conservation laws.

Findings

Components of angular momentum can be stored separately in reference frames.

The results extend to other non-commuting conserved quantities.

Implications for quantum thermodynamics of non-commuting observables.

Abstract

Even in the presence of conservation laws, one can perform arbitrary transformations on a system if given access to a suitable reference frame, since conserved quantities may be exchanged between the system and the frame. Here we explore whether these quantities can be separated into different parts of the reference frame, with each part acting as a `battery' for a distinct quantity. For systems composed of spin-$\frac12$ particles, we show that the components of angular momentum $S_x$, $S_y$ and $S_z$ (non-commuting conserved quantities) may be separated in this way, and also provide several extensions of this result. These results also play a key role in the quantum thermodynamics of non-commuting conserved quantities.

Equations

$${\Vert {\mathrm{tr}}_R\left\{V{\rho }_S\otimes {\rho }_R{V}^{†}\right\}-U_S{\rho }_S{U}_S^{†}\Vert }_1\le ϵ.$$

$$\left|\Delta s_j+\Delta S_j^{(j)}\right|$$

$$\left|\Delta S_j^{(k)}\right|$$

$$T=\mathbf{s}\cdot ({\mathbf{s}}^{\prime }\times {\mathbf{s}}^{\prime \prime })=\sum _{j,k,\ell \in \{x,y,z\}}{ϵ}_{jk\ell }{s}_j{s}_k^{\prime }{s}_{\ell }^{\prime \prime },$$

$$V_{\alpha }=\exp \{-i\frac{4\alpha }{N}T\}.$$

$${\mathrm{tr}}_R\left\{V_{\alpha }{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{V}_{\alpha }^{†}\right\}$$

$$+\mathcal{O}\left(\frac{1}{N^2}\right).$$

$$\Delta s_z^{\prime }$$

$$\Delta s_y^{\prime \prime }$$

$$\Delta s_x$$

$$\Delta s_z$$

$${\tau }_R={\tau }_y^{(z)}\otimes {\tau }_z^{(y)}\otimes {\tau }_z^{(x)}\otimes {\tau }_x^{(z)}\otimes {\tau }_x^{(y)}\otimes {\tau }_y^{(x)}.$$

$$\begin{gathered} {\mathrm{tr}}_R\left\{V_{{\alpha }_z}{V}_{{\alpha }_y}{V}_{{\alpha }_x}{\rho }_S\otimes {\tau }_R{V}_{{\alpha }_x}^{†}{V}_{{\alpha }_y}^{†}{V}_{{\alpha }_z}^{†}\right\} \\ =U_H{\rho }_S{U}_H^{†}+\mathcal{O}\left(\frac{1}{N^2}\right). \end{gathered}$$

$${\Vert {\mathrm{tr}}_R\left\{V{\rho }_S\otimes {\rho }_R{V}^{†}\right\}-U_S{\rho }_S{U}_S^{†}\Vert }_1\le \mathcal{O}\left(\frac{1}{N}\right),$$

$$\left|\Delta s_j+\Delta s_j^{(j)}\right|$$

$$\left|\Delta s_j^{(k)}\right|$$

$$U=\sum _{n,m=0}^1{X}^n{Z}^m\otimes |n⟩⟨n|\otimes |m⟩⟨m|$$

$${\mathrm{tr}}_R\left\{V_{\alpha }{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{V}_{\alpha }^{†}\right\}$$

$$={\rho }_S-i\frac{4\alpha }{N}\sum _{j,k,\ell \in \{x,y,z\}}{ϵ}_{jk\ell }[s_j,{\rho }_S]\mathrm{tr}\left\{s_k^{\prime }{\tau }_y^{\prime }\right\}\mathrm{tr}\left\{s_{\ell }^{\prime \prime }{\tau }_z^{\prime \prime }\right\}+\mathcal{O}\left(\frac{1}{N^2}\right),$$

$$={\rho }_S-i\frac{\alpha }{N}[s_x,{\rho }_S]+\mathcal{O}\left(\frac{1}{N^2}\right),$$

$$=U_{\alpha ,x}{\rho }_S{U}_{\alpha ,x}^{†}+\mathcal{O}\left(\frac{1}{N^2}\right).$$

$$\xi \equiv {\Vert {\mathrm{tr}}_R\left\{V_{\alpha }{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{V}_{\alpha }^{†}\right\}-({\rho }_S-i\frac{4\alpha }{N}{\mathrm{tr}}_R\left\{[T,{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }])\right\}\Vert }_1\le \mathcal{O}\left(\frac{1}{N^2}\right)$$

$$\xi$$

$$\le \sum _{n=2}^{\infty }{\left(\frac{4\alpha }{N}\right)}^n\sum _{k=0}^n\frac{1}{k!(n-k)!}{\Vert T^k{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{T}^{n-k}\Vert }_1$$

$$\le \sum _{n=2}^{\infty }{\left(\frac{4\alpha }{N}\right)}^n\sum _{k=0}^n\frac{1}{k!(n-k)!}\Vert T^k\Vert \Vert {\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{\Vert }_1\Vert T^{n-k}\Vert$$

$$\le \sum _{n=2}^{\infty }{\left(\frac{4\alpha }{N}\right)}^n\sum _{k=0}^n\frac{1}{k!(n-k)!}\Vert T{\Vert }^n$$

$$\le \sum _{n=2}^{\infty }\frac{1}{n!}{\left(\frac{3\alpha }{N}\right)}^n\sum _{k=0}^n\frac{n!}{k!(n-k)!}$$

$$=\sum _{n=2}^{\infty }\frac{1}{n!}{\left(\frac{6\alpha }{N}\right)}^n$$

$$=36{\left(\frac{\alpha }{N}\right)}^2\sum _{n=2}^{\infty }\frac{1}{n!}{\left(\frac{6\alpha }{N}\right)}^{n-2}.$$

$$\xi \le 36(e-2){\left(\frac{\alpha }{N}\right)}^2.$$

$${\Vert U_{\alpha ,x}{\rho }_S{U}_{\alpha ,x}^{†}-\left({\rho }_S-i\frac{\alpha }{N}[s_x,{\rho }_S]\right)\Vert }_1\le 4(e-2){\left(\frac{\alpha }{N}\right)}^2,$$

$${\Vert {\mathrm{tr}}_R\left\{V_{\alpha }{\rho }_S\otimes {\tau }_y^{\prime }\otimes {\tau }_z^{\prime \prime }{V}_{\alpha }^{†}\right\}-U_{\alpha ,x}{\rho }_S{U}_{\alpha ,x}^{†}\Vert }_1\le 40(e-2){\left(\frac{\alpha }{N}\right)}^2,$$

$$\Delta s_i^{\prime }$$

$$=-i\frac{4\alpha }{N}\sum _{j,k,\ell \in \{x,y,z\}}{ϵ}_{jk\ell }\mathrm{tr}\left\{s_j{\rho }_S\right\}\mathrm{tr}\left\{s_i^{\prime }[s_k^{\prime },{\tau }_y^{\prime }]\right\}\mathrm{tr}\left\{s_{\ell }^{\prime \prime }{\tau }_z^{\prime \prime }\right\}+\mathcal{O}\left(\frac{1}{N^2}\right),$$

$$=-\frac{\alpha }{N}{\delta }_{i,z}\mathrm{tr}\left\{s_y{\rho }_S\right\}+\mathcal{O}\left(\frac{1}{N^2}\right),$$

$$\left|\Delta s_i^{\prime }+\frac{\alpha }{N}{\delta }_{i,z}\mathrm{tr}\left\{s_y{\rho }_S\right\}\right|\le 18(e-2){\left(\frac{\alpha }{N}\right)}^2,$$

$$\left|\Delta s_i^{\prime \prime }-\frac{\alpha }{N}{\delta }_{i,y}\mathrm{tr}\left\{s_z{\rho }_S\right\}\right|\le 18(e-2){\left(\frac{\alpha }{N}\right)}^2.$$

$$\zeta \equiv {\Vert {\mathrm{tr}}_R\left\{V_{{\alpha }_z}{V}_{{\alpha }_y}{V}_{{\alpha }_x}({\rho }_S\otimes {\tau }_R)V_{{\alpha }_x}^{†}{V}_{{\alpha }_y}^{†}{V}_{{\alpha }_z}^{†}\right\}-({\rho }_S-i\frac{1}{N}[H,{\rho }_S])\Vert }_1.$$

$$\zeta \le \sum _{n=2}^{\infty }{\left(\frac{18\pi }{N}\right)}^n.$$

Full text

Conservation laws are amongst the most important and widely used aspects of physics, greatly restricting the possible transformations that an isolated system can undergo Wig52 ; Ara60 ; Marvian . However, when a system is not isolated but allowed to interact with other systems, with conservation laws applying only globally, much greater freedom is possible AharonovSusskind1 ; AharonovKaufherr ; RobRTA ; Ahmadi ; Marvian13 ; QRF18 . The situation is particularly interesting in quantum theory, where different conserved quantities may not commute (such as the different components of angular momentum), and interference effects are crucial.

Perhaps surprisingly, it has been shown that any transformation of a quantum system can be implemented, as long as one has access to an appropriate ancillary system RobRTA ; Ahmadi ; Marvian13 ; QRF18 . This additional system plays a dual role of providing a reference frame for the transformation, and acting as a reservoir which can exchange conserved quantities with the system.

While undergoing its transformation, the system will generally exchange many different conserved quantities with the reference frame. An interesting question is whether the exchange of each of these different conserved quantities can be separated into different parts of the reference system, with each part effectively acting as a ‘battery’ for a specific conserved quantity. When the conserved quantities commute, this is possible NCCQ . The main question we raise - and answer - in the present paper, is whether this is possible when the conserved quantities do not commute. In this situation one might expect that it is impossible to separate the various conserved quantities into different reservoires, because one cannot even measure one of them without disturbing the other. Surprisingly, we show that this is possible.

As a concrete example, consider rotations of a spin-$\frac{1}{2}$ particle in the presence of angular momentum conservation. We would like to be able to localise any changes in the three components of spin $s_{x}$, $s_{y}$ and $s_{z}$ of the spin-$\frac{1}{2}$ particle within different subsystems in the reference frame (see Fig. 1). In this paper, we show that this is indeed possible. Our approach will generalise to any unitary transformation on any number of spin-$\frac{1}{2}$ systems. Furthermore, as an interesting example of such a protocol, we show how to completely extract the three components of angular momentum of an unknown spin state into three distinct systems (up to arbitrary precision). Finally, in the Supplementary material, we extend these results for arbitrary conserved quantities of any system of dimension $2^{n}$ and to conservation of angular momentum under rotations for arbitrary spin.

These results address a fundamental aspect of conservation laws in quantum theory. Furthermore, they are of particular importance in quantum thermodynamics, which has recently been extended to multiple non-commuting conserved quantities vaccaro ; barnett ; halpern2016 ; halpern2018 ; lostaglio2017 ; NCCQ ; jonathan ; gour ; ito ; Manzano ; Sparaciari ; Yunger , and where batteries for storing conserved quantities are of particular interest.

*Separate batteries for spin-$\frac{1}{2}$ systems.—*We consider the case of spin-$\frac{1}{2}$ particles, and the possibility of separating the different conserved components of angular momentum $s_{x}$, $s_{y}$ and $s_{z}$ (see Fig. 1). We will show that by considering a fixed reference frame composed of multiple spin-$\frac{1}{2}$ particles, and interacting with it in a particular way: (i) the total angular momentum of the system and reference frame is conserved, (ii) any unitary transformation can be implemented on the system with arbitrary precision, and (iii) any changes in the average angular momentum components of the system are stored in different parts of the reference system (up to arbitrarily small correction terms). Essentially, we can think of the reference system as being partitioned into three separate batteries, each of which stores a different component of angular momentum.

The reference system must: (a) indicate the $x$, $y$ and $z$ directions in order to allow rotations of the system about these directions and (b) do this via a rotationally invariant interaction so that the total angular momentum is conserved. One way to construct a reference frame for the $x$-direction would be to prepare a number of spins all pointing in the $x$-direction (as in QRF18 ). This could be used to implement a rotation of the system about $x$, but each spin in the reference frame would generally accumulate changes in both $s_{y}$ and $s_{z}$, thus not separating the conserved quantities. The key intuition behind our approach is that there is an alternative way to define the $x$-direction. Instead of aligning each spin in the reference frame along the $x$-direction, we prepare pairs of spins pointing in the $y$ and $z$ direction, and implement a cross product via the interaction. In this way, each spin in the reference frame accumulates changes in only one component of spin, and thus the different conserved quantities can be separated.

Our result is formalised in the following theorem. Using lower case $s$ for the spin of individual spin-$\frac{1}{2}$ particles and capital $S$ for the spin of systems composed by many spins, we have the following. Let our system be a spin-$\frac{1}{2}$ particle, whose spin is denoted by s, with components $\{s_{k}\}_{k=x,y,z}$. Let the total spin of the $j$-part of the reference frame (i.e. the part which is intended to store spin in the $j$-direction) be denoted by $\textbf{S}^{(j)}$, $j=x,y,z$, and has components $\{S_{k}^{(j)}\}_{k:x,y,z}$; and finally, the total spin operator of the system and frame is denoted by ${\bf S}^{\text{tot}}=\textbf{s}+\textbf{S}^{(x)}+\textbf{S}^{(y)}+\textbf{S}^{(z)}$, and its components given by $S^{\text{tot}}_{k}=s_{k}+S_{k}^{(x)}+S_{k}^{(y)}+S_{k}^{(z)}$, $k=x,y,z$.

• •

Conservation:* $V$ conserves all components of total angular momentum $\textbf{S}^{\text{tot}}$, i.e., $[V,\textbf{S}^{\text{tot}}]=0$.*

• •

Accuracy:* $V$ effectively implements $U_{S}$ on the system with precision $\epsilon$, i.e., for any initial system state $\rho_{S}$,*

[TABLE]

• •

Separation:* Each component of angular momentum of the system is exchanged only with the corresponding part of the reference system, up to precision $\delta$:*

[TABLE]

where $\Delta s_{j}$ is equal to the change in the average angular momentum of the system in the $j$-direction, and $\Delta S_{j}^{(k)}$ is equal to the change in the average angular momentum of the $k$-part of the reference system in the $j$-direction.

The key is to consider the following operator acting on three spin-$\frac{1}{2}$ particles

[TABLE]

where $\mathbf{s}$, $\mathbf{s}^{\prime}$ and $\mathbf{s}^{\prime\prime}$ are spin operators of the three particles. In the right-hand side of Eq. (4), the sub-index $j$ in the spin operators denotes the spin component in the $j$ direction, and $\epsilon_{jk\ell}$ the Levi-Civita tensor. For simplicity we have set $\hbar=1$, so each spin operator is equal to half the corresponding Pauli operator.

The operator $T$ is invariant under rotations, as it is a “scalar”, being defined as a dot product of two vectors, $\mathbf{s}$ and $(\mathbf{s}^{\prime}\times\mathbf{s}^{\prime\prime})$. We can now construct a unitary interaction between three qubits given by

[TABLE]

Since $T$ is invariant under rotations, it commutes with the total spin $\textbf{S}^{\text{tot}}=\mathbf{s}+\mathbf{s}^{\prime}+\mathbf{s}^{\prime\prime}$. In particular this means that also $V_{\alpha}$ preserves the conserved quantities $S_{x}$, $S_{y}$ and $S_{z}$, and hence satisfies $[V_{\alpha},\textbf{S}]=0$ (see Appendix).

Let us define $\tau_{j}$ as the density matrix of a spin-$\frac{1}{2}$ particle pointing in the $j$ direction (i.e. $\tau_{j}={I\over 2}+s_{j}$); for example $\tau_{z}=\ket{\uparrow_{z}}\!\bra{\uparrow_{z}}={I\over 2}+s_{z}$,

Suppose that we want to implement a small rotation of the system about the $x$-direction, given by $U_{\alpha,x}=\exp\left(-i\frac{\alpha}{N}s_{x}\right)$. To do this we prepare the spins $\mathbf{s}^{\prime}$ and $\mathbf{s}^{\prime\prime}$ in the density matrix $\tau_{y}^{\prime}\otimes\tau_{z}^{\prime\prime}$ and act with $V_{\alpha}$ on the system and these two spins.

We show in the Appendix that

[TABLE]

In this way, the state $\tau_{y}^{\prime}\otimes\tau_{z}^{\prime\prime}$ defines a reference frame for the direction $x$.

We now consider how the angular momentum in the reference system changes under this transformation.

[TABLE]

while all other components of the reference spins are left unchanged, up to $\mathcal{O}\left(\tfrac{1}{N^{2}}\right)$. These equations are proven in the Appendix. Hence to leading order, the first reference system only picks up $z$-spin, and the second reference system only picks up $y$-spin. Therefore, in the context of the overall reference frame, $\mathbf{s}^{\prime}$ belongs to the $z$-part, and $\mathbf{s}^{\prime\prime}$ to its $y$-part.

As total angular momentum is conserved, it follows that the change in the system’s angular momentum obeys

[TABLE]

Similarly, due to the cyclic symmetry of the spin operators, we can generate a small rotation of the system about the $y$-direction or $z$-direction by acting with $V_{\alpha}$ on the system and a reference frame in the state $\tau_{z}^{\prime}\otimes\tau_{x}^{\prime\prime}$ or $\tau_{x}^{\prime}\otimes\tau_{y}^{\prime\prime}$ respectively. In each case, the two components of angular momentum that change (perpendicular to the axis of rotation) are separated into the two different reference spins. For example, when performing a small $y$-rotation with the frame $\tau_{z}^{\prime}\otimes\tau_{x}^{\prime\prime}$, to first order in $\frac{1}{N}$ only the $x$-spin of the first reference particle and the $z$-spin of the second reference particle are modified. Intuitively, neither reference system changes its spin-component parallel to the axis of rotation of the system, and each reference spin does not change its spin-component parallel to the direction it was originally pointing in, as it is maximal in this direction and we are considering only first order changes.

A small rotation about a general axis, given by the unitary $U_{H}=\exp\left(-i\frac{H}{N}\right)$ where $H=\sum_{k=x,y,z}\alpha_{k}\,s_{k}$ can be generated by performing three subsequent small rotations, around the $x$, $y$ and $z$-directions as described above. In particular, we use 6 spin-$\frac{1}{2}$ particles, in the state

[TABLE]

where the superscripts denote which part of the global reference frame the spins are in, i.e. which angular momentum component they will store. We then implement $U_{H}$ by first applying $V_{\alpha_{x}}$ to the system and the first two reference spins, then $V_{\alpha_{y}}$ to the system and the next two reference spins, then $V_{\alpha_{z}}$ to the system and the last two reference spins. Following a similar approach to Eq. (Proof.), we thereby obtain

[TABLE]

Finally, we iterate this procedure $N$ times using a new set of six reference spins in the state $\tau_{R}$ each time. The overall reference frame is therefore $\rho_{R}=\tau_{R}^{\otimes N}$, which consists of 6$N$ spins (2$N$ in each of the three parts).

In this way we will approximately implement the desired transformation $U_{S}=(U_{H})^{N}=\exp\left(-iH\right)$, which is (up to a global phase) the most general unitary transformation on a spin-$\frac{1}{2}$ system.

Defining the full sequence of transformations by $V$, we note that as this is a sequence of $V_{\alpha}$ transformations, each of which conserve the three components of angular momentum, $V$ will satisfy the conservation property $[V,\textbf{S}^{\text{tot}}]=0$.

The error in implementing $U_{S}$ is bounded by the sum of the errors from each step, giving a total error of $N\mathcal{O}\left(\frac{1}{N^{2}}\right)=\mathcal{O}\left(\frac{1}{N}\right)$ QRF18 . It follows that

[TABLE]

which proves the accuracy property in Eq. (1) by suitably choosing a sufficiently large $N$ according to the value of $\epsilon$.

To prove the separation property, we use Eq. (Proof.) and its equivalents for $y$ and $z$ rotations, together with the fact that there are only $2N$ spins in each part of the reference system, imply that

[TABLE]

For any $\delta$, we can therefore choose a sufficiently large $N$ such that Eqs. (2) and (3) hold. Explicit bounds for the $\mathcal{O}\left(\frac{1}{N}\right)$ and $\mathcal{O}\left(\frac{1}{N^{2}}\right)$ terms in this section may be found in the Appendix. ∎

These results can be extended to systems composed of any number of spin-$\frac{1}{2}$ particles, as we argue in the following. First, the above proof shows that we can implement any unitary on a single spin. Then, we can also implement interactions between two spins inside the system via an interacting unitary such as $\sqrt{SWAP}$ or $e^{-i\theta\,\mathbf{s}^{1}\cdot\mathbf{s}^{2}}$. These commute with all extensive conserved quantities and do not require the use of a reference system. Thinking of our spin-$\frac{1}{2}$ systems as qubits, we know that the ability to perform all single-qubit unitaries plus any particular interacting two-qubit unitary is computationally universal DupontDupond . Hence we can construct a circuit to approximately implement any unitary transformation on any number of spin-half systems, whilst storing any changes to angular momentum in different batteries.

*Extracting the angular momentum components of an unknown spin.—*An interesting possibility enabled by the above procedure is to take an unknown spin-$\frac{1}{2}$ state with average spin $\langle{\bf s}\rangle=(\langle s_{x}\rangle,\langle s_{y}\rangle,\langle s_{z}\rangle)$, and three other systems, and completely extract the different components of spin into the three systems (up to arbitrary accuracy). That is, we can perform a unitary transformation such that, up to arbitrary accuracy, the spin-$\frac{1}{2}$ particle finally has average spin zero, and the average spin of the three systems has increased by $(\langle s_{x}\rangle,0,0)$, $(0,\langle s_{y}\rangle,0)$, and $(0,0,\langle s_{z}\rangle)$ respectively.

To do this, we use the $x$, $y$ and $z$-parts of our reference frame as the three systems, and include two ancillary spin-$\frac{1}{2}$ particles, each in a maximally mixed state, in one of the systems. We then perform a unitary $V$ on the entire state which is given by the circuit construction above, and which approximately implements a unitary

[TABLE]

on the initial spin and the two maximally mixed spins, where $X$ and $Z$ are the Pauli unitaries on the system. The net effect of $U$ is to completely decohere the state of the spin-$\frac{1}{2}$ particle and transform it into the maximally mixed state tonyFn . Because the ancillary spins are also left in maximally mixed states, all of the average angular momentum in the initial state of the spin must have been transferred to the three components of the reference frame, and hence to the three desired systems.

*Conclusions.—*We have shown that it is possible to perform an arbitrary unitary transformation on any number of spin-$\frac{1}{2}$ particles whilst respecting angular momentum conservation, in such a way that any changes in the three components of angular momentum are separated into different ‘batteries’. Any errors in this procedure can be made arbitrarily small by making these batteries sufficiently large.

Importantly, the use of the cross product technique is more than a simple technical development. It has, we believe, a deep conceptual meaning. It shows that in the realm of quantum mechanics one can build frames of reference in various ways, all equally good for acting as references (for specifying a direction in our case), but which have fundamentally different properties. We have established this result in the particular case of specifying a direction, but we expect this to be a general property of all quantum frames.

Our results also allow one to completely extract the different components of angular momentum of an unknown spin state into distinct systems (up to arbitrary precision). It would be interesting to investigate the ultimate limits of such a procedure, such as whether it can be made exact, and the smallest possible implementation.

The study of quantum thermodynamics has recently been extended to other conserved quantities besides energy, in particular to non-commuting conserved quantities, and this result allows one to consider explicit batteries for the angular momentum. The fact that the different components of angular momentum can be separated in this way is particularly surprising given that they do not commute, and measurements of one component would disturb the others.

The protocols we have described for spin-$\tfrac{1}{2}$ systems may be generalised to higher spins, when the objective is to implement a spatial rotation under total angular momentum conservation. However, such a higher-dimensional quantum system has many more observables, each of which could be a potential conserved quantity, up to the maximum of $d^{2}-1$ for a system with $d$-dimensional Hilbert space (for instance $d=2\sigma+1$ for a spin-$\sigma$ particle). One may ask whether it is possible to construct a reference frame, with a part for each of of the observables to be conserved, such that an arbitrary transformation on the system can be implemented by an operation that globally obeys all potential conservation laws, while the change of the $k$-th quantity on the system is offset by the same change of only the $k$-th part of the frame, all to arbitrary precision. In the case that the dimension is a power of $2$, we show in the Supplemental Material that a complete set of conserved quantities can be constructed for which this is possible, by an adaptation of our spin-$\frac{1}{2}$ methods. The question remains open, however, for other dimensions and general sets of quantum observables.

*Acknowledgements.—*SP acknowledges support from The Institute for Theoretical Studies, ETH Zurich. ABS acknowledges support by the Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). AW acknowledges support by the Spanish MINECO (project FIS2016-86681-P) with the support of FEDER funds, and the Generalitat de Catalunya (project 2017-SGR-1127). This research was partially supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

Appendix A Explicit bounds for $\mathcal{O}\left(\frac{1}{N^{2}}\right)$ and

$\mathcal{O}\left(\frac{1}{N}\right)$ terms for spin-$\frac{1}{2}$ systems

Here we provide more detailed technical proofs giving explicit bounds for the $\mathcal{O}\left(\frac{1}{N^{2}}\right)$ and $\mathcal{O}\left(\frac{1}{N}\right)$ terms in our bounds for spin-$\frac{1}{2}$ systems, obtained using similar techniques to those introduced in Ref. QRF18 .

A.1 Proof of Eq. (Proof.)

First notice that

[TABLE]

What we need to do now is to provide a rigorous computation for the $\mathcal{O}\left(\tfrac{1}{N^{2}}\right)$ term. To do this, we must first show that

[TABLE]

where $V_{\alpha}=\exp\{-i\tfrac{4\alpha}{N}T\}$. Expanding the exponentials we obtain

[TABLE]

where in the fifth line we have used the fact that $T=\sum_{j,k,\ell\in\{x,y,z\}}\epsilon_{jk\ell}\,s_{j}\otimes s^{\prime}_{k}\otimes s^{\prime\prime}_{\ell}$ has 6 non-zero terms, each of which has operator norm $\frac{1}{8}$ (as $\|s_{j}\|=\frac{1}{2}$), giving $\|T\|\leq\frac{3}{4}$. Hence for $N\geq 6\alpha$,

[TABLE]

On the other hand, applying Eq. (D5) of Ref. QRF18 to our setup yields

[TABLE]

where $U_{\alpha,x}=\exp\left(-i\frac{\alpha}{N}s_{x}\right)$.

Combining Eqs. (19) and (20) through the triangle inequality, we obtain

[TABLE]

where we have used the fact that $\mathrm{tr}_{R}\left\{[T,\rho_{S}\otimes\tau_{y}^{\prime}\otimes\tau_{z}^{\prime\prime}]\right\}=\frac{1}{4}[s_{x},\rho_{S}]$.

A.2 Proof of Eqs. (7) and (8)

Here we will discuss the case of Eq. (7), since Eq. (8) follows similarly. First notice that

[TABLE]

where in the last step we have noted that the expression is only non-zero when $\ell=z$, $k=x$ and $j=y$, and used the spin commutation relation $[s_{a},s_{b}]=i\epsilon_{abc}s_{c}$.

Now, applying similar arguments to those in Eqs. (17)–(19) to the changes in spin given by Eq. (22) we obtain

[TABLE]

A.3 Proofs of Eqs. (12), (13) and (14)

Let us start by proving Eq. (12). When considering a general small rotation, we must obtain a bound on

[TABLE]

Following a similar argument to that below Eq. (D7) of Ref. QRF18 , expanding the unitaries and collecting terms in $\left(\frac{1}{N}\right)^{n}$ for $n\geq 2$ we obtain at most $6^{n}$ contributions (as each of the 6 unitaries could provide each power of $\frac{1}{N}$), with a constant coefficient upper bounded by $(4\alpha_{\max})^{n}$ where $\alpha_{\max}\leq\pi$. Each term also contains the trace norm of an operator with $n$ copies of $T$, acting on different parties, distributed either before or after the initial state. Following the argument above Eq. (19), such a term is upper bounded by $\left(\frac{3}{4}\right)^{n}$. Combining all of these observations we obtain

[TABLE]

For $N\geq 36\pi$ we therefore obtain $\zeta\leq 648\pi^{2}\left(\frac{1}{N}\right)^{2}$. Combining this with Eq. (D11) of Ref. QRF18 , for $D=3$, we obtain

[TABLE]

for sufficiently large $N$. Iterating this transformation $N$ times and using the inductive argument in Appendix C of Ref. QRF18 , gives

[TABLE]

This completes the proof of Eq. (12).

To now prove Eqs. (13) and (14), we can apply a similar argument as above. This leads to

[TABLE]

from which the argument follows.

Appendix B Separating conserved quantities for higher dimensional systems

We will now consider whether we can separate changes in conserved quantities for higher dimensional systems, with dimension $d>2$.

B.1 Introductory thoughts on systems of dimension $d>2$ and spin conservation laws

Let us first consider angular momentum conservation of spin $s$ systems, with dimension $d=2s+1$. In this case, we can use essentially the same procedure as in the main text (now taking $\tau_{j}$ as the eigenstate of $s_{j}$ with maximum spin in the $j$-direction, and $V_{\alpha}=\exp\{-i\tfrac{\alpha}{s^{2}N}\}$) to approximately implement any single system rotation of the form $e^{-i\theta\,\mathbf{n}\cdot\mathbf{s}}$ on a spin-$s$ system, whilst conserving the three components of total angular momentum $S_{x}$, $S_{y}$ and $S_{z}$. Any changes in angular momentum will be separated into three different ‘batteries’ as before. Furthermore, we could again implement interacting unitaries between different spins such as $e^{-i\theta\,\mathbf{s}^{(1)}\cdot\mathbf{s}^{(2)}}$, as this is rotationally invariant.

However, unlike in the case of spin-$\frac{1}{2}$ particles, spatial rotations of the form $e^{-i\theta\,\mathbf{n}\cdot\mathbf{s}}$ no longer represent the complete set of local unitary transformations for spin-$s$ particles. It would be interesting to explore what additional transformations are required in order to give a universal gate set, and whether these can be implemented in such a way as to localise any changes to angular momentum.

Moving from angular momentum conservation to more general conservation laws, an interesting question is whether we can construct a complete basis of conserved quantities, such that arbitrary unitary transformations of a system can be performed whilst separating the changes in all of these conserved quantities into different ‘batteries’. Later in this section we generalise the proofs to show that this is possible when the dimension of the system is $2^{n}$.

B.2 Generalisation to arbitrary dimension: what we can and may not do (yet)

Here we present a generalisation of the reference frame defined in the main text for spin-$\frac{1}{2}$ systems, and of the operator $T$ of Eq. (4), to higher dimensional systems. In particular, we give sufficient conditions to be able to construct a complete basis of extensive conserved quantities for such systems, and a reference frame that allows us to perform arbitrary unitary transformations of the system whilst storing any changes in these conserved quantities in different batteries (up to arbitrarily small corrections).

Consider a system $S$, of dimension $d>2$. In this case there exist $K=d^{2}-1$ possible linearly independent conserved quantities, where without loss of generality we do not include the scalar. In what follows, we will consider the case in which there exists a particular choice for these conserved quantities given by K Hermitian operators $\mathcal{M}=\{O_{k}\}_{k=0\ldots K-1}$ with the following properties

1. (i)

Traceless: $\mathrm{tr}\left\{O_{k}\right\}=0$ for all $k$ 2. (ii)

Orthogonal: $\mathrm{tr}\left\{O_{k}O_{\ell}\right\}=0$ if $k\neq\ell$ 3. (iii)

Closed under commutation: For each $k,\ell$ either $[O_{k},O_{\ell}]=0$, or there exists an $m$ such that $[O_{k},O_{\ell}]\propto O_{m}$.

Together with the identity these operators form an orthogonal operator basis for the $d$-dimensional system. In (iii), note that the constant of proportionality could depend on $k$ and $\ell$, and that $m\neq\ell$ as $\mathrm{tr}\left\{[O_{k},O_{\ell}]O_{\ell}\right\}=\mathrm{tr}\left\{O_{k}[O_{\ell},O_{\ell}]\right\}=0$ (and similarly $m\neq k$).

First let us consider how to perform the small unitary transformation $U_{\alpha,0}=\exp(-i\frac{\alpha}{N}\,O_{0})$. To do this, we first prepare a reference frame consisting of $D=K-1$ particles of the same type as the system, initialised in the state

[TABLE]

where $\rho^{(k)}=\frac{1}{d}\left(\mathbb{I}+\frac{1}{\|O_{k}\|}\,O_{k}\right)$. Given that the operators $O_{k}$ are traceless and orthogonal, it follows that $\mathrm{tr}\left\{O_{k}\rho^{(r)}\right\}=\eta_{k}\delta_{k,r}$ where $\eta_{k}=\mathrm{tr}\left\{O_{k}^{2}\right\}/(d\|O_{k}\|)$.

A generalisation of the interaction $T$ of Eq. (4) to systems of arbitrary dimensions is

[TABLE]

where $f_{aa_{1}\ldots a_{D}}=0$ unless all the sub-indices are different, in which case $f_{aa_{1}\ldots a_{D}}=\pm 1$, where the sign indicates if the sub-indices are an even or odd permutation of $0,1,\ldots D$.

Notice that for $d=2$, $f$ is just the antisymmetric symbol $\epsilon_{ijk}$. In addition, when $d=2$ and $\mathcal{M}=\{s_{k}\}_{k=x,y,z}$, we recover the interaction defined in Eq. (4) and the batteries defined in the main text for spin-$\frac{1}{2}$ systems.

The main requirement that $T$ should satisfy is $[T,\mathbf{C}]=0$ for any extensive conserved quantity $\mathbf{C}$. We will show this in the next subsection, but for now we focus on using it to perform transformations of the system. The operator $T$ allows us to define a global unitary interaction among $K$ systems given by

[TABLE]

similarly to Eq. (5).

The effective action of this global unitary $V$ on our system $A$ is given by $\mathrm{tr}_{R}\left\{V\,\rho_{S}\otimes\rho_{R}\,V^{\dagger}\right\}$. Computing this trace, we find that only the term in $T$ containing $f_{012\ldots D}$ gives a non-zero contribution to first order in $\frac{1}{N}$, leading to

[TABLE]

Similarly we can perform the small unitary transformation $U_{\alpha,r}=\exp(-i\frac{\alpha}{N}\,O_{r})$, generated by an arbitrary operator $O_{r}$ by preparing the reference frame such that $\rho^{(k)}=\frac{\mathbb{I}}{d}+c_{(k+r)\textrm{mod}D}\,O_{(k+r)\textrm{mod}D}$ and applying $V$, such that the only non-zero term in the trace corresponds to $f_{r\,(r+1)\ldots D\,0\,1\ldots(r-1)}.$ As in the main paper, by considering a sequence of small transformations generated by each operator $O_{r}$ we can implement an arbitrary small transformation $\exp(-i\frac{H}{N})$. By iterating this process $N$ times, we can then implement an arbitrary transformation $\exp(-iH)$ on the system. As the error per step is $\mathcal{O}\left(\frac{1}{N^{2}}\right)$ and there are $\mathcal{O}\left(N\right)$ steps, the overall error can be made as small as desired by choosing $N$ sufficiently large.

Next, let us see how the the conserved quantities are stored in the batteries (i.e., the reference frame). For simplicity, we again consider implementing the small transformation $U_{\alpha,0}$ generated by $O_{0}$ on the system. The change in the conserved quantity $O_{k}$ for particle $r$ in the reference frame is given by

[TABLE]

where ${O}_{k}^{r}:=\mathbb{I}\otimes...\otimes O_{k}\otimes\ldots\otimes\mathbb{I}$ is the operator that is $\mathbb{I}$ everywhere except for the $r$-th frame particle where it is $O_{k}$. Expanding $V$ to first order in $\frac{1}{N}$ we obtain

[TABLE]

Now notice that the sum in Eq. (35) consists of two terms: $\{a=0,a_{r}=r\}$ and $\{a=r,a_{r}=0\}$, since any other assignment of values to $a$ or $a_{r}$ will render $f_{a,1,\ldots,r-1,a_{r},r+1,D}=0$. Hence,

[TABLE]

Using $\rho^{(r)}=\frac{1}{d}\left(\mathbb{I}+\frac{1}{\|O_{k}\|}\,O_{k}\right)$ and $f_{r,1,\ldots,r-1,0,r+1,D}=-1$, we obtain

[TABLE]

So far, we have only used the first two properties of the operators $O_{k}$. Given the third property that the operators are closed under commutation, we find that either $[O_{0},O_{r}]=0$, in which case the $r^{th}$ subsystem in the reference frame accumulates no changes in any conserved quantity (i.e., $\Delta O_{k}^{r}=0$ for all $k$), or $[O_{0},O_{r}]\propto O_{m}$. In the latter, the $r^{th}$ subsystem will only accumulate changes in the conserved quantity $O_{m}$ (i.e., $\Delta O_{k}^{r}=0$ unless $k=m$). As each subsystem accumulates changes in at most one conserved quantity, it is possible to separate all of the conserved quantities into different batteries. A similar result will apply for small rotations generated by any $O_{k}$ and thus for the overall transformation.

B.3 $T$ preserves extended conserved quantities

In this subsection, we show that $T$ commutes with all extensive conserved quantities. Let us begin by revisiting the case with $d=2$, where our conserved quantities are $s_{x}$, $s_{y}$ and $s_{z}$, as presented in the main text for spin-$\frac{1}{2}$ systems. Here, T acts on three particles as

[TABLE]

Now consider the conserved quantity corresponding to the total angular momentum in the $x$-direction, $S_{x}=s_{x}\otimes\mathbb{I}\otimes\mathbb{I}+\mathbb{I}\otimes s_{x}\otimes\mathbb{I}+\mathbb{I}\otimes\mathbb{I}\otimes s_{x}$, which is the sum of the angular momentum of the three individual particles in the $x$-direction. The commutator $[S_{x},T]$ may be expressed as

[TABLE]

We will now see how each term in each sum is either zero or cancelled out by a similar term appearing in a different sum but with the opposite sign. First, note that all terms in the final answer must contain the same spin operator on exactly two particles and a different spin operator on the third particle. This is because $a,a_{1}$, and $a_{2}$ must all be distinct in order for $f_{aa_{1}a_{2}}$ to be non-zero, and the spin operator generated by the commutator is different from the spin operators appearing within it. As an example, a term proportional to $s_{y}\otimes s_{y}\otimes s_{x}$ can be generated by $[s_{x},s_{z}]\otimes s_{y}\otimes s_{x}$ and $s_{y}\otimes[s_{x},s_{z}]\otimes s_{x}$. The former of these will contribute to $[S_{x},T]$ with a coefficient $f_{zyx}$ via the first sum, whereas the latter will contribute with a coefficient $f_{yzx}$ via the second sum. As $f_{zyx}=-f_{yzx}$ these two terms will cancel out, and the same applies to all other terms, giving a total commutator of zero.

Now let us see how a similar reasoning applies to the higher dimensional case. We want to show that the total conserved quantity $O_{k}^{\mathrm{tot}}=O_{k}+O_{k}^{R}$ is preserved by $T$, i.e., that $[O_{k}^{\mathrm{tot}},T]=0$, where

[TABLE]

with

[TABLE]

We will see how each term in $[{O}_{k}^{r},T]$, for every $r$, is either [math] or cancelled out by a term in $[{O}_{k}^{u},T]$, for some other $u$. To begin, take a fixed but arbitrary $r$, and a non-zero term in $[{O}_{k}^{r},T]$. This term is identified by its indices $aa_{1}\ldots a_{D}$. Given the ‘closed under commutation’ property of the operators in $\mathcal{M}$, $[O_{k},O_{a_{r}}]=\xi\,O_{b}$, where $b\in\{a,a_{1},\ldots,a_{D}\}\setminus\{a_{r}\}$, and $\xi$ is a constant that may depend on $k$ and $a_{r}$. Let $u$ be such that $a_{u}=b$, and for simplicity, and without loss of generality, assume $u>r$. Hence,

[TABLE]

Now, the term with indices $(aa_{1}\ldots a_{r}\ldots a_{u}\ldots a_{D})$ will contribute to $[O_{k}^{\mathrm{tot}},T]$ via the sum in the term $[{O}_{k}^{r},T]$ with coefficient $f_{aa_{1}\ldots a_{r}\ldots a_{u}\ldots a_{D}}$. The same term will however appear in the sum in the term $[{O}_{k}^{u},T]$, with indices $(aa_{1}\ldots a_{u}\ldots a_{r}\ldots a_{D})$, and will contribute to $[\mathcal{O}_{k},T]$ with coefficient $f_{aa_{1}\ldots a_{u}\ldots a_{r}\ldots a_{D}}$. By the properties of $f$, $f_{aa_{1}\ldots a_{r}\ldots a_{u}\ldots a_{D}}=-f_{aa_{1}\ldots a_{u}\ldots a_{r}\ldots a_{D}}$, and the two terms cancel out. As every term in the final answer can be generated in exactly two ways with opposite coefficients, the result that $[O_{k}^{\mathrm{tot}},T]=0$ then follows. Although we will think of the operators $O_{k}^{\mathrm{tot}}$ as the conserved quantities of interest, note that any other extensive conserved quantity can be expressed as a linear combination of the operators $O_{k}^{\mathrm{tot}}$ and the identity, this means that $T$ commutes with all extensive conserved quantities.

Interestingly, one can also show that $[O_{k}^{\mathrm{tot}},T]=0$ without the ‘closed under commutation’ property, by using the fact that $[O_{k},O_{\ell}]=\sum_{m\neq k,\ell}w_{k\ell m}O_{m}$ for some coefficients $w_{k\ell m}$. As above, every non-zero term in $[O_{k}^{\mathrm{tot}},T]=0$ can be generated in exactly two ways with opposite coefficients. Hence for any complete set of traceless orthonormal operators $O_{k}$, the operator $T$ constructed from those operators commutes with all extensive conserved quantities.

B.4 Dimension $2^{n}$

Let us consider the case where we have a system $S$ of dimension $d=2^{n}$. Here, we can indeed find an operator basis with the properties specified above, and can thus separate any changes to a complete basis of conserved quantities. In particular, we take the set $\mathcal{M}$ of $K=4^{n}-1$ operators $\{O_{k}\}$ to consist of products of spin-$\frac{1}{2}$ operators. That is, each $O\in\mathcal{M}$ has the form $O=s_{1}\otimes\ldots\otimes s_{n}$, where $s_{k}\in\left\{\frac{1}{2}\mathbb{I},\frac{1}{2}\sigma_{x},\frac{1}{2}\sigma_{y},\frac{1}{2}\sigma_{z}\right\}$, but where not all $s_{k}$ may equal the identity.

It is easy to verify that these operators are traceless and orthogonal. Furthermore, the product of any two distinct operators in $\mathcal{M}$ is proportional to another operator in the set, i.e., there exists an $m$ such that $O_{k}O_{\ell}\propto O_{m}$. By taking the Hermitian conjugate, we also have $O_{\ell}O_{k}\propto O_{m}$. Hence either $[O_{k},O_{\ell}]=0$, or $[O_{k},O_{\ell}]\propto O_{m}$ and these operators are hence closed under commutation as defined above.

B.5 Explicit bounds

As in the previous case, we can also calculate explicit bounds on the $\mathcal{O}\left(\frac{1}{N^{2}}\right)$ terms in the bounds above, using similar techniques to those in Appendix A and in Ref. QRF18 .

In particular, we find

[TABLE]

where we have used an approach similar to that in the derivation of Eq. (A.1), assuming that $N>\left(2K!\,\frac{\|O_{0}\|\|O_{1}\|\ldots\|O_{D}\|}{\eta_{1}\eta_{2}\ldots\eta_{D}}\right)\alpha$.

Applying now Eq. (D5) of Ref. QRF18 to this setup yields

[TABLE]

Combining Eqs. (40) and (41) through the triangle inequality, we find

[TABLE]

Finally, following a similar approach we obtain

[TABLE]

In the particular case considered above of dimension $d=2^{n}$, in which the operators $O_{k}$ are products of spin-$\frac{1}{2}$ operators and $\mathbb{I}/2$, note that $\eta_{k}=\|O_{k}\|=\frac{1}{d}$ and hence these expressions simplify considerably.